Operator theory, analysis and the state space approach. In honor of R.Kaashoek 9783030042684, 9783030042691

Table of contents :
Preface......Page 6
Contents......Page 7
Curriculum Vitae of M.A. Kaashoek......Page 10
Research monographs......Page 14
Papers in professional journals......Page 15
Edited books and journal issues......Page 27
Other publications......Page 28
Ph.D. students of M.A. Kaashoek......Page 31
Marinus A. Kaashoek: Guide, Companion and Friend......Page 32
Amsterdam......Page 36
Potchefstroom......Page 38
My PhD time with Rien Kaashoek......Page 39
Personal reminiscences: working with Rien......Page 40
References......Page 41
Working with Rien Kaashoek......Page 42
Rien Kaashoek: Mentor, colleague and friend......Page 43
Introduction......Page 46
1. Five types of tangent......Page 49
2. Tangents with a unique extremal Φω......Page 50
3. Royal tangents......Page 57
4. Flat tangents......Page 60
5. Purely balanced tangents......Page 61
6. Relation to a result of L. Kosiński and W. Zwonek......Page 63
References......Page 65
Standard versus strict Bounded Real Lemma with infinite-dimensional state space III: The dichotomous and bicausal cases......Page 67
1. Introduction......Page 68
2. Dichotomous system theory......Page 73
3. Bicausal systems......Page 81
4. Storage functions......Page 88
5. The available storage and required supply......Page 96
6. Storage functions for bicausal systems......Page 102
7. Dichotomous and bicausal bounded real lemmas......Page 107
8. Bounded real lemma for nonstationary systems with dichotomy......Page 112
References......Page 115
1. Introduction......Page 118
2.1. Terminology, notation and statement of basic results......Page 120
2.2. Verification via reduction to the upper triangular case......Page 123
2.3. An algorithm for constructing canonical forms......Page 124
2.4. An example......Page 126
3.1. Zero pattern subspaces and algebras......Page 128
3.3. L-free directed bipartite graphs: definition and related notions......Page 129
3.4. In-diagrams and out-diagrams......Page 132
3.5. L-closures......Page 136
4. L-free directed graphs: canonical forms......Page 137
5.2. Relationship of the in/out-diagrams to in/out-ultra closures......Page 143
5.3. L-closures of upper triangular-type partial orders......Page 144
5.4. Canonical forms......Page 147
6. Application: counting equivalence classes......Page 148
7. Concluding remarks and open problems......Page 156
7.1. L-free directed bipartite graphs: characterization......Page 157
7.2. Issues for further research......Page 158
References......Page 159
1. Introduction......Page 161
2. Main results......Page 163
3. An example......Page 166
4. Proof of the main results......Page 169
References......Page 183
1. A personal note......Page 186
2.1. Wiener–Hopf factorization......Page 187
2.2. Finite section method......Page 189
3. Operators on the Bergman space......Page 190
3.1. Finite section method......Page 191
3.3. Polynomial collocation......Page 192
3.4. The Bergman metric......Page 193
3.5. Analytic element collocation......Page 194
3.6. Test computations......Page 196
3.7. General domains......Page 203
Appendix: Phase plots......Page 204
References......Page 205
1. Introduction......Page 208
2. Approximate identities and VMO......Page 212
3.1. Fibers over......Page 214
3.2. Fibers over......Page 216
4. Localization of PQC OVER QC......Page 218
References......Page 220
1. Introduction......Page 221
2.1. The general quasiseparable structure......Page 224
2.2. The diagonal plus small rank representation......Page 225
3.1. The Sturm sequences property for a Hermitian matrix......Page 226
3.3. The basic eigenvalue algorithm......Page 227
3.4. The computation of eigenvectors......Page 228
4. The inverse problem......Page 230
5.1. The quasiseparable generators of A = (A0)*A0......Page 232
5.2. Computation in linear time of the standard matrix norm before SVD......Page 233
6. Numerical experiments......Page 234
6.1.1. Eigenvalues known in advance.......Page 235
6.1.3. Eigenvalues of Hermitian band matrices.......Page 236
6.2. Results for singular values......Page 237
7. Conclusions and future work......Page 238
References......Page 239
1. Introduction......Page 241
2. Preliminaries......Page 242
3. Inner-outer factorization......Page 243
4. An auxiliary lemma......Page 248
5. Proof of the inner-outer factorization......Page 249
References......Page 253
1. Introduction......Page 255
2. On a special kind of Schur complement......Page 258
3. On a restricted extension problem for a finite Hankel non-negative definite extendable sequence......Page 263
4. On equivalence classes of truncated matricial moment problems of type MP[R; (sj)2n j=0, ≤]......Page 266
5. On truncated matricial [α, ∞)-Stieltjes moment problems......Page 269
6. On a restricted extension problem for a finite α-Stieltjes right-sided non-negative definite extendable sequence......Page 270
7. On equivalence classes of truncated matricial moment problems of type MP[[α, ∞); (sj)m j=0, ≤]......Page 274
References......Page 277
1. Introduction......Page 279
2. Basic properties of Tω......Page 284
3. Intermezzo: Division with remainder by a polynomial in Hp......Page 286
4. Fredholm properties of Tω for ω ∈ Rat(T)......Page 290
5. Fredholm properties of Tω: General case......Page 295
6. Matrix representation......Page 300
7. Examples......Page 304
References......Page 307
1. Introduction......Page 309
2.1. Complex case......Page 312
2.2. Real case......Page 315
3.1. Eigenvalue one......Page 318
3.3. Eigenvalues not ±1......Page 322
4.2. Complex unimodular eigenvalues......Page 323
4.2.1. The case when n is odd......Page 325
4.2.2. The case when......Page 327
5. The complex H-symplectic case......Page 328
References......Page 329
1. Introduction......Page 331
2. Properties of the Mellin transformation......Page 332
3.1. Boundedness......Page 342
3.2. Fredholmness......Page 350
References......Page 364
1. Introduction......Page 366
Problem 2.1 (Problems of system theory of rational systems).......Page 368
3.1. Example Glycolysis in Yeast......Page 369
3.2. A Rational System of an Economic Control Problem......Page 374
4.1. Concepts......Page 376
5. Canonical Forms of Rational Systems......Page 380
6. Decompositions and Subclasses of Rational Systems......Page 384
6.1. A Line Network of Rational Control Systems......Page 385
6.2. A Ring Network of Rational Systems......Page 386
7. System Approximation......Page 388
8. Control Synthesis of Rational Systems......Page 390
9. Computer Algebra for Rational Systems......Page 392
10. Concluding Remarks......Page 394
References......Page 395
1. Introduction......Page 399
2. A brief survey of recent results......Page 402
3. Some interesting proof techniques......Page 410
4. Open problems......Page 417
5. A new result......Page 419
References......Page 423
1. Introduction......Page 426
2. GBDT of discrete skew-selfadjoint Dirac systems......Page 427
3. Explicit solutions of the corresponding non-stationary systems......Page 433
References......Page 434
1. Introduction......Page 436
2. Review of some known results......Page 438
3. Direct extensions......Page 443
4. Proofs......Page 445
5. Equivalence after extension......Page 447
6. The case X = Y and A = I......Page 449
7.2. Equivalence after extension (EAE)......Page 450
7.4. Delta-relations......Page 451
7.6. Schur coupling (SC)......Page 452
8. Open problems......Page 453
References......Page 454
1. Introduction......Page 457
2. Proof of Main Result with Consequences......Page 460
3. Semidefinite Toeplitz Completions......Page 468
4. The Singularity Degree of Some Toeplitz Cycles......Page 470
References......Page 475
1. Introduction......Page 478
2. Preliminaries......Page 481
3. Tm-invariant symbols......Page 482
4. Toeplitz operator action and algebras......Page 487
5. C*-algebras......Page 490
6. C*-algebras of Toeplitz operators on the Fock space F2(Cn)......Page 497
References......Page 499

Citation preview

Operator Theory Advances and Applications 271

Harm Bart Sanne ter Horst André C.M. Ran Hugo J. Woerdeman Editors

Operator Theory, Analysis and the State Space Approach In Honor of Rien Kaashoek

Operator Theory: Advances and Applications Volume 271 Founded in 1979 by Israel Gohberg

Editors: Joseph A. Ball (Blacksburg, VA, USA) Albrecht Böttcher (Chemnitz, Germany) Harry Dym (Rehovot, Israel) Heinz Langer (Wien, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Wolfgang Arendt (Ulm, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Kenneth R. Davidson (Waterloo, ON, Canada) Fritz Gesztesy (Waco, TX, USA) Pavel Kurasov (Stockholm, Sweden) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Ilya Spitkovsky (Abu Dhabi, UAE)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Marinus A. Kaashoek (Amsterdam, NL) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Orange, CA, USA) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany)

More information about this series at http://www.springer.com/series/4850

Harm Bart • Sanne ter Horst • André C.M. Ran Hugo J. Woerdeman Editors

Operator Theory, Analysis and the State Space Approach In Honor of Rien Kaashoek

Editors Harm Bart Econometric Institute Erasmus University Rotterdam, The Netherlands André C.M. Ran Department of Mathematics Vrije Universiteit Amsterdam, The Netherlands

Sanne ter Horst Department of Mathematics Unit for BMI North-West University Potchefstroom, South Africa Hugo J. Woerdeman Department of Mathematics Drexel University Philadelphia, PA, USA

ISSN 0255-0156 ISSN 2296-4878 (electronic) Operator Theory: Advances and Applications ISBN 978-3-030-04268-4 ISBN 978-3-030-04269-1 (eBook) https://doi.org/10.1007/978-3-030-04269-1 Library of Congress Control Number: 2018966530 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface On November 10, 2017, M.A. (Rien) Kaashoek turned eighty. Several initiatives were taken to celebrate this event. At the Vrije Universiteit in Amsterdam a one-afternoon workshop was organized on October 14; at the 2017 IWOTA conference in Chemnitz there was a special session and a plenary lecture in honor of Rien’s upcoming birthday. The present volume in the Operator Theory: Advances and Applications series is also dedicated to Rien’s eightieth birthday and in celebration of his over fifty years of contributions to the field of operator theory. Rien retired from his position as full professor at the Vrije Universiteit in 2002 at the mandatory retirement age of sixty-five, already having left a legacy of fifteen Ph.D. students, more than 150 papers, and six research monographs. For some this would have been a good moment to stop and fully enjoy other aspects of life, but not for Rien! On the contrary, even today, fifteen years after retirement, he is still a very active researcher, as can be seen from his curriculum vitae, publication list, and list of Ph.D. students, all of which are included in this volume. For his many friends and colleagues this is wonderful. With relevant questions, excellent research problems and discussions, Rien still stimulates a wide group of researchers. It was therefore no surprise to us when our invitations to contribute to this volume were met with unanimous enthusiasm. The present volume consists of two parts. The first part contains biographical information on Rien, his curriculum vitae, publication list, and list of Ph.D. students referred to above, as well as a few personal reminiscences from some of his students and collaborators. The second part comprises eighteen research papers by Rien’s colleagues, collaborators, students, and friends. We thank all the authors for the time and efforts they have put into their contributions, and their enthusiastic responses. We are also very appreciative of the dedicated referees, whose input greatly improved the research papers. This volume is dedicated to Rien, with admiration and gratitude from his many friends and colleagues around the world. The editors

Harm Bart, Sanne ter Horst, Andr´e Ran, Hugo Woerdeman

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Contents Curriculum Vitae of M.A. Kaashoek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Publication List of M.A. Kaashoek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv List of Ph.D. students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiii H. Bart, S. ter Horst, D.Pik, A. Ran, F. van Schagen and H.J. Woerdeman Personal reminiscenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv J. Agler, Z.A. Lykova and N.J. Young Carath´eodory extremal functions on the symmetrized bidisc . . . . . . . . . . . . . .1 J.A. Ball, G.J. Groenewald and S. ter Horst Standard versus strict Bounded Real Lemma with infinite-dimensional state space III: The dichotomous and bicausal cases . . . . . . . . . . . . . . . . . . . . .23 H. Bart, T. Ehrhardt and B. Silbermann L-free directed bipartite graphs and echelon-type canonical forms . . . . . . . 75 J.M. Bogoya, S.M. Grudsky and I.S. Malysheva Extreme individual eigenvalues for a class of large Hessenberg Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A. B¨ ottcher and E. Wegert How to solve an equation with a Toeplitz operator? . . . . . . . . . . . . . . . . . . . . 145 T. Ehrhardt and Z. Zhou On the maximal ideal space of even quasicontinuous functions on the unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Y. Eidelman and I. Haimovici Bisection eigenvalue method for Hermitian matrices with quasiseparable representation and a related inverse problem . . . . . . . . . . . 181 A.E. Frazho and A.C.M. Ran A note on inner-outer factorization of wide matrix-valued functions . . . . 201 B. Fritzsche, B. Kirstein and C. M¨ adler An application of the Schur complement to truncated matricial power moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 G.J. Groenewald, S. ter Horst, J. Jaftha and A.C.M. Ran A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 vii

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G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran Canonical form for H-symplectic matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 P. Junghanns and R. Kaiser A note on the Fredholm theory of singular integral operators with Cauchy and Mellin kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 J. Nˇemcov´ a, M. Petreczky and J.H. van Schuppen Towards a system theory of rational systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 ˇ L. Plevnik and P. Semrl Automorphisms of effect algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 A.L. Sakhnovich GBDT of discrete skew-selfadjoint Dirac systems and explicit solutions of the corresponding non-stationary problems . . . . . . . . . . . . . . . . . . . . . . . . . . 389 F.-O. Speck On the reduction of general Wiener–Hopf operators . . . . . . . . . . . . . . . . . . . . 399 S. Sremac, H.J. Woerdeman and H. Wolkowicz Maximum determinant positive definite Toeplitz completions . . . . . . . . . . 421 N. Vasilevski On commutative C ∗ -algebras generated by Toeplitz operators with Tm -invariant symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443

Marinus A. (Rien) Kaashoek in his office at home, 2018

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Curriculum Vitae of M.A. Kaashoek Personal data: Name: Marinus (Rien) A. Kaashoek Born: Ridderkerk, The Netherlands; November 10, 1937 Degree: Ph.D., University of Leiden, The Netherlands, 1964; supervisor: A.C. Zaanen Married: with Wilhelmina (Wies) E.J. Bakema; three children Research interests: Analysis and operator theory, and various connections between operator theory, matrix theory and mathematical systems theory. In particular, Wiener– Hopf integral equations and Toeplitz operators and their nonstationary variants; state space methods for problems in analysis; also metric constrained interpolation problems, and various extension and completion problems for partially defined matrices or operators, including relaxed commutant lifting problems; more recently, inverse problems for orthogonal functions determined by Toeplitz plus Hankel operators. Positions: – Assistant, University of Leiden, 1959–1962. – Junior staff member, University of Leiden, 1962–1965. – Postdoc, University of California at Los Angeles, 1965–1966. – Senior Lecturer, Vrije Universiteit, Amsterdam, 1966–1969. – Professor, Vrije Universiteit, Amsterdam, 1969–2002. – Professor Emeritus, Vrije Universiteit, Amsterdam, 2002–present. Visiting professorships: – University of Maryland at College Park, MD, USA, January–July 1975. – University of Calgary, Calgary, Alberta, Canada, August–September 1981. – Ben Gurion University of the Negev, Beer Sheva, Israel, April 1987. – Tel-Aviv University, Tel-Aviv, Israel, various periods.

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Curriculum Vitae of M.A. Kaashoek

Fellowships: – Z.W.O. Fellowship for postdoctoral studies at the University of California at Los Angeles, 1965–1966. – Royal Irish Academy Senior Visiting Fellowship, Spring 1970. – British Science Research Council Senior Visiting Fellowship, May–June 1976. Honors: – Toeplitz Lecturer, Tel-Aviv, April 1991. – Knight in the Royal Order of the Dutch Lion, November 2002. – Doctor Honoris Causa from North West University, South Africa, May 27, 2014. – Elected honorary member Royal Dutch Mathematical Society, March 22, 2016. – Member of the Honorary and Advisory Editorial Board of the book series Operator Theory: Advances and Applications, since 2018. – Member of the Honorary and Advisory Editorial Board of Integral Equations and Operator Theory, since 2018. Publications: – Author or co-author of more than 200 papers, many of which are written jointly with I. Gohberg. – Co-author of 9 research monographs and textbooks. – Co-editor of 28 conference proceedings, special volumes and special issues of journals. Editorial work: Member of the editorial board of – Integral Equations and Operator Theory – until end of 2017. – Operator Theory: Advances and Applications – until end of 2017. – Asymptotic Analysis – until end of 2006. – Mathematische Nachrichten – until end of 2017. – Contributions to Encyclopaedia of Mathematics. – Indagationes Mathematicae.

Curriculum Vitae of M.A. Kaashoek

xiii

Conference organization: – Co-organizer of the third International Workshop on Operator Theory and its Applications (IWOTA), Amsterdam 1985. – Member Tagungsleitung Wiener–Hopf-Probleme, Oberwolfach, 1986 and 1989. – Co-organizer of the international conference Operator Theory: Advances and Applications, Calgary, 1988. – Co-chairman of the ninth International Symposium on the Mathematical Theory of Networks and Systems (MTNS), Amsterdam, 1989. – Member of the Organizing Committee of the workshop on Matrix and Operator Theory, Rotterdam, 1989. – Member of the Steering Committee of MTNS, 1989–2006. – Chairman of the Steering Committee of the MTNS, 1989–1991. – Member and vice president of the IWOTA Steering Committee, since 1991. – Chair of the board of IWOTA vice presidents, 2010–2017, and Past Chair since 2018. – Member of the Organizing Committees of MTNS at – Kobe, Japan, 1991 – Regensburg, Germany, 1993 – Padua, Italy, 1998 – Perpignan, France, 2000 – Kyoto, Japan, 2006 Other services: – Member of various committees of the Vrije Universiteit and of the Dutch mathematical community. – Member, Board of the Faculty of Mathematics and Natural Sciences, Vrije Universiteit, 1973–1974. – Chairman, Research Committee of the Department of Mathematics and Computer Science, Vrije Universiteit, 1978–1995. – Chairman, Board of the Dutch Mathematical Society, 1980–1981. – Member and lecturer, Dutch Network System and Control Theory, 1987– 1993. – Chairman, Board of the Thomas Stieltjes Institute for Mathematics, 1992–2002. – Member, Advisory Board of the Mathematical Institute of Leiden University, 1996–2005. – Coordinator, INTAS projects 93-0249, 93-0249-EXT and 97-M016.

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Curriculum Vitae of M.A. Kaashoek

– Member, Board of Supervisors for the Special Chair of Mathematics, in particular, Applied Mathematics, at Leiden University, 1998–2007. – Dean, Faculty of Mathematics and Computer Science of the Vrije Universiteit, October 1996–February 1998. – Dean, Faculty of Sciences of the Vrije Universiteit, March 1998–December 2000. – Netherlands Coordinator, European Research Network Analysis and Operators, 2000–2004. – Co-author of New dimensions and broader range. A national strategy for mathematical research and related master programs (in Dutch), 2002. – Member, Program Board for Mathematics of the Lorentz Center, 2004– 2005. – Chair, Advisory Committee Wiskunde in actie, Akademie Raad voor de Wiskunde, 2005–2006. – Member, midterm review committee Departments of Applied Mathematics at Delft University of Technology, Eindhoven University of Technology, and the University of Twente, 2007. – Member research review committee Departments of Applied Mathematics at Delft University of Technology, Eindhoven University of Technology, the University of Twente, and of one programme at the University of Groningen, 2009–2010. – Chairman (technical), assessment committee of the NWO Mathematics of Planet Earth program, September 2014.

Publication List of M.A. Kaashoek

Dissertation [1] M.A. Kaashoek, Closed linear operators on Banach spaces, University of Leiden, Amsterdam, 1964.

Research monographs [1] M.A. Kaashoek and T.T. West, Locally compact semi-algebras, North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974. With applications to spectral theory of positive operators; North-Holland Mathematics Studies, No. 9. [2] H. Bart, I. Gohberg, and M. A. Kaashoek, Minimal factorization of matrix and operator functions, Operator Theory: Advances and Applications, vol. 1, Birkh¨ auser Verlag, Basel–Boston, Mass., 1979. [3] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of linear operators. Vol. I, Operator Theory: Advances and Applications, vol. 49, Birkh¨ auser Verlag, Basel, 1990. [4] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of linear operators. Vol. II, Operator Theory: Advances and Applications, vol. 63, Birkh¨ auser Verlag, Basel, 1993. [5] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Partially specified matrices and operators: classification, completion, applications, Operator Theory: Advances and Applications, vol. 79, Birkh¨ auser Verlag, Basel, 1995. [6] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, Operator Theory: Advances and Applications, vol. 100, Birkh¨ auser Verlag, Basel, 1998. [7] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Basic classes of linear operators, Birkh¨ auser Verlag, Basel, 2003. [8] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of matrix and operator functions: the state space method, Operator Theory: Advances and Applications, vol. 178, Birkh¨ auser Verlag, Basel, 2008. Linear Operators and Linear Systems. [9] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, A state space approach to canonical factorization with applications, Operator Theory: Advances and Applications, vol. 200, Birkh¨ auser Verlag, Basel, 2010. Linear Operators and Linear Systems.

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Publication List of M.A. Kaashoek

Papers in professional journals [1] M.A. Kaashoek, Stability theorems for closed linear operators, Nederl. Akad. Wetensch. Proc. Ser. A 68=Indag. Math. 27 (1965), 452–466. [2] M.A. Kaashoek, Closed linear operators on Banach spaces, Nederl. Akad. Wetensch. Proc. Ser. A 68=Indag. Math. 27 (1965), 405–414. [3] M.A. Kaashoek and D.C. Lay, On operators whose Fredholm set is the complex plane, Pacific J. Math. 21 (1967), 275–278. [4] M.A. Kaashoek, Ascent, descent, nullity and defect: A note on a paper by A. E. Taylor, Math. Ann. 172 (1967), 105–115. [5] M.A. Kaashoek, On the Riesz set of a linear operator, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 46–53. [6] M.A. Kaashoek and T.T. West, Locally compact monothetic semi-algebras, Proc. London Math. Soc. (3) 18 (1968), 428–438. [7] M.A. Kaashoek and T.T. West, Semi-simple locally compact monothetic semialgebras, Proc. Edinburgh Math. Soc. (2) 16 (1968/1969), 215–219. [8] M.A. Kaashoek, Locally compact semi-algebras and spectral theory, Nieuw Arch. Wisk. (3) 17 (1969), 8–16. [9] M.A. Kaashoek and T.T. West, Compact semigroups in commutative Banach algebras, Proc. Cambridge Philos. Soc. 66 (1969), 265–274. [10] M.A. Kaashoek and D.C. Lay, Ascent, descent, and commuting perturbations, Trans. Amer. Math. Soc. 169 (1972), 35–47. [11] M.A. Kaashoek and M.R.F. Smyth, On operators T such that f (T ) is Riesz or meromorphic, Proc. Roy. Irish Acad. Sect. A 72 (1972), 81–87. [12] M.A. Kaashoek, On the peripheral spectrum of an element in a strict closed semialgebra, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970), North-Holland, Amsterdam, 1972, pp. 319–332. Colloq. Math. Soc. J´ anos Bolyai, 5. [13] M.A. Kaashoek, A note on the spectral properties of linear operators leaving invariant a convex set, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973), 254–262. [14] H. Bart, M.A. Kaashoek, and D.C. Lay, Stability properties of finite meromorphic operator functions. I, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 217–230. [15] H. Bart, M.A. Kaashoek, and D.C. Lay, Stability properties of finite meromorphic operator functions. II, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 231–243. [16] H. Bart, M.A. Kaashoek, and D.C. Lay, Stability properties of finite meromorphic operator functions. III, Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36 (1974), 244–259. [17] K.-H. F¨ orster and M.A. Kaashoek, The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator, Proc. Amer. Math. Soc. 49 (1975), 123–131. [18] H. Bart, M.A. Kaashoek, and D.C. Lay, Relative inverses of meromorphic operator functions and associated holomorphic projection functions, Math. Ann. 218 (1975), no. 3, 199–210. [19] I. Gohberg, M.A. Kaashoek, and D.C. Lay, Spectral classification of operators and operator functions, Bull. Amer. Math. Soc. 82 (1976), no. 4, 587–589. [20] H. Bart, I. Gohberg, and M.A. Kaashoek, Operator polynomials as inverses of characteristic functions, Integral Equations Operator Theory 1 (1978), no. 1, 1–18. [21] I.C. Gohberg, M.A. Kaashoek, and D.C. Lay, Equivalence, linearization, and decomposition of holomorphic operator functions, J. Functional Analysis 28 (1978), no. 1, 102–144. [22] H. Bart, M.A. Kaashoek, and D.C. Lay, The integral formula for the reduced algebraic multiplicity of meromorphic operator functions, Proc. Edinburgh Math. Soc. (2) 21 (1978/79), no. 1, 65–71.

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[23] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Common multiples of operator polynomials with analytic coefficients, Manuscripta Math. 25 (1978), no. 3, 279–314. [24] I. Gohberg and M.A. Kaashoek, Unsolved problems in matrix and operator theory. I. Partial multiplicities and additive perturbations, Integral Equations Operator Theory 1 (1978), no. 2, 278–283. [25] H. Bart, I. Gohberg, and M.A. Kaashoek, Stable factorizations of monic matrix polynomials and stable invariant subspaces, Integral Equations Operator Theory 1 (1978), no. 4, 496–517. [26] I. Gohberg, M.A. Kaashoek, and L. Rodman, Spectral analysis of families of operator polynomials and a generalized Vandermonde matrix. II. The infinite-dimensional case, J. Funct. Anal. 30 (1978), no. 3, 358–389. [27] I. Gohberg, M.A. Kaashoek, and L. Rodman, Spectral analysis of families of operator polynomials and a generalized Vandermonde matrix. I. The finite-dimensional case, Topics in functional analysis (essays dedicated to M. G. Kre˘in on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, Academic Press, New YorkLondon, 1978, pp. 91–128. [28] I. Gohberg and M.A. Kaashoek, Unsolved problems in matrix and operator theory. II. Partial multiplicities for products, Integral Equations Operator Theory 2 (1979), no. 1, 116–120. [29] M.A. Kaashoek, Recent developments in the spectral analysis of matrix and operator polynomials, Proceedings, Bicentennial Congress Wiskundig Genootschap (Vrije Univ., Amsterdam, 1978), Math. Centre Tracts, vol. 101, Math. Centrum, Amsterdam, 1979, pp. 233–247. [30] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Similarity of operator blocks and canonical forms. I. General results, feedback equivalence and Kronecker indices, Integral Equations Operator Theory 3 (1980), no. 3, 350–396. [31] H. Bart, I. Gohberg, M.A. Kaashoek, and P. Van Dooren, Factorizations of transfer functions, SIAM J. Control Optim. 18 (1980), no. 6, 675–696. [32] M.A. Kaashoek and M.P.A. van de Ven, A linearization for operator polynomials with coefficients in certain operator ideals, Ann. Mat. Pura Appl. (4) 125 (1980), 329–336. [33] I. Gohberg, M.A. Kaashoek, L. Lerer, and L. Rodman, Common multiples and common divisors of matrix polynomials. I. Spectral method, Indiana Univ. Math. J. 30 (1981), no. 3, 321–356. [34] M.A. Kaashoek, C.V.M. van der Mee, and L. Rodman, Analytic operator functions with compact spectrum. I. Spectral nodes, linearization and equivalence, Integral Equations Operator Theory 4 (1981), no. 4, 504–547. [35] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Similarity of operator blocks and canonical forms. II. Infinite-dimensional case and Wiener–Hopf factorization, Topics in modern operator theory (Timi¸soara/Herculane, 1980), Operator Theory: Adv. Appl., vol. 2, Birkh¨ auser, Basel–Boston, Mass., 1981, pp. 121–170. [36] H. Bart, I. Gohberg, and M.A. Kaashoek, Convolution equations and linear systems, Integral Equations Operator Theory 5 (1982), no. 3, 283–340. [37] M.A. Kaashoek, Symmetrisable operators and minimal factorization, From A to Z (Leiden, 1982), Math. Centre Tracts, vol. 149, Math. Centrum, Amsterdam, 1982, pp. 27–38. [38] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener–Hopf integral equations, Toeplitz matrices and linear systems, Toeplitz centennial (Tel Aviv, 1981), Operator Theory: Adv. Appl., vol. 4, Birkh¨ auser, Basel–Boston, Mass., 1982, pp. 85–135. [39] I. Gohberg, M.A. Kaashoek, L. Lerer, and L. Rodman, Common multiples and common divisors of matrix polynomials. II. Vandermonde and resultant matrices, Linear and Multilinear Algebra 12 (1982/83), no. 3, 159–203.

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[40] M.A. Kaashoek, C.V.M. van der Mee, and L. Rodman, Analytic operator functions with compact spectrum. II. Spectral pairs and factorization, Integral Equations Operator Theory 5 (1982), no. 6, 791–827. [41] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Rational matrix and operator functions with prescribed singularities, Integral Equations Operator Theory 5 (1982), no. 5, 673–717. [42] M.A. Kaashoek, C.V.M. van der Mee, and L. Rodman, Analytic operator functions with compact spectrum. III. Hilbert space case: inverse problem and applications, J. Operator Theory 10 (1983), no. 2, 219–250. [43] I. Gohberg and M.A. Kaashoek, Time varying linear systems with boundary conditions and integral operators. I. The transfer operator and its properties, Integral Equations Operator Theory 7 (1984), no. 3, 325–391. [44] H. Bart, I. Gohberg, and M.A. Kaashoek, The coupling method for solving integral equations, Topics in operator theory systems and networks (Rehovot, 1983), Oper. Theory Adv. Appl., vol. 12, Birkh¨ auser, Basel, 1984, pp. 39–73. [45] I. Gohberg, M.A. Kaashoek, L. Lerer, and L. Rodman, Minimal divisors of rational matrix functions with prescribed zero and pole structure, Topics in operator theory systems and networks (Rehovot, 1983), Oper. Theory Adv. Appl., vol. 12, Birkh¨ auser, Basel, 1984, pp. 241–275. [46] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Noncompact integral operators with semiseparable kernels and their discrete analogues: inversion and Fredholm properties, Integral Equations Operator Theory 7 (1984), no. 5, 642–703. [47] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener–Hopf factorization and realization, Mathematical theory of networks and systems (Beer Sheva, 1983), Lect. Notes Control Inf. Sci., vol. 58, Springer, London, 1984, pp. 42–62. [48] M.A. Kaashoek, Minimal factorization, linear systems and integral operators, Operators and function theory (Lancaster, 1984), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 153, Reidel, Dordrecht, 1985, pp. 41–86. [49] H. Bart, I. Gohberg, and M.A. Kaashoek, Fredholm theory of Wiener–Hopf equations in terms of realization of their symbols, Integral Equations Operator Theory 8 (1985), no. 5, 590–613. [50] I. Gohberg and M.A. Kaashoek, On minimality and stable minimality of timevarying linear systems with well-posed boundary conditions, Internat. J. Control 43 (1986), no. 5, 1401–1411. [51] M.A. Kaashoek, Analytic equivalence of the boundary eigenvalue operator function and its characteristic matrix function, Integral Equations Operator Theory 9 (1986), no. 2, 275–285. [52] I. Gohberg, M.A. Kaashoek, and L. Lerer, Minimality and irreducibility of timeinvariant linear boundary value systems, Internat. J. Control 44 (1986), no. 2, 363– 379. [53] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators, J. Funct. Anal. 68 (1986), no. 1, 1–42. [54] I. Gohberg and M.A. Kaashoek, Similarity and reduction for time varying linear systems with well-posed boundary conditions, SIAM J. Control Optim. 24 (1986), no. 5, 961–978. [55] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener-Hopf equations with symbols analytic in a strip, Constructive methods of Wiener–Hopf factorization, Oper. Theory Adv. Appl., vol. 21, Birkh¨ auser, Basel, 1986, pp. 39–74. [56] I. Gohberg, M.A. Kaashoek, L. Lerer, and L. Rodman, On Toeplitz and Wiener– Hopf operators with contourwise rational matrix and operator symbols, Constructive methods of Wiener–Hopf factorization, Oper. Theory Adv. Appl., vol. 21, Birkh¨ auser, Basel, 1986, pp. 75–126.

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[57] I. Gohberg and M.A. Kaashoek, Minimal factorization of integral operators and cascade decompositions of systems, Constructive methods of Wiener–Hopf factorization, Oper. Theory Adv. Appl., vol. 21, Birkh¨ auser, Basel, 1986, pp. 157–230. [58] H. Bart, I. Gohberg, and M.A. Kaashoek, Explicit Wiener–Hopf factorization and realization, Constructive methods of Wiener–Hopf factorization, Oper. Theory Adv. Appl., vol. 21, Birkh¨ auser, Basel, 1986, pp. 235–316. [59] H. Bart, I. Gohberg, and M.A. Kaashoek, Invariants for Wiener–Hopf equivalence of analytic operator functions, Constructive methods of Wiener–Hopf factorization, Oper. Theory Adv. Appl., vol. 21, Birkh¨ auser, Basel, 1986, pp. 317–355. [60] H. Bart, I. Gohberg, and M.A. Kaashoek, Multiplication by diagonals and reduction to canonical factorization, Constructive methods of Wiener–Hopf factorization, Oper. Theory Adv. Appl., vol. 21, Birkh¨ auser, Basel, 1986, pp. 357–372. [61] M.A. Kaashoek and A.C.M. Ran, Symmetric Wiener–Hopf factorization of selfadjoint rational matrix functions and realization, Constructive methods of Wiener– Hopf factorization, Oper. Theory Adv. Appl., vol. 21, Birkh¨ auser, Basel, 1986, pp. 373–409. [62] I. Gohberg and M.A. Kaashoek, Various minimalities for systems with boundary conditions and integral operators, Modelling, identification and robust control (Stockholm, 1985), North-Holland, Amsterdam, 1986, pp. 181–196. [63] I. Gohberg and M.A. Kaashoek, An inverse spectral problem for rational matrix functions and minimal divisibility, Integral Equations Operator Theory 10 (1987), no. 3, 437–465. [64] I. Gohberg and M.A. Kaashoek, Minimal representations of semiseparable kernels and systems with separable boundary conditions, J. Math. Anal. Appl. 124 (1987), no. 2, 436–458. [65] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Szeg˝ o–Kac–Achiezer formulas in terms of realizations of the symbol, J. Funct. Anal. 74 (1987), no. 1, 24–51. [66] H. Bart, I. Gohberg, and M.A. Kaashoek, The state space method in problems of analysis, Proceedings of the first international conference on industrial and applied mathematics (ICIAM 87) (Paris, 1987), CWI Tract, vol. 36, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1987, pp. 1–16. [67] I. Gohberg, M.A. Kaashoek, and L. Lerer, On minimality in the partial realization problem, Systems Control Lett. 9 (1987), no. 2, 97–104. [68] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Rational contractive and unitary interpolants in realized form, Integral Equations Operator Theory 11 (1988), no. 1, 105–127. [69] M.A. Kaashoek and H.J. Woerdeman, Unique minimal rank extensions of triangular operators, J. Math. Anal. Appl. 131 (1988), no. 2, 501–516. [70] I. Gohberg, M.A. Kaashoek, and L. Lerer, Nodes and realization of rational matrix functions: minimality theory and applications, Topics in operator theory and interpolation, Oper. Theory Adv. Appl., vol. 29, Birkh¨ auser, Basel, 1988, pp. 181–232. [71] I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Interpolation problems for rational matrix functions with incomplete data and Wiener–Hopf factorization, Topics in interpolation theory of rational matrix-valued functions, Oper. Theory Adv. Appl., vol. 33, Birkh¨ auser, Basel, 1988, pp. 73–108. [72] I. Gohberg and M.A. Kaashoek, Regular rational matrix functions with prescribed pole and zero structure, Topics in interpolation theory of rational matrix-valued functions, Oper. Theory Adv. Appl., vol. 33, Birkh¨ auser, Basel, 1988, pp. 109–122. [73] I. Gohberg, M.A. Kaashoek, and P. Lancaster, General theory of regular matrix polynomials and band Toeplitz operators, Integral Equations Operator Theory 11 (1988), no. 6, 776–882. [74] I. Gohberg and M.A. Kaashoek, Block Toeplitz operators with rational symbols, Contributions to operator theory and its applications (Mesa, AZ, 1987), Oper. Theory Adv. Appl., vol. 35, Birkh¨ auser, Basel, 1988, pp. 385–440.

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[75] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral Equations Operator Theory 12 (1989), no. 3, 343–382. [76] I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Partial pole and zero displacement by cascade connection, SIAM J. Matrix Anal. Appl. 10 (1989), no. 3, 316–325. [77] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Eigenvalues of completions of submatrices, Linear and Multilinear Algebra 25 (1989), no. 1, 55–70. [78] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, The band method for positive and contractive extension problems, J. Operator Theory 22 (1989), no. 1, 109–155. [79] M.A. Kaashoek and H.J. Woerdeman, Minimal lower separable representations: characterization and construction, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988), Oper. Theory Adv. Appl., vol. 41, Birkh¨ auser, Basel, 1989, pp. 329–344. [80] M.A. Kaashoek and J.N.M. Schermer, Inversion of convolution equations on a finite interval and realization triples, Integral Equations Operator Theory 13 (1990), no. 1, 76–103. [81] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, The band method for extension problems and maximum entropy, Signal processing, Part I, IMA Vol. Math. Appl., vol. 22, Springer, New York, 1990, pp. 75–94. [82] I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Regular rational matrix functions with prescribed null and pole data except at infinity, Linear Algebra Appl. 137/138 (1990), 387–412. [83] A. Ben-Artzi, I. Gohberg, and M.A. Kaashoek, Invertibility and dichotomy of singular difference equations, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkh¨ auser, Basel, 1990, pp. 157–184. [84] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, A maximum entropy principle in the general framework of the band method, J. Funct. Anal. 95 (1991), no. 2, 231–254. [85] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, A note on extensions of band matrices with maximal and submaximal invertible blocks, Proceedings of the First Conference of the International Linear Algebra Society (Provo, UT, 1989), 1991, pp. 157–166. [86] I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Matrix polynomials with prescribed zero structure in the finite complex plane, Topics in matrix and operator theory (Rotterdam, 1989), Oper. Theory Adv. Appl., vol. 50, Birkh¨ auser, Basel, 1991, pp. 241–266. [87] I. Gohberg and M.A. Kaashoek, The Wiener–Hopf method for the transport equation: a finite-dimensional version, Modern mathematical methods in transport theory (Blacksburg, VA, 1989), Oper. Theory Adv. Appl., vol. 51, Birkh¨ auser, Basel, 1991, pp. 20–33. [88] I. Gohberg, M.A. Kaashoek, and L. Lerer, A directional partial realization problem, Systems Control Lett. 17 (1991), no. 4, 305–314. [89] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, Time variant extension problems of Nehari type and the band method, H∞ -control theory (Como, 1990), Lecture Notes in Math., vol. 1496, Springer, Berlin, 1991, pp. 309–323. [90] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, The band method for several positive extension problems of nonband type, J. Operator Theory 26 (1991), no. 1, 191–218. [91] I. Gohberg and M.A. Kaashoek, The state space method for solving singular integral equations, Mathematical system theory, Springer, Berlin, 1991, pp. 509–523. [92] I. Gohberg and M.A. Kaashoek, Asymptotic formulas of Szeg˝ o–Kac–Achiezer type, Asymptotic Anal. 5 (1992), no. 3, 187–220. [93] I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorizations of and extensions to J-unitary rational matrix functions on the unit circle, Integral Equations Operator Theory 15 (1992), no. 2, 262–300.

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[94] M.A. Kaashoek and S.M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc. 334 (1992), no. 2, 479–517. [95] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Nevanlinna–Pick interpolation for timevarying input-output maps: the discrete case, Time-variant systems and interpolation, Oper. Theory Adv. Appl., vol. 56, Birkh¨ auser, Basel, 1992, pp. 1–51. [96] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Nevanlinna–Pick interpolation for timevarying input-output maps: the continuous time case, Time-variant systems and interpolation, Oper. Theory Adv. Appl., vol. 56, Birkh¨ auser, Basel, 1992, pp. 52–89. [97] I. Gohberg, M.A. Kaashoek, and L. Lerer, Minimality and realization of discrete time-varying systems, Time-variant systems and interpolation, Oper. Theory Adv. Appl., vol. 56, Birkh¨ auser, Basel, 1992, pp. 261–296. [98] I. Gohberg and M.A. Kaashoek, The band extension on the real line as a limit of discrete band extensions. II. The entropy principle, Continuous and discrete Fourier transforms, extension problems and Wiener–Hopf equations, Oper. Theory Adv. Appl., vol. 58, Birkh¨ auser, Basel, 1992, pp. 71–92. [99] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Reduction of the abstract four block problem to a Nehari problem, Continuous and discrete Fourier transforms, extension problems and Wiener–Hopf equations, Oper. Theory Adv. Appl., vol. 58, Birkh¨ auser, Basel, 1992, pp. 121–141. [100] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Time-varying systems: Nevanlinna–Pick interpolation and sensitivity minimization, Recent advances in mathematical theory of systems, control, networks and signal processing, I (Kobe, 1991), Mita, Tokyo, 1992, pp. 53–58. [101] I. Gohberg, M.A. Kaashoek, and L. Lerer, Minimal rank completion problems and partial realization, Recent advances in mathematical theory of systems, control, networks and signal processing, I (Kobe, 1991), Mita, Tokyo, 1992, pp. 65–70. [102] I. Gohberg and M.A. Kaashoek, The band extension on the real line as a limit of discrete band extensions. I. The main limit theorem, Operator theory and complex analysis (Sapporo, 1991), Oper. Theory Adv. Appl., vol. 59, Birkh¨ auser, Basel, 1992, pp. 191–220. [103] M.A. Kaashoek, A.C.M. Ran, and L. Rodman, Local minimal factorizations of rational matrix functions in terms of null and pole data: formulas for factors, Integral Equations Operator Theory 16 (1993), no. 1, 98–130. [104] A. Ben-Artzi, I. Gohberg, and M.A. Kaashoek, Invertibility and dichotomy of differential operators on a half-line, J. Dynam. Differential Equations 5 (1993), no. 1, 1–36. [105] I. Gohberg, M.A. Kaashoek, and F. van Schagen, On the local theory of regular analytic matrix functions, Linear Algebra Appl. 182 (1993), 9–25. [106] A. Ben-Artzi, I. Gohberg, and M.A. Kaashoek, A time-varying generalization of the canonical factorization theorem for Toeplitz operators, Indag. Math. (N.S.) 4 (1993), no. 4, 385–405. [107] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Bitangential interpolation for inputoutput operators of time-varying systems: the discrete time case, New aspects in interpolation and completion theories, Oper. Theory Adv. Appl., vol. 64, Birkh¨ auser, Basel, 1993, pp. 33–72. [108] I. Gohberg, M.A. Kaashoek, and H.J. Woerdeman, The band method for bordered algebras, Contributions to operator theory and its applications, Oper. Theory Adv. Appl., vol. 62, Birkh¨ auser, Basel, 1993, pp. 85–97. [109] J.A. Ball, I. Gohberg, and M.A. Kaashoek, The time-varying two-sided Nudelman interpolation problem and its solution, Challenges of a generalized system theory (Amsterdam, 1992), Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, vol. 40, North-Holland, Amsterdam, 1993, pp. 45–58.

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[110] A. Ben-Artzi, I. Gohberg, and M.A. Kaashoek, Exponentially dominated infinite block matrices of finite Kronecker rank, Integral Equations Operator Theory 18 (1994), no. 1, 30–77. [111] M.A. Kaashoek and J. Kos, The Nehari–Takagi problem for input-output operators of time-varying continuous time systems, Integral Equations Operator Theory 18 (1994), no. 4, 435–467. [112] J.A. Ball, M.A. Kaashoek, G. Groenewald, and J. Kim, Column reduced rational matrix functions with given null-pole data in the complex plane, Linear Algebra Appl. 203/204 (1994), 67–110. [113] J.A. Ball, I. Gohberg, and M.A. Kaashoek, H∞ -control and interpolation for timevarying systems, Systems and networks: mathematical theory and applications, Vol. I (Regensburg, 1993), Math. Res., vol. 77, Akademie-Verlag, Berlin, 1994, pp. 33–48. [114] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Bitangential interpolation for inputoutput maps of time-varying systems: the continuous time case, Integral Equations Operator Theory 20 (1994), no. 1, 1–43. [115] M.A. Kaashoek and S.M. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Differential Equations 112 (1994), no. 2, 374– 406. [116] I. Gohberg and M.A. Kaashoek, Projection method for block Toeplitz operators with operator-valued symbols, Toeplitz operators and related topics (Santa Cruz, CA, 1992), Oper. Theory Adv. Appl., vol. 71, Birkh¨ auser, Basel, 1994, pp. 79–104. [117] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Two-sided Nudelman interpolation for input-output operators of discrete time-varying systems, Integral Equations Operator Theory 21 (1995), no. 2, 174–211. [118] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Input-output operators of J-unitary timevarying continuous time systems, Operator theory in function spaces and Banach lattices, Oper. Theory Adv. Appl., vol. 75, Birkh¨ auser, Basel, 1995, pp. 57–94. [119] A. Ben-Artzi, I. Gohberg, and M.A. Kaashoek, Discrete nonstationary bounded real lemma in indefinite metrics, the strict contractive case, Operator theory and boundary eigenvalue problems (Vienna, 1993), Oper. Theory Adv. Appl., vol. 80, Birkh¨ auser, Basel, 1995, pp. 49–78. [120] J.A. Ball, I. Gohberg, and M.A. Kaashoek, A frequency response function for linear, time-varying systems, Math. Control Signals Systems 8 (1995), no. 4, 334–351. [121] A. B¨ ottcher, A. Dijksma, H. Langer, M.A. Dritschel, J. Rovnyak, and M.A. Kaashoek, Lectures on operator theory and its applications, Fields Institute Monographs, vol. 3, American Mathematical Society, Providence, RI, 1996. Lectures presented at the meeting held at the Fields Institute for Research in Mathematical Sciences, Waterloo, Ontario, September 1994; Edited by Peter Lancaster. [122] M.A. Kaashoek, State space theory of rational matrix functions and applications, Lectures on operator theory and its applications (Waterloo, ON, 1994), Fields Inst. Monogr., vol. 3, Amer. Math. Soc., Providence, RI, 1996, pp. 233–333. [123] J.A. Ball, I. Gohberg, and M.A. Kaashoek, The band method and Grassmannian approach for completion and extension problems, Recent developments in operator theory and its applications (Winnipeg, MB, 1994), Oper. Theory Adv. Appl., vol. 87, Birkh¨ auser, Basel, 1996, pp. 17–60. [124] I. Gohberg, M.A. Kaashoek, and J. Kos, Classification of linear time-varying difference equations under kinematic similarity, Integral Equations Operator Theory 25 (1996), no. 4, 445–480. [125] I. Gohberg, M.A. Kaashoek, and J. Kos, The asymptotic behavior of the singular values of matrix powers and applications, Linear Algebra Appl. 245 (1996), 55–76. [126] I. Gohberg, M.A. Kaashoek, and L. Lerer, Factorization of banded lower triangular infinite matrices, Linear Algebra Appl. 247 (1996), 347–357.

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[127] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, Discrete time-variant interpolation as classical interpolation with an operator argument, Integral Equations Operator Theory 26 (1996), no. 4, 371–403. [128] M.A. Kaashoek, C.V.M. van der Mee, and A.C.M. Ran, Weighting operator patterns of Pritchard–Salamon realizations, Integral Equations Operator Theory 27 (1997), no. 1, 48–70. [129] J.A. Ball, I. Gohberg, and M.A. Kaashoek, Nudelman interpolation and the band method, Integral Equations Operator Theory 27 (1997), no. 3, 253–284. [130] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener-Hopf equations and linear systems, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994), Proc. Sympos. Appl. Math., vol. 52, Amer. Math. Soc., Providence, RI, 1997, pp. 115–128. [131] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, A time-variant version of the commutant lifting theorem and nonstationary interpolation problems, Integral Equations Operator Theory 28 (1997), no. 2, 158–190. [132] D.Z. Arov, M.A. Kaashoek, and D.R. Pik, Minimal and optimal linear discrete time-invariant dissipative scattering systems, Integral Equations Operator Theory 29 (1997), no. 2, 127–154. [133] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, Parameterization of all solutions of the three chains completion problem, Integral Equations Operator Theory 29 (1997), no. 4, 455–490. [134] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Operator blocks and quadruples of subspaces: classification and the eigenvalue completion problem, Linear Algebra Appl. 269 (1998), 65–89. [135] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, The maximum principle for the three chains completion problem, Integral Equations Operator Theory 30 (1998), no. 1, 67–82. [136] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Sturm–Liouville systems with rational Weyl functions: explicit formulas and applications, Integral Equations Operator Theory 30 (1998), no. 3, 338–377. [137] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Pseudo-canonical systems with rational Weyl functions: explicit formulas and applications, J. Differential Equations 146 (1998), no. 2, 375–398. [138] M.A. Kaashoek and D.R. Pik, Factorization of lower triangular unitary operators with finite Kronecker index into elementary factors, Recent progress in operator theory (Regensburg, 1995), Oper. Theory Adv. Appl., vol. 103, Birkh¨ auser, Basel, 1998, pp. 183–217. [139] D.Z. Arov, M.A. Kaashoek, and D.R. Pik, Optimal time-variant systems and factorization of operators. I. Minimal and optimal systems, Integral Equations Operator Theory 31 (1998), no. 4, 389–420. [140] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Canonical systems with rational spectral densities: explicit formulas and applications, Math. Nachr. 194 (1998), 93– 125. [141] M.A. Kaashoek, C.V.M. van der Mee, and A.C.M. Ran, Wiener–Hopf factorization of transfer functions of extended Pritchard–Salamon realizations, Math. Nachr. 196 (1998), 71–102. [142] M.A. Kaashoek and A.C.M. Ran, Norm bounds for Volterra integral operators and time-varying linear systems with finite horizon, Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995), Oper. Theory Adv. Appl., vol. 106, Birkh¨ auser, Basel, 1998, pp. 275–290. [143] M.A. Kaashoek and C.G. Zeinstra, The band method and generalized Carath´ eodory– Toeplitz interpolation at operator points, Integral Equations Operator Theory 33 (1999), no. 2, 175–210.

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[144] I. Gohberg and M.A. Kaashoek, State space methods for analysis problems involving rational matrix functions, Dynamical systems, control, coding, computer vision (Padova, 1998), Progr. Systems Control Theory, vol. 25, Birkh¨ auser, Basel, 1999, pp. 93–109. [145] I. Gohberg, M.A. Kaashoek, and J. Kos, Classification of linear periodic difference equations under periodic or kinematic similarity, SIAM J. Matrix Anal. Appl. 21 (1999), no. 2, 481–507. [146] D.Z. Arov, M.A. Kaashoek, and D.R. Pik, Optimal time-variant systems and factorization of operators. II. Factorization, J. Operator Theory 43 (2000), no. 2, 263–294. [147] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Finite section method for linear ordinary differential equations, J. Differential Equations 163 (2000), no. 2, 312–334. [148] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Canonical systems on the line with rational spectral densities: explicit formulas, Differential operators and related topics, Vol. I (Odessa, 1997), Oper. Theory Adv. Appl., vol. 117, Birkh¨ auser, Basel, 2000, pp. 127–139. [149] D. Alpay, I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Direct and inverse scattering problem for canonical systems with a strictly pseudo-exponential potential, Math. Nachr. 215 (2000), 5–31. [150] M.A. Kaashoek and H.J. Woerdeman, Positive extensions and diagonally connected patterns, Recent advances in operator theory (Groningen, 1998), Oper. Theory Adv. Appl., vol. 124, Birkh¨ auser, Basel, 2001, pp. 287–305. [151] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Bound states of a canonical system with a pseudo-exponential potential, Integral Equations Operator Theory 40 (2001), no. 3, 268–277. [152] C. Foias, A.E. Frazho, and M.A. Kaashoek, A weighted version of almost commutant lifting, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), Oper. Theory Adv. Appl., vol. 129, Birkh¨ auser, Basel, 2001, pp. 311–340. [153] A.E. Frazho and M.A. Kaashoek, A band method approach to a positive expansion problem in a unitary dilation setting, Integral Equations Operator Theory 42 (2002), no. 3, 311–371. [154] C. Foias, A.E. Frazho, and M.A. Kaashoek, Relaxation of metric constrained interpolation and a new lifting theorem, Integral Equations Operator Theory 42 (2002), no. 3, 253–310. [155] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Scattering problems for a canonical system with a pseudo-exponential potential, Asymptot. Anal. 29 (2002), no. 1, 1–38. [156] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Finite section method for difference equations, Linear operators and matrices, Oper. Theory Adv. Appl., vol. 130, Birkh¨ auser, Basel, 2002, pp. 197–207. [157] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Finite section method for linear ordinary differential equations revisited, Toeplitz matrices and singular integral equations (Pobershau, 2001), Oper. Theory Adv. Appl., vol. 135, Birkh¨ auser, Basel, 2002, pp. 183–191. [158] C. Foias, A.E. Frazho, and M.A. Kaashoek, Contractive liftings and the commutator, C.R. Math. Acad. Sci. Paris 335 (2002), no. 5, 431–436. [159] I. Gohberg, M.A. Kaashoek, and F. van Schagen, Finite section method for linear ordinary differential equations on the full line, Interpolation theory, systems theory and related topics (Tel Aviv/Rehovot, 1999), Oper. Theory Adv. Appl., vol. 134, Birkh¨ auser, Basel, 2002, pp. 209–224. [160] C. Foias, A.E. Frazho, and M.A. Kaashoek, The distance to intertwining operators, contractive liftings and a related optimality result, Integral Equations Operator Theory 47 (2003), no. 1, 71–89.

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[161] I. Gohberg, M.A. Kaashoek, and I.M. Spitkovsky, An overview of matrix factorization theory and operator applications, Factorization and integrable systems (Faro, 2000), Oper. Theory Adv. Appl., vol. 141, Birkh¨ auser, Basel, 2003, pp. 1–102. [162] I. Gohberg, M.A. Kaashoek, and F. van Schagen, On inversion of convolution integral operators on a finite interval, Operator theoretical methods and applications to mathematical physics, Oper. Theory Adv. Appl., vol. 147, Birkh¨ auser, Basel, 2004, pp. 277–285. [163] I. Gohberg, M.A. Kaashoek, and F. van Schagen, On inversion of finite Toeplitz matrices with elements in an algebraic ring, Linear Algebra Appl. 385 (2004), 381– 389. [164] A.E. Frazho and M.A. Kaashoek, A Naimark dilation perspective of Nevanlinna–Pick interpolation, Integral Equations Operator Theory 49 (2004), no. 3, 323–378. [165] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Schur complements and state space realizations, Linear Algebra Appl. 399 (2005), 203–224. [166] D.Z. Arov, M.A. Kaashoek, and D.R. Pik, Minimal representations of a contractive operator as a product of two bounded operators, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 313–336. [167] O. Iftime, M. Kaashoek, and A. Sasane, A Grassmannian band method approach to the Nehari–Takagi problem, J. Math. Anal. Appl. 310 (2005), no. 1, 97–115. [168] I. Gohberg, M.A. Kaashoek, and A.L. Sakhnovich, Taylor coefficients of a pseudoexponential potential and the reflection coefficient of the corresponding canonical system, Math. Nachr. 278 (2005), no. 12-13, 1579–1590. [169] M.A. Kaashoek and A.L. Sakhnovich, Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model, J. Funct. Anal. 228 (2005), no. 1, 207– 233. [170] M.A. Kaashoek, Metric constrained interpolation and control theory, Not. S. Afr. Math. Soc. 36 (2005), no. 2, 114–143. [171] G.J. Groenewald and M.A. Kaashoek, A new proof of an Ellis–Gohberg theorem on orthogonal matrix functions related to the Nehari problem, Recent advances in operator theory and its applications, Oper. Theory Adv. Appl., vol. 160, Birkh¨ auser, Basel, 2005, pp. 217–232. [172] I. Gohberg, I. Haimovici, M.A. Kaashoek, and L. Lerer, The Bezout integral operator: main property and underlying abstract scheme, The state space method generalizations and applications, Oper. Theory Adv. Appl., vol. 161, Birkh¨ auser, Basel, 2006, pp. 225–270. [173] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, Coupling and relaxed commutant lifting, Integral Equations Operator Theory 54 (2006), no. 1, 33–67. [174] G.J. Groenewald and M.A. Kaashoek, A Gohberg–Heinig type inversion formula involving Hankel operators, Interpolation, Schur functions and moment problems, Oper. Theory Adv. Appl., vol. 165, Birkh¨ auser, Basel, 2006, pp. 291–302. [175] D.Z. Arov, M.A. Kaashoek, and D.R. Pik, The Kalman–Yakubovich–Popov inequality for discrete time systems of infinite dimension, J. Operator Theory 55 (2006), no. 2, 393–438. [176] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (2006), no. 1-2, 299–318. [177] I. Gohberg, M.A. Kaashoek, and L. Lerer, Quasi-commutativity of entire matrix functions and the continuous analogue of the resultant, Modern operator theory and applications, Oper. Theory Adv. Appl., vol. 170, Birkh¨ auser, Basel, 2007, pp. 101– 106. [178] I. Gohberg, M.A. Kaashoek, and L. Lerer, The continuous analogue of the resultant and related convolution operators, The extended field of operator theory, Oper. Theory Adv. Appl., vol. 171, Birkh¨ auser, Basel, 2007, pp. 107–127. [179] I. Gohberg, M.A. Kaashoek, and L. Lerer, On a class of entire matrix function equations, Linear Algebra Appl. 425 (2007), no. 2-3, 434–442.

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[180] I. Gokhberg, M.A. Kaaskhuk, and L. Lerer, The inverse problem for orthogonal Kre˘in matrix functions, Funktsional. Anal. i Prilozhen. 41 (2007), no. 2, 44–57, 111 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 41 (2007), no. 2, 115–125. [181] I. Gohberg, M.A. Kaashoek, and L. Lerer, The resultant for regular matrix polynomials and quasi commutativity, Indiana Univ. MatH.J. 57 (2008), no. 6, 2793–2813. [182] M.A. Kaashoek, L. Lerer, and I. Margulis, Krein orthogonal entire matrix functions and related Lyapunov equations: a state space approach, Integral Equations Operator Theory 65 (2009), no. 2, 223–242. [183] D. Alpay, I. Gohberg, M.A. Kaashoek, L. Lerer, and A. Sakhnovich, Krein systems, Modern analysis and applications. The Mark Krein Centenary Conference. Vol. 2: Differential operators and mechanics, Oper. Theory Adv. Appl., vol. 191, Birkh¨ auser Verlag, Basel, 2009, pp. 19–36. [184] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, Relaxed commutant lifting: an equivalent version and a new application, Recent advances in operator theory and applications, Oper. Theory Adv. Appl., vol. 187, Birkh¨ auser, Basel, 2009, pp. 157–168. [185] A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle, Integral Equations Operator Theory 66 (2010), no. 2, 215–229. [186] D. Alpay, I. Gohberg, M.A. Kaashoek, L. Lerer, and A.L. Sakhnovich, Krein systems and canonical systems on a finite interval: accelerants with a jump discontinuity at the origin and continuous potentials, Integral Equations Operator Theory 68 (2010), no. 1, 115–150. [187] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, A time-variant norm constrained interpolation problem arising from relaxed commutant lifting, Operator algebras, operator theory and applications, Oper. Theory Adv. Appl., vol. 195, Birkh¨ auser Verlag, Basel, 2010, pp. 139–166. [188] A.E. Frazho and M.A. Kaashoek, A contractive operator view on an inversion formula of Gohberg-Heinig, Topics in operator theory. Volume 1. Operators, matrices and analytic functions, Oper. Theory Adv. Appl., vol. 202, Birkh¨ auser Verlag, Basel, 2010, pp. 223–252. [189] M.A. Kaashoek and L. Lerer, Quasi commutativity of regular matrix polynomials: resultant and Bezoutian, Topics in operator theory. Volume 1. Operators, matrices and analytic functions, Oper. Theory Adv. Appl., vol. 202, Birkh¨ auser Verlag, Basel, 2010, pp. 297–314. [190] A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, I: The least squares solution, Integral Equations Operator Theory 70 (2011), no. 3, 395–418. [191] M.A. Kaashoek and F. van Schagen, On inversion of certain structured linear transformations related to block Toeplitz matrices, A panorama of modern operator theory and related topics, Oper. Theory Adv. Appl., vol. 218, Birkh¨ auser/Springer Basel AG, Basel, 2012, pp. 377–386. [192] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, Optimal solutions to matrix-valued Nehari problems and related limit theorems, Mathematical methods in systems, optimization, and control, Oper. Theory Adv. Appl., vol. 222, Birkh¨ auser/Springer Basel AG, Basel, 2012, pp. 151–172. [193] M.A. Kaashoek and F. van Schagen, Ellis–Gohberg identities for certain orthogonal functions I: Block matrix generalizations and 2 -setting, Indag. Math. (N.S.) 23 (2012), no. 4, 777–795. [194] M.A. Kaashoek and L. Lerer, The band method and inverse problems for orthogonal matrix functions of Szeg˝ o–Kreˇin type, Indag. Math. (N.S.) 23 (2012), no. 4, 900–920. [195] A.E. Frazho and M.A. Kaashoek, Canonical factorization of rational matrix functions. A note on a paper by P. Dewilde [MR2991932], Indag. Math. (N.S.) 23 (2012), no. 4, 1154–1164.

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[196] A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, II: Description of all solutions, Oper. Matrices 6 (2012), no. 4, 833–857. [197] M.A. Kaashoek and L. Lerer, On a class of matrix polynomial equations, Linear Algebra Appl. 439 (2013), no. 3, 613–620. [198] M.A. Kaashoek and F. van Schagen, Ellis-Gohberg identities for certain orthogonal functions II: Algebraic setting and asymmetric versions, Math. Proc. R. Ir. Acad. 113A (2013), no. 2, 107–130. [199] A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, Rational matrix solutions of a Bezout type equation on the half-plane, Advances in structured operator theory and related areas, Oper. Theory Adv. Appl., vol. 237, Birkh¨ auser/Springer, Basel, 2013, pp. 145– 160. [200] M.A. Kaashoek and F. van Schagen, Inverting structured operators related to Toeplitz plus Hankel operators, Advances in structured operator theory and related areas, Oper. Theory Adv. Appl., vol. 237, Birkh¨ auser/Springer, Basel, 2013, pp. 161–187. [201] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, State space formulae for stable rational matrix solutions of a Leech problem, Indag. Math. (N.S.) 25 (2014), no. 2, 250–274. [202] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, State space formulas for a suboptimal rational Leech problem I: Maximum entropy solution, Integral Equations Operator Theory 79 (2014), no. 4, 533–553. [203] M.A. Kaashoek and F. van Schagen, The inverse problem for Ellis–Gohberg orthogonal matrix functions, Integral Equations Operator Theory 80 (2014), no. 4, 527–555. [204] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, State space formulas for a suboptimal rational Leech problem II: Parametrization of all solutions, Recent advances in inverse scattering, Schur analysis and stochastic processes, Oper. Theory Adv. Appl., vol. 244, Birkh¨ auser/Springer, Cham, 2015, pp. 149–179. [205] G.J. Groenewald, S. ter Horst, and M.A. Kaashoek, The Bezout-corona problem revisited: Wiener space setting, Complex Anal. Oper. Theory 10 (2016), no. 1, 115– 139. [206] D.Z. Arov, M.A. Kaashoek, and D.R. Pik, Generalized solutions of Riccati equalities and inequalities, Methods Funct. Anal. Topology 22 (2016), no. 2, 95–116. [207] B. Fritzsche, M.A. Kaashoek, B. Kirstein, and A.L. Sakhnovich, Skew-selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations, Math. Nachr. 289 (2016), no. 1415, 1792–1819. [208] M.A. Kaashoek and F. van Schagen, The Ellis–Gohberg inverse problem for matrixvalued Wiener functions on the line, Oper. Matrices 10 (2016), no. 4, 1009–1042. [209] S. Ter Horst, M.A. Kaashoek, and F. van Schagen, The discrete twofold Ellis–Gohberg inverse problem, J. Math. Anal. Appl. 452 (2017), no. 2, 846–870. [210] G.J. Groenewald, M.A. Kaashoek, and A.C.M. Ran, Wiener-Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators, Indag. Math. (N.S.) 28 (2017), no. 3, 694–710. [211] G.J. Groenewald, S. ter Horst, and M.A. Kaashoek, The B´ ezout equation on the right half-plane in a Wiener space setting, Large truncated Toeplitz matrices, Toeplitz operators, and related topics, Oper. Theory Adv. Appl., vol. 259, Birkh¨ auser/Springer, Cham, 2017, pp. 395–411. [212] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to an operator Nevanlinna–Pick interpolation problem, Operator theory in different settings and related applications, Oper. Theory Adv. Appl., vol. 262, Birkh¨ auser/Springer, Cham, 2018, pp. 139–220.

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Edited books and journal issues [1] C.B. Huijsmans, W.A.J. Luxemburg, M.A. Kaashoek, and W.K. Vietsch (eds.), From A to Z, Mathematical Centre Tracts, vol. 149, Mathematisch Centrum, Amsterdam, 1982. [2] H. Bart, I. Gohberg, and M.A. Kaashoek (eds.), Operator theory and systems, Operator Theory: Advances and Applications, vol. 19, Birkh¨ auser Verlag, Basel, 1986. [3] I. Gohberg and M.A. Kaashoek (eds.), Constructive methods of Wiener–Hopf factorization, Operator Theory: Advances and Applications, vol. 21, Birkh¨ auser Verlag, Basel, 1986. [4] B. N. Datta, C.R. Johnson, M.A. Kaashoek, R.J. Plemmons, and E.D. Sontag (eds.), Linear algebra in signals, systems, and control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. [5] H. Dym, S. Goldberg, M.A. Kaashoek, and P. Lancaster (eds.), The Gohberg anniversary collection. Vol. I, Operator Theory: Advances and Applications, vol. 40, Birkh¨ auser Verlag, Basel, 1989. [6] H. Dym, S. Goldberg, M.A. Kaashoek, and P. Lancaster (eds.), The Gohberg anniversary collection. Vol. II, Operator Theory: Advances and Applications, vol. 41, Birkh¨ auser Verlag, Basel, 1989. [7] M.A. Kaashoek, J.H. van Schuppen, and A.C.M. Ran (eds.), Realization and modelling in system theory, Progress in Systems and Control Theory, vol. 3, Birkh¨ auser Boston, Inc., Boston, MA, 1990. [8] M.A. Kaashoek, J.H. van Schuppen, and A.C.M. Ran (eds.), Robust control of linear systems and nonlinear control, Progress in Systems and Control Theory, vol. 4, Birkh¨ auser Boston, Inc., Boston, MA, 1990. [9] M.A. Kaashoek, J.H. van Schuppen, and A.C.M. Ran (eds.), Signal processing, scattering and operator theory, and numerical methods, Progress in Systems and Control Theory, vol. 5, Birkh¨ auser Boston, Inc., Boston, MA, 1990. [10] H. Bart, I. Gohberg, and M.A. Kaashoek (eds.), Topics in matrix and operator theory, Operator Theory: Advances and Applications, vol. 50, Birkh¨ auser Verlag, Basel, 1991. [11] F. van der Blij, H. Duistermaat, R. Kaashoek, R. Tijdeman, and J. Wiegerink (eds.), Indag. Math. (N.S.), vol. 4, 1993. Special Issue: Papers dedicated to Jaap Korevaar. [12] P. Dewilde, M.A. Kaashoek, and M. Verhaegen (eds.), Challenges of a generalized system theory, Koninklijke Nederlandse Akademie van Wetenschappen. Verhandelingen, Afd. Natuurkunde. Eerste Reeks [Royal Netherlands Academy of Sciences. Proceedings, Physics Section. Series 1], vol. 40, North-Holland Publishing Co., Amsterdam, 1993. [13] C.B. Huijsmans, M.A. Kaashoek, W.A.J. Luxemburg, and B. de Pagter (eds.), Operator theory in function spaces and Banach lattices, Operator Theory: Advances and Applications, vol. 75, Birkh¨ auser Verlag, Basel, 1995. Essays dedicated to A.C. Zaanen on the occasion of his 80th birthday. [14] A. Dijksma, I. Gohberg, M.A. Kaashoek, and R. Mennicken (eds.), Contributions to operator theory in spaces with an indefinite metric, Operator Theory: Advances and Applications, vol. 106, Birkh¨ auser Verlag, Basel, 1998. The Heinz Langer anniversary volume; Papers from the colloquium held at the Technical University of Vienna, Vienna, October 12–13, 1995. [15] V.M. Adamyan, I. Gohberg, M. Gorbachuk, V. Gorbachuk, M.A. Kaashoek, H. Langer, and G. Popov (eds.), Differential operators and related topics. Vol. I, Operator Theory: Advances and Applications, vol. 117, Birkh¨ auser Verlag, Basel, 2000. [16] V.M. Adamyan, I. Gohberg, M. Gorbachuk, V. Gorbachuk, M.A. Kaashoek, H. Langer, and G. Popov (eds.), Operator theory and related topics. Vol. II, Operator Theory: Advances and Applications, vol. 118, Birkh¨ auser Verlag, Basel, 2000. [17] A. Dijksma, M.A. Kaashoek, and A.C.M. Ran (eds.), Recent advances in operator theory, Operator Theory: Advances and Applications, vol. 124, Birkh¨ auser Verlag,

Publication List of M.A. Kaashoek

[18]

[19]

[20]

[21]

[22] [23]

[24] [25]

[26] [27]

[28]

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Basel, 2001. The Israel Gohberg anniversary volume; Papers from the International Workshop on Operator Theory and its Applications (IWOTA-98) held at the University of Groningen, Groningen, June 30–July 3, 1998. R. Curtain and R. Kaashoek (eds.), Infinite-dimensional systems theory and operator theory, University of Zielona G´ ora, Zielona G´ ora, 2001. Selected papers from the 14th International Conference on Mathematical Theory of Networks and Systems (MTNS’2000) held in Perpignan, June 19–23, 2000; Int. J. Appl. Math. Comput. Sci. 11 (2001), no. 6. A. B¨ ottcher, M.A. Kaashoek, A.B. Lebre, A.F. dos Santos, and F.-O. Speck (eds.), Singular integral operators, factorization and applications, Operator Theory: Advances and Applications, vol. 142, Birkh¨ auser Verlag, Basel, 2003. Papers from the 12th International Workshop on Operator Theory and Applications (IWOTA 2000) held at the University of Algarve, Faro, September 12–15, 2000. R.B. Bapat, R. Kaashoek, R. Mathias, T.Y. Tam, and F. Uhlig (eds.), Tenth Conference of the International Linear Algebra Society, Elsevier B. V., Amsterdam, 2004. Linear Algebra Appl. 379 (2004). M.A. Kaashoek, S. Seatzu, and C. van der Mee (eds.), Recent advances in operator theory and its applications, Operator Theory: Advances and Applications, vol. 160, Birkh¨ auser Verlag, Basel, 2005. The Israel Gohberg anniversary volume; Selected papers from the 14th International Workshop on Operator Theory and its Applications (IWOTA 2003) held in Cagliari, June 24–27, 2003. H. Bart, T. Hempfling, and M.A. Kaashoek (eds.), Israel Gohberg and friends, Birkh¨ auser Verlag, Basel, 2008. On the occasion of his 80th birthday. H. Dym, M.A. Kaashoek, P. Lancaster, H. Langer, and L. Lerer (eds.), A panorama of modern operator theory and related topics, Operator Theory: Advances and Applications, vol. 218, Birkh¨ auser/Springer Basel AG, Basel, 2012. The Israel Gohberg memorial volume. H.W. Broer and M.A. Kaashoek (eds.), Indag. Math. (N.S.), vol. 23, 2012. Special Issue: Indagationes Mathematicae honoring Israel Gohberg. M.A. Kaashoek, L. Rodman, and H.J. Woerdeman (eds.), Advances in structured operator theory and related areas, Operator Theory: Advances and Applications, vol. 237, Birkh¨ auser/Springer, Basel, 2013. The Leonid Lerer anniversary volume. A. B¨ ottcher, H. Dym, M.A. Kaashoek, and A.C.M. Ran (eds.), Linear Algebra Appl., vol. 438, 2013. Special Issue in Honor of Harm Bart. M. Cepedello Boiso, H. Hedenmalm, M.A. Kaashoek, A. Montes Rodr´ıguez, and S. Treil (eds.), Concrete operators, spectral theory, operators in harmonic analysis and approximation, Operator Theory: Advances and Applications, vol. 236, Birkh¨ auser/Springer, Basel, 2014. Including papers from the 22nd International Workshop in Operator Theory and its Applications (IWOTA 2011) held at the University of Sevilla, Sevilla, July 2011. R. Duduchava, M.A. Kaashoek, N. Vasilevski, and V. Vinnikov (eds.), Operator theory in different settings and related applications, Operator Theory: Advances and Applications, vol. 262, Birkh¨ auser/Springer, Cham, 2018. 26th IWOTA, Tbilisi, July 2015; Selected papers from the International Workshop on Operator Theory and its Applications held at the Georgian National Academy of Sciences and Ivane Javakhishvili Tbilisi State University.

Other publications [1] M.A. Kaashoek, Over de wiskundige analyse, Openbare les, 5 mei 1967, Scheltema en Holkema N.V., Amsterdam, 1967 (Dutch). [2] M.A. Kaashoek, Abstract en konkreet in de wiskunde, Oratie, 19 december 1969, Scheltema en Holkema N.V., Amsterdam, 1969 (Dutch).

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[3] H. Bart, M.A. Kaashoek, H.G.J. Pijls, W.J. de Schipper, and J. de Vries, Colloquium Halfalgebras en Positieve Operatoren, Mathematisch Centrum, Amsterdam, 1971 (Dutch). [4] M.A. Kaashoek, Matrix- en operatorfuncties, enkele aspecten van het onderzoek in de wiskundige analyse aan de Vrije Universiteit, Rapport aan P. Mullender bij zijn afscheid als hoogleraar, Wiskundig Seminarium der Vrije Universiteit, 1981, pp. 51– 60 (Dutch). [5] M.A. Kaashoek, Review of Factorization of matrix functions and singular integral operators, by K. Clancey and I. Gohberg, Bull. Amer. Math. Soc. 10 (1984), 123–128. [6] H. Bart, M.A. Kaashoek, and L. Lerer, Review of Matrix Polynomials, by I. Gohberg, P. Lancaster and L. Rodman, Lin. Alg. Appl. 64 (1985), 267–272. [7] I. Gohberg and M.A. Kaashoek, Review of Introduction to the spectral theory of polynomial operator pencils, by A.S. Markus, Bull. Amer. Math. Soc. 21 (1989), 350–354. [8] M.A. Kaashoek, Review of Analysis of Toeplitz Operators, by A. B¨ ottcher and B. Silbermann, Jahresbericht der Deutschen Mathematiker-Vereinigung Bd. 95, Heft 2 (1993). [9] M.A. Kaashoek, Review of Interpolation of rational matrix functions, by J.A. Ball, I. Gohberg and L. Rodman, Bull. Amer. Math. Soc. 28 (1993), 426–434. [10] M.A. Kaashoek, Review of Time-varying discrete linear systems, by A. Halanay and V. Ionescu, GAMM Mitteilungen. [11] C.B. Huijsmans, M.A. Kaashoek, W.A.J. Luxemburg, and B. de Pagter, Biographical notes, Operator theory in function spaces and Banach lattices, Oper. Theory Adv. Appl., vol. 75, Birkh¨ auser, Basel, 1995, pp. 1–5. [12] M. Kaashoek, Laudatio for Ciprian Foias, Nieuw Arch. Wiskd. (5) 2 (2001), no. 2, 107–108 (Dutch). [13] M.A. Kaashoek, A review of the mathematical work of Israel Gohberg, Recent advances in operator theory (Groningen, 1998), Oper. Theory Adv. Appl., vol. 124, Birkh¨ auser, Basel, 2001, pp. xxvii–xxxii. [14] M. A. Kaashoek, Wiskunde: inspirerend, vernieuwend, betrokken. Afscheidscollege Vrije Universiteit, Amsterdam, 29 November 2002, Nieuw Arch. Wiskd. (4/5) 1 (2003), 18–23 (Dutch). [15] I. Gohberg and M.A. Kaashoek, Our meetings with Erhard Meister, Operator theoretical methods and applications to mathematical physics, Oper. Theory Adv. Appl., vol. 147, Birkh¨ auser, Basel, 2004, pp. 73–75. [16] M.A. Kaashoek, Many happy returns, Lin. Alg. Appl. 385 (2004), 14. [17] M. A. Kaashoek and H. van der Vorst, Kick-off wiskundeclusters, Nieuw Arch. Wiskd. (5/6) 2 (2005), 164–167 (Dutch). [18] R. Kaashoek, Congratulations from Amsterdam, Indiana Univ. MatH.J. 57 (2008), no. 6, vii–viii. [19] M.A. Kaashoek and L. Lerer, Gohberg’s mathematical work in the period 1998–2008, Israel Gohberg and Friends, Birkh¨ auser, Basel, 2008, pp. 111–115. [20] M.A. Kaashoek, Professor at Amsterdam: Thank you Israel!, Israel Gohberg and Friends, Birkh¨ auser, Basel, 2008, pp. 303–307. [21] A.E. Frazho, M.A. Kaashoek, and L. Rodman, Israel Gohberg, IEEE Control Syst. Mag. 30 (2010), no. 6, 135–139. [22] H. Bart, H. Dym, R. Kaashoek, P. Lancaster, A. Markus, and L. Rodman, In memoriam Israel Gohberg August 23, 1928–October 12, 2009, Linear Algebra Appl. 433 (2010), no. 5, 877–892. [23] R. Kaashoek, In memoriam Israel Gohberg (1928–2009). Hij kwam, zag en overwon, Nieuw Arch. Wiskd. (5) 11 (2010), no. 3, 163–166 (Dutch). [24] M.A. Kaashoek, In memoriam Israel Gohberg, Methods Funct. Anal. Topology 16 (2010), no. 4, 291–297.

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[25] H. Bart and M.A. Kaashoek, Linear Algebra and Operator Theory: Hand in Hand. Israel Gohberg’s Mathematics, Image 48 (2012), 3-4. [26] A. B¨ ottcher, R. Brualdi, H. Dym, R. Kaashoek, and A. Ran, Preface – Special issue in honor of Harm Bart, Linear Algebra Appl. 439 (2013), no. 3, 511–512. [27] M.A. Kaashoek, Leonia Lerer’s mathematical work and Amsterdam visits, Advances in structured operator theory and related areas, Oper. Theory Adv. Appl., vol. 237, Birkh¨ auser/Springer, Basel, 2013, pp. 1–7. [28] R. Kaashoek, L. Rodman, and H.J. Woerdeman, Leonid Arie Lerer: 70th Anniversary, Image 51 (2013), 17. [29] A. B¨ ottcher, H. Dym, M.A. Kaashoek, and A.C.M. Ran, Preface – Special Issue in Honor of Harm Bart 438 (2013), ii–iii. [30] M.A. Kaashoek, Pieter Mullender: 1917–2014, Newsletter London Math. Soc., (November, 2014). [31] M.A. Kaashoek and J. Rovnyak, On the preceding paper by R.B. Leech, Integral Equations Operator Theory 78 (2014), no. 1, 75–77. [32] M. Kaashoek, W. Luxemburg, and B. de Pagter, Adriaan Cornelis Zaanen, Indag. Math. (N.S.) 25 (2014), no. 2, 164–169. Translated from the Dutch [Nieuw Arch. Wiskd. (5) 5, 2004]. [33] K. Grobler, M.A. Kaashoek, A. Schep, and P. Zaanen, Adriaan Cornelis Zaanen, Ordered structures and Applications, Birkh¨ auser Verlag, Basel, 2016, pp. xi-xxii. [34] R. Kaashoek, Linear Algebra in the Netherlands, Image 56 (2016), 9. [35] R. Kaashoek, Brouwer, een wiskundige van wereldformaat, belicht van diverse kanten, Nieuw Arch. Wiskd. (5) 17 (2016), no. 4, 245–246 (Dutch). [36] M.A. Kaashoek, Living apart together, Liber Amicorum, Jan van Mill on 65th birthday, 2016, pp. 44–45. [37] J.A. Ball, M.A. Kaashoek, A.C.M. Ran, and I.M. Spitkovsky, Remembering Leiba Rodman 1949–2015, at IWOTA 2015, Operator theory in different settings and related applications, Oper. Theory Adv. Appl., vol. 262, Birkh¨ auser/Springer, Cham, 2018, pp. 3–12.

Ph.D. students of M.A. Kaashoek 1. N.P. Dekker, 1969 (official promotor: P.C. Baayen); subject: Joint numerical range and joint spectrum of Hilbert space operators. 2. H. Bart, 1973; subject: Meromorphic operator valued functions. 3. G.Ph.A. Thijsse, 1978; subject: Decomposition theorems for finite meromorphic operator functions. 4. H. den Boer, 1981; subject: Block diagonalization of matrix functions. 5. C.V.M. van der Mee, 1981 (co-supervisor: I. Gohberg); subject: Semigroup and factorization methods in transport theory. 6. A.C.M. Ran, 1984; subject: Semidefinite invariant subspaces, stability and applications. 7. L. Roozemond, 1987 (co-supervisor: I. Gohberg); subject: Systems of non-normal and first kind Wiener-Hopf equations. 8. H.J. Woerdeman, 1989 (co-supervisor: I. Gohberg); subject: Matrix and operator extensions. 9. R. Vreugdenhil, 1990 (co-supervisors: I. Gohberg, A.C.M. Ran); subject: Spectral theory of selfadjoint Wiener-Hopf and Toeplitz operators with rational symbols. 10. A.B. Kuijper, 1992 (co-supervisor: I. Gohberg); subject: The state space method for integro-differential equations of convolution type with rational matrix symbols. 11. G.J. Groenewald, 1993; subject: Wiener-Hopf factorization of rational matrix functions in terms of realization: an alternative approach. 12. J. Kos, 1995 (co-supervisor: I. Gohberg); subject: Time-dependent problems in linear operator theory. 13. D. Temme, 1996 (co-supervisors: H. Langer and A.C.M. Ran); subject: Dissipative operators in indefinite scalar product spaces. 14. D.R. Pik, 1999 (co-supervisor: D.Z. Arov); subject: Block lower triangular operators and optimal contractive systems. 15. P. Beneker, 2002 (co-supervisor and main research leader: J.J.O.O Wiegerinck at University of Amsterdam); subject: Strongly exposed points in unit balls of Banach spaces of holomorphic functions. 16. Z.D. Arova, 2003; subject: Operator nodes with strongly regular characteristic functions. 17. S. ter Horst, 2007 (jointly with A.C.M. Ran); subject: Relaxed commutant lifting and Nehari interpolation. xxxiii

Personal reminiscences Harm Bart, Sanne ter Horst, Derk Pik, Andr´e Ran, Freek van Schagen and Hugo J. Woerdeman Abstract. This contribution contains personal notes from some of Rien’s students and colleagues on the occasion of his eightieth birthday.

Marinus A. Kaashoek: Guide, Companion and Friend Harm Bart

Rien Kaashoek (2011)

Marinus A. Kaashoek (Rien) came into my life in 1966. At that time I was still an undergraduate student at the Vrije Universiteit in Amsterdam. Rien had studied in Leiden, got his PhD there in 1964 with Zaanen as his supervisor, and had spent the academic year 1965/1966 in California at UCLA as a postdoc. When we met, he had just been appointed as a very young lector, associate professor one might say. The wind of change was going through the ‘Wiskundig Seminarium’ as the Department of Mathematics at the Vrije Universiteit was then called. xxxv

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Bart, ter Horst, Pik, Ran, van Schagen and Woerdeman

Piet Mullender, the director of the institute, was aware of this. In the social sphere, he ordained that every morning at about 11 o’clock there should be coffee in the secretaries office, and that everybody in the house should in principle be there. In that way a strong interaction developed between the relatively small group of staff members, not only the permanent ones, but also those that were in temporary positions. This included even the student assistants, and I was one of them. It guaranteed easy access to the professors – not something that could be taken for granted at Dutch universities in the middle of the 1960s. It is in that light the ‘anecdote’ which follows should be read. Rien’s courses in operator theory were crisp, precise and interesting. Questions were taken seriously. No vague answers but attempts to reach the bottom. Let me give an example. In material on functional analysis, I found that the dual of the sequence spaces c0 and c are both norm isomorphic to 1 . It was emphasized (in the form of a corollary) that different Banach spaces can have the same dual. This is right, but in this form it is also superficial: the spaces c0 and c might be norm isomorphic (instead of only, which they are, just topologically isomorphic). After mentioning this to Rien, the answer came within a day: c0 and c are not norm isomorphic. Indeed, the unit ball of c0 does not have extreme points, that of c has. I was impressed. And, as a result, I abandoned my first love, topology. It was a good choice. After finishing my undergraduate study I became Rien’s PhD student, the first that he guided from the beginning to the end. And guiding he did! Good memories are connected with this. Even in concrete form: I still have the notepad in which I recorded things that came up in our frequent conversations. The topic Rien suggested to me was ‘analytic operator-valued functions‘. More specifically, the aim was to investigate under what circumstances results on the resolvent of Riesz operators have counterparts for generally nonlinear analytic operator-valued functions. Meromorphic behavior – poles in contrast to essential singularities – was a special focus point here. At UCLA Rien had met Lothrop Mittenthal, who had been working on analytic operator-valued functions too. As a first assignment, Rien gave me Mittenthal’s thesis to read. Studying it carefully, I found an imperfection in one of the proofs. Fixing it up led to my first publishable mathematical result. Later, much later, underlying mathematical issues led to a prolonged cooperation with Bernd Silbermann and his (former) student Torsten Ehrhardt. The work is going on up to the present day. At the time, however, this particular line of investigation stopped and Rien guided me to other problems in the field. After having obtained my PhD in 1973, we became companions doing joint work with David Lay from the University of Maryland at College park. Rien went there for half a year in 1975 and had the good fortune to get into close contact with Israel Gohberg who had just left the Soviet Union. He, Rien and David wrote a paper on

Personal reminiscences

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linearization by extension of analytic operator functions which rightly can be called a landmark in the field. In the Fall of 1975, I went to College Park myself and worked there with David Lay and Seymour Goldberg. With the latter I wrote a paper on almost periodic strongly operator semigroups. When I returned to Amsterdam, it was my intention to go on in this direction. But Rien’s actions interfered. He had brought Israel Gohberg to the Vrije Universiteit as a guest professor and with that the world changed. One day, in the Fall of 1976, I was ‘summoned’ to Rien’s office. Israel was also there. He asked me whether I was familiar with the theory of characteristic operator functions. The answer was: no. Then he described it in rough outline and at the end announced that he saw possibilities to advance the subject in the direction of systems theory and factorization problems. He intended to work on that with Rien and invited me to join the project. I did, again deviating from a path I had plotted for myself. So, for the time being, no semigroups. For the time being, indeed. Because a few years later they made a strong comeback when we hit upon the notions of bisemigroup and exponentially dichotomous operator. Fascinating to be involved in all of this! Not very long after we embarked on the project, we already compiled what we had in a book with the title ‘Minimal Factorization of Matrix and Operator Functions’. It came out as OT 1, the first volume in a Birkh¨auser series started by Israel Gohberg. Working on it was a frantic affair. When the book was ready to be reproduced by what was called offset printing, Israel suddenly announced that a considerable piece of material should be added. I still see Rien and myself, cutting and pasting, pieces of paper all over the tables and the floor. How glad we can be that nowadays we have LATEX.

Rien Kaashoek with OT 1 (Photograph by Dorothy Mazlum)

There is another thing that should be mentioned. After coming back from College Park in 1975, Rien started a seminar on operator theory and analysis. It was held practically every Thursday morning and lasted for about 25 years. In the course of the years, virtually all leading figures in the field could be welcomed. Many new insights and results were communicated there for the first time. It was an enormous stimulus for both staff and students.

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Bart, ter Horst, Pik, Ran, van Schagen and Woerdeman

Being companions in research lasted quite a while. But in the 1990s things began to change. For one, my cooperation with Bernd Silbermann and Torsten Ehrhardt, referred to above already, got form. Second, I became more and more involved in the administrative affairs of the Erasmus University in Rotterdam where I had been appointed in 1984. Here also Rien served as a role model. I had taken good notice of how he acted in his leadership role in Amsterdam. But then, in the first decade of the present century, we became scientific companions again. This involved writing two research monographs, updating and following up on OT 1. They were written jointly with Israel Gohberg and Andre Ran. Sadly enough, Israel passed away in 2009 just before the second of the two books actually came out. The memory of him is kept alive in a Birkh¨ auser book edited by Thomas Hempfling, Rien and myself that was published at the occasion of Israel’s eightieth birthday in 2008.

Andre Ran, Rien Kaashoek, Israel Gohberg, Harm Bart (2009 Haifa Matrix Theory Conference)

The work on the two research monographs was the last Rien and I did in the way of being companions in research. With their publication it came to an end – more than 35 years after it had started. But the friendship – from house to house! – remained. Therefore I was very happy to be given the honor to lecture on Rien’s achievements at the opening of IWOTA 2017 in Chemnitz. Impressive achievements! Among them the guiding role in the IWOTA community he took over from his close friend and companion Israel Gohberg in the past ten years. Marinus Kaashoek, guide, companion and friend: thank you very much!

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Reminiscences on Rien Kaashoek: Amsterdam and Potchefstroom Sanne ter Horst Amsterdam The first encounter I had with Rien Kaashoek – though not in person – was when I was at the VU Amsterdam in 2003 for an interview for the Ph.D. position I would start on later that year. After the interview, the proposed supervisor, Andr´e Ran, briefly explained some of the projects I could choose from. My choice was clear, and Andr´e explained to me that this would involve working with Rien, that is, Mr. Kaashoek at that stage. I must admit that before this visit to the VU I did not hear about Rien. My M.Sc. was done at the Radboud University in Nijmegen, then still Catholic University Nijmegen, under supervision of Arnoud van Rooij on a topic in topological vector spaces of measurable cardinality. The transfer to (applied) operator theory was not obvious, nor trivial, however, it went very smoothly. A few months later I showed up at the VU, installed myself in an office with two other Ph.D. students, and soon after the discussions with Rien began. I was to work on a relaxation of the famous commutant lifting theory of B´ela Sz.-Nagy and Ciprian Foia¸s that Rien had introduced in a recent paper with Ciprian Foia¸s and Art Frazho, a continuation of the work in their 1998 monograph with Israel Gohberg, referred to by us simply as OT 100. Many of my meetings with Rien followed a more or less standard pattern. I would send him the latest versions of my notes or a paper we were working on, usually late, often the evening before. He would come in the next morning by bus from Haarlem, and during the approximately half-hour bus ride he had managed to read through the notes in detail. The meeting would usually begin with Rien saying that he had enjoyed reading my notes, after which he got his copy, marked up extensively with corrections, in red. Although I think I had a very decent training in logical deduction and writing proofs in Nijmegen, writing a paper is something different altogether, as I found out, and Rien taught me how to do it. During these meetings many other mathematical topics were discussed as well, not necessarily directly related to the topic of my project. These could be related to other projects Rien was involved in, many variations on metric constrained interpolation, mathematical systems and control problems, other topics that were currently relevant in the operator theory community, or something that just came up in our discussions. I still have more than ten ring binders filled with papers from that time, many of which Rien suggested me to read. Much of what I learned during these discussions, still plays an important role in the research I do today. The most intense working periods were when Art Frazho was visiting Rien, typically in June, which happened every year for over two decades now. Art is an engineer by profession, but those who know his book with Ciprian

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Foia¸s (OT 44) or the one with Wisuwat Bhosri (OT 204) know he also is an excellent mathematician. Nonetheless, many of the discussions between Rien, Art and myself were about the importance of computability of the formulas and solutions we derived. Often my initial solutions showed a severe lack of computability. In these discussions Rien would usually take the role of mediator. He knew most of the literature and the operator theory techniques to our disposal, but was also open to proofs by direct computation. Occasionally, Art would walk in saying: “I have a theorem”, which he would write down, and motivate by the fact that for all examples, often many, that he had tried with Matlab it was true. Some of our nicest results started in this way, some of these ‘theorems’ we never managed to prove or disprove. One of the things Rien tried to instill in his students from the start is the importance of good examples, not necessarily difficult ones, to illustrate the relevance of the results. Although the operator theory group at the VU had shrunk a bit in the years towards Rien’s retirement in 2002, there was still a vibrant atmosphere with a weekly seminar and many international visitors. In the months after I started, we had visits from Gilbert Groenewald and Mark Petersen from North West University in South Africa, Olof Staffans from Abo Akademi in Finland and Martin Smith, a recent Ph.D. student of Johnathan Partington from Leeds, UK, mostly for several weeks or even months. And of course there were many visits from Israel Gohberg. In the years that followed I learned how to be an academic. Not just writing papers and teaching. Rien was always involved in an overwhelming number of activities, editorial work, the OT series, research monographs, IWOTA and MTNS, seminars, workshops, even the restructuring of the national strategy for mathematical research in the Netherlands, including the related mathematics master education, and much, much more. Always well planned and well organized. Several of these activities I have also tried, some successful, some less successful. In these matters, Rien was always a great example and source of inspiration. There was one obligatory part of doing a Ph.D. in the VU operator theory group, taking graduate courses at the Dutch Institute of Systems and Control (DISC). For someone with a background in topological vector spaces, and working on commutant lifting theory, the use of this was not entirely clear. However, Rien did not like the (strict) distinction between applied and pure mathematics, and so I had no choice but to go. In most of these courses I was the only mathematician, among predominantly engineering students. In hindsight, I learned many new things that became relevant later in my career, and I am very grateful now that I participated in these classes. I gained a lot of appreciation for the mathematical control theory that underlies the operator theory we have been working on, and in the course of time I managed to understand more and more of the talks at the MTNS conferences we attended. While the fact that Rien was retired had great benefits, he had no teaching obligations and very limited administrative responsibilities, it came

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back to haunt me at the very end of my time in Amsterdam. I am Rien’s last student. There is a rule in the Netherlands which says that a professor can only supervise Ph.D. students till five years after retirement. My graduation was on November 30, 2007, the last day of this five year period. Potchefstroom After finishing my Ph.D., Rien and I stayed in contact and still collaborated, though less intense. I did a postdoctoral fellowship at Virginia Tech, followed by some assistant professor positions after which I settled in 2012 at the Potchefstroom campus of North West University in South Africa. Soon after, we picked up our collaboration and started working on some new projects, involving Leech factorization and Wiener space matrix functions, amongst others. Several visits followed, three from Rien to Potchefstroom, sometimes with Wiesje, three from myself to Amsterdam, usually with my family. Our working relation grew into a friendship, and some of my children refer to Rien as granddad (opa Rien) now. Rien has a reputation at NWU, and a good one! Two of his Ph.D. students made it to full professor at NWU, Gilbert Groenewald and myself. Another, Andr´e Ran, is an extraordinary professor and visits us every year. Rien’s mathematical brother, Koos Grobler, both are students of Prof. A.C. Zaanen, has been at NWU since 1966. The visit of Rien in 2014 was special. During this visit he received an honorary doctorate from NWU for his exceptional contributions to mathematics, the mathematical community, and NWU in particular.

Honorary doctorate ceremony, 2014 S. ter Horst, M.A. Kaashoek, G.J. Groenewald, J.J. Grobler

The ceremony was not only attended by the higher management of NWU, the Rector Magnificus of the VU, Frank van der Duyn Schouten, had come over from the Netherlands, and at the dinner that followed the ceremony he gave a lively speech in which, among others, he shared his pleasant memories

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of the mathematics classes he had taken from Rien in the 1960s and 1970s, which he still remembered vividly. Rien, I want to thank you for providing a stimulating environment during my Ph.D. studies at the VU, being an inspiring teacher and role model for my academic career, and being a great friend and mentor in my life. It has been a true privilege to work with you all these years.

My PhD time with Rien Kaashoek Derk Pik Rien Kaashoek has been of decisive importance for my development as a mathematician as well as for my personal growth. The first time I met Rien I was a student and he lectured a course in analysis. It surprised me then that he did everything by heart, including all complicated formulas for estimations. That looked like magic at the time. Later I understood this was the result of a good preparation. Now, twentyfive years later, I take care of a similar lecture in analysis at the University of Amsterdam (for MasterMath). I try to follow his example and I experience how good it is to know everything by heart, which makes it easier to freely communicate with the students. When I became a PhD student, I had the choice to take on geometry or operator theory. I have to admit that the main reason to choose operator theory was the cheerful and lively character of the operator theory group at the Free University, led by Rien together with Israel Gohberg. There were always mathematical discussions going on, in the corridors of the math department and on their way to their collective lunches in the main building. The weekly seminar was outstanding and an excellent way for PhD students (and everybody else as well) to learn new things. Each week at least one speaker came along, often from abroad. If special topics were to be studied, specialists where invited, for instance, Heinz Langer on Krein spaces. I have nowhere else seen a colloquium of this intensity. A very happy consequence of this colloquium was that I became linked to Damir Arov, at the start of my PhD time. The three of us have enjoyed a long and fruitful collaboration. In the beginning we had to learn to understand each other. For me it is striking how much patience both Arov and Rien had when establishing precise definitions where we all could agree on, and from which a whole theory could grow, in a for me miraculous way. For Rien, my time as a PhD student was probably the busiest period of his life. He was dean of the faculty and had to attend meetings every day. I was allowed to disturb him in between two meetings, whenever I wanted to. I think he even enjoyed this. Each PhD student of Rien had to report weekly on his progress and on the problems encountered. This was another stimulating aspect of his guidance. Whenever you encountered a problem, he was very patient. On the

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other hand, when you came up with a new part of a proof of something, he put a lot of energy into it to improve it, make it clearer and simpler. He also taught me how to formulate theorems in the most simple and unambiguous way. He was very strict in this respect. In our research this has put us often on the right track. Later, as Editor-in-Chief of Pythagoras, the Dutch youth mathematics magazine, this came into good use. When writing for children, clarity and simplicity is essential. The operator theory group of Rien and Israel Gohberg was founded on generosity. People walked in and out ones offices to ask for help or just to participate in a discussion. There was an incredibly stimulating open atmosphere. Rien and Israel and with them the others shared everything they knew and that was enormous. I admire Rien very much for his outstanding quality as a mathematician. Yet I think that this is not the only factor to establish a successful research group. I firmly believe that his unselfishness, his wisdom to generously share knowledge, his willingness to listen to other mathematicians, especially from the area of applications, has created the operator theory group in Amsterdam, and made it such an inspiring place. That I had the luck to be a part of this group, has made me a different, better person, and I am very grateful for that.

Personal reminiscences: working with Rien Andr´e Ran In preparation for writing this piece I consulted not only my memory, but also several OT volumes. The volume 122 was dedicated to Rien’s 60th birthday, and Harm, Hugo and I wrote short personal reminiscences there. Volume 195, the proceedings of IWOTA 2007 in Potchefstroom, contains a printed version of the opening speech dedicated to the 60th birthday of Joe Ball and the 70th birthday of Rien. Many of the things that were written in those two volumes obviously still apply, so I will refrain from repeating what has already been written down there. Instead, let me focus on a more recent experience. In the autumn of 2016 Gilbert Groenewald (North West University, South Africa, also a PhD student of Rien) was visiting for a period of three weeks in Amsterdam. On almost the first day of his visit, Rien joined us for a coffee and discussion in my office at the VU, and he proposed an interesting project, connected to a paper which he had written with Art Frazho some time earlier [3]. The project had to do with Wiener–Hopf indices of a unitary rational matrix function in terms of a realization of the matrix function. That was a topic with which we all were very familiar. In fact, the very first paper I wrote together with Rien was concerned with describing the Wiener–Hopf indices of a selfadjoint rational matrix function in terms of the matrices appearing in a minimal realization of the function [6]. Likewise, the first paper Gilbert

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wrote with Rien was also connected to indices [1], and the second chapter of his thesis [4] was devoted to non-canonical Wiener–Hopf factorization. Over the course of an exciting three weeks we had very intense discussions between the three of us, which we thoroughly enjoyed. As a result, after three weeks we had a finished paper in our hands, which was submitted, after one more round of proofreading, to Indagationes Mathematicae. At our final meeting in those three weeks, Rien professed his amazement at what we had accomplished together in such a short time, and told us that he did not recollect ever finishing a paper in three weeks. Both Gilbert and I shared that feeling of amazement, but thruth be told: a large part of the effectiveness of this period was due to Rien’s talent at asking the right questions, pointing us in the right direction, and encouraging us to work very hard. To me it felt like being a PhD student all over again, and I enjoyed the experience tremendously. I had the pleasure of being able to present the results of the paper at the IWOTA in Chemnitz in 2017, in a session dedicated to Rien. One of the techniques used in the proof of the main theorem in [5] is matricial coupling between a Toeplitz operator and a matrix that plays an important role in the story, and which was introduced in [3]. The matricial coupling technique originated in work of Rien with Harm Bart and Israel Gohberg, [2]. It is also a topic which was discussed quite extensively in the seminar Analysis and Operator Theory at the VU in the mid-1980s, and which is now again a focus of attention for current research, in cooperation with another PhD student of Rien, Sanne ter Horst (North West University) among others (see, e.g., [7]). Both in terms of the topic and in terms of the experience the three weeks in October 2016 felt like coming full circle back to the time of doing my PhD, and I thank Rien for that, and for all that he has been to me (as well as to many of his PhD students): mentor, supervisor, advisor, inspiring colleague, and above all, friend.

References [1] J.A. Ball, M.A. Kaashoek, G.J. Groenewald, J. Kim, Column reduced rational matrix functions with given null-pole data in the complex plane, Linear Algebra Appl. 203/204 (1994), 67–110. [2] H. Bart, I. Gohberg, M.A. Kaashoek, The coupling method for solving integral equations, in: Topics in Operator Theory, Systems and Networks, OT 12, Birkh¨auser Verlag, Basel, 1984, pp. 39–73. [3] A.E. Frazho, M.A. Kaashoek, Canonical factorization of rational matrix functions. A note on a paper by P. Dewilde, Indag. Math. (N.S.) 23 (2012) 1154–1164. [4] G.J. Groenewald, Wiener–Hopf Factorization of Rational Matrix Functions in Terms of Realizations: An Alternative Version, Ph.D. thesis, VU, Amsterdam, 1993. [5] G.J. Groenewald, M.A. Kaashoek, A.C.M. Ran, Wiener–Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators, Indag. Math. (N.S.) 28 (2017), 694–710.

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[6] M.A. Kaashoek, A.C.M. Ran, Symmetric Wiener–Hopf factorization of selfadjoint rational matrix functions and realization, in: Constructive methods of Wiener–Hopf factorization, OT 21, Birkh¨auser Verlag, Basel, 1986, pp. 373–409. [7] S. ter Horst, M. Messerschmidt, A.C.M. Ran, M. Roelands, M. Wortel, Equivalence after extension and Schur coupling coincide for inessential operators, Indag. Math. (N.S.) 29 (2018), 1350–1361.

Working with Rien Kaashoek Freek van Schagen Rien Kaashoek and I met for the first time very briefly in 1963 when Rien was a PhD student at Leiden University and I was a young student. Later on we met again at the Vrije Universiteit in Amsterdam, where I got a teaching position and Rien was a full professor. In the fall of 1976 Israel Gohberg visited our department on invitation of Rien. Israel Gohberg gave an inspiring series of lectures on singular integral equations and Toeplitz operators and a seminar on matrix polynomials. This seminar was to be my starting point of working in research in operator theory with Rien and Israel Gohberg. As my main task over the years has been in teaching and in the organization of bachelor and master programs, often the time I had available for research was very restricted. It was the inspiring influence of Rien and Israel that kept my involvement in research existing. One of the projects that we worked on together was on completion problems and canonical forms. A series of papers evolved. Later on the results of this project were laid down in the book ‘Partially Specified Matrices and Operators: Classification, Completion, Applications’ by Israel Gohberg, Marinus A. Kaashoek and Frederik van Schagen, OT 79, Birkh¨auser Verlag, Basel, 1995. After Israel Gohberg passed away in 2009, Rien and I kept on working on research projects. Working with Rien in research is a great pleasure. He has inspiring ideas for projects. Usually Rien is simultaneously involved in several projects, of which in general I am actively taking part in only one. He has a broad knowledge of mathematics. In our projects we used methods originating from operator theory, integral equations, mathematical systems theory, the theory of complex functions, and linear algebra. He requires a high standard for the presentation of the results and knows how to achieve this standard. He helped me to improve my presentation of results greatly both in lectures and in writing and I am sure that I was not the only one to profit from his lessons. Rien has had much influence on me over the years also in other aspects than research in mathematics. We both were, be it in different positions, involved in the cooperation of mathematics departments at the Dutch Universities. For instance, we both worked on the development of the joint program for the master students in mathematics of nine Dutch universities. I often asked Rien for advice about my tasks, and that way I learned important

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lessons from him. At occasions I observed him chairing meetings and saw a lot that was useful later on, when I was acting as a chairman. I take this opportunity to write: Thank you, Rien, for all and in particular for the inspiration that kept my research alive.

Rien Kaashoek: Mentor, colleague and friend Hugo J. Woerdeman I have known Rien for over 30 years: first as my Ph.D. advisor at the Vrije Universiteit in Amsterdam, later as a colleague, once I became a faculty member myself, and perhaps most importantly, as a friend, as we have enjoyed delightful dinners and visits with each other that have included our wives, Wiesje and Dara, as well as our children. As a thesis advisor, Rien encouraged me every step of the way and always treated me with the utmost respect. Once graduated, he continued his guidance by involving me in activities such as editorial work, the IWOTA steering committee, etc. The pictures are a testimony to the evolution of our relationship over time: while we started out in a more “orthogonal” relationship, 30 years later, we are now very much in “parallel” with each other as close friends and respected colleagues.

Rien Kaashoek (standing) and Hugo Woerdeman. Calgary, Alberta, August 1988. Picture taken by Andr´e Ran.

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Wiesje and Rien Kaashoek and Hugo Woerdeman. Merion Station, Pennsylvania, April 2015. Picture taken by Dara Woerdeman.

Rien, I would like to thank you for all your wonderful guidance and the great personal friendship. Dara and I wish you and Wiesje many many more healthy years together!

————————————————————————————–

Harm Bart Econometric Institute, Erasmus University Rotterdam P.O. Box 1738, 3000 DR Rotterdam The Netherlands e-mail: [email protected] Sanne ter Horst Department of Mathematics, Unit for BMI, North-West University Potchefstroom, 2531 South Africa e-mail: [email protected] Derk Pik Faculty of Social and Behavioural Sciences, University of Amsterdam Amsterdam The Netherlands e-mail: [email protected]

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Andr´e Ran Department of Mathematics, Faculty of Science, VU Amsterdam De Boelelaan 1081a, 1081 HV Amsterdam The Netherlands and Unit for BMI, North-West University, Potchefstroom South Africa e-mail: [email protected] Freek van Schagen Department of Mathematics, VU Amsterdam De Boelelaan 1081a, 1081 HV Amsterdam The Netherlands e-mail: [email protected] Hugo J. Woerdeman Department of Mathematics, Drexel University 3141 Chestnut Street, Philadelphia, PA 19104 USA e-mail: [email protected]

Carath´eodory extremal functions on the symmetrized bidisc Jim Agler, Zinaida A. Lykova and N.J. Young To Rien Kaashoek in esteem and friendship

Abstract. We show how realization theory can be used to find the solutions of the Carath´eodory extremal problem on the symmetrized bidisc def

G = {(z + w, zw) : |z| < 1, |w| < 1}. We show that, generically, solutions are unique up to composition with automorphisms of the disc. We also obtain formulae for large classes of extremal functions for the Carath´eodory problems for tangents of non-generic types. Mathematics Subject Classification (2010). 32A07, 53C22, 54C15, 47A57, 32F45, 47A25, 30E05. Keywords. Carath´eodory extremal functions, symmetrized bidisc, model formulae, realization formulae.

Introduction A constant thread in the research of Marinus Kaashoek over several decades has been the power of realization theory applied to a wide variety of problems in analysis. Among his many contributions in this area we mention his monograph [6], written with his longstanding collaborators Israel Gohberg and Harm Bart, which was an early and influential work in the area, and his more recent papers [13, 10]. Realization theory uses explicit formulae for functions in terms of operators on Hilbert space to prove function-theoretic results. In this paper we continue along the Bart–Gohberg–Kaashoek path by using realization theory to prove results in complex geometry. Specifically, Partially supported by National Science Foundation Grants DMS 1361720 and 1665260, the UK Engineering and Physical Sciences Research Council grant EP/N03242X/1, the London Mathematical Society Grant 41730 and Newcastle University.

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_1

1

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J. Agler, Z.A. Lykova and N.J. Young

we are interested in the geometry of the symmetrized bidisc def

G = {(z + w, zw) : |z| < 1, |w| < 1}, a domain in C2 that has been much studied in the last two decades: see [8, 9, 11, 7, 17, 18, 2], along with many other papers. We shall use realization theory to prove detailed results about the Carath´eodory extremal problem on G, defined as follows (see [14, 12]). Consider a domain (that is, a connected open set) Ω in Cn . For domains Ω1 , Ω2 , we denote by Ω2 (Ω1 ) the set of holomorphic maps from Ω1 to Ω2 . A point in the complex tangent bundle T Ω of Ω will be called a tangent (to def

Ω). Thus if δ = (λ, v) is a tangent to Ω, then λ ∈ Ω and v is a point in the complex tangent space Tλ Ω ∼ Cn of Ω at λ. We say that δ is a nondegenerate tangent if v = 0. We write | · | for the Poincar´e metric on T D: def

|(z, v)| =

|v| 1 − |z|2

for z ∈ D, v ∈ C.

The Carath´eodory or Carath´eodory-Reiffen pseudometric [12] on Ω is the Finsler pseudometric | · |car on T Ω defined for δ = (λ, v) ∈ T Ω by def

|δ|car =

sup |F∗ (δ)|

F ∈D(Ω)

|Dv F (λ)| . 2 1 F ∈D(Ω) − |F (λ)|

= sup

(0.1)

Here F∗ is the standard notation for the pushforward of δ by the map F to an element of T D, given by g, F∗ (δ) = g ◦ F, δ for any analytic function g in a neighbourhood of F (λ). The Carath´eodory extremal problem Car δ on Ω is to calculate |δ|car for a given δ ∈ T Ω, and to find the corresponding extremal functions, which is to say, the functions F ∈ D(Ω) for which the supremum in equation (0.1) is attained. We shall also say that F solves Car δ to mean that F is an extremal function for Car δ. For a general domain Ω one cannot expect to find either | · |car or the corresponding extremal functions explicitly. In a few cases, however, there are more or less explicit formulae for |δ|car . In particular, when Ω = G, | · |car is a metric on T G (it is positive for nondegenerate tangents) and the following result obtains [4, Theorem 1.1 and Corollary 4.3]. We use the co-ordinates (s1 , s2 ) for a point of G. Theorem 0.1. Let δ be a nondegenerate tangent vector in T G. There exists ω ∈ T such that the function in D(G) given by 2ωs2 − s1 2 − ωs1 is extremal for the Carath´eodory problem Car δ in G. def

Φω (s1 , s2 ) =

(0.2)

Carath´eodory extremal functions

3

It follows that |δ|car can be obtained as the maximum modulus of a 1 fractional  1 2 quadratic function over the unit circle [4, Corollary 4.4] : if δ = (s , s ), v ∈ T G, then |δ|car = sup |(Φω )∗ (δ)| ω∈T     v1 (1 − ω 2 s2 ) − v2 ω(2 − ωs1 ) . = sup   1 1 2 2 2 2 1 2 1 ω∈T (s − s s )ω − 2(1 − |s | )ω + s − s s Hence |δ|car can easily be calculated numerically to any desired accuracy. In the latter equation we use superscripts (in s1 , s2 ) and squares (of ω, |s2 |). The question arises: what are the extremal functions for the problem Car δ? By Theorem 0.1, there is an extremal function for Car δ of the form Φω for some ω in T, but are there others? It is clear that if F is an extremal function for Car δ, then so is m ◦ F for any automorphism m of D, by the invariance of the Poincar´e metric on D. We shall say that the solution of Car δ is essentially unique if, for every pair of extremal functions F1 , F2 for Car δ, there exists an automorphism m of D such that F2 = m ◦ F1 . We show in Theorem 2.1 that, for any nondegenerate tangent δ ∈ T G, if there is a unique ω in T such that Φω solves Car δ, then the solution of Car δ is essentially unique. Indeed, for any point λ ∈ G, the solution of Car(λ, v) is essentially unique for generic directions v (Corollary 2.7). We also derive (in Section 3) a parametrization of all solutions of Car δ in the special case that δ is tangent to the ‘royal variety’ (s1 )2 = 4s2 in G, and in Sections 4 and 5 we obtain large classes of Carath´eodory extremals for two other classes of tangents, called flat and purely balanced tangents. The question of the essential uniqueness of solutions of Car δ in domains including G was studied by L. Kosi´ nski and W. Zwonek in [15]. Their terminology and methods differ from ours; we explain the relation of their Theorem 5.3 to our Theorem 2.1 in Section 6. Incidentally, the authors comment that very little is known about the set of all Carath´eodory extremals for a given tangent in a domain. As far as the domain G goes, in this paper we derive a substantial amount of information, even though we do not achieve a complete description of all Carath´eodory extremals on G. The main tool we use is a model formula for analytic functions from G to the closed unit disc D− proved in [5] and stated below as Definition 2.2 and Theorem 2.3. Model formulae and realization formulae for a class of functions are essentially equivalent: one can pass back and forth between them by standard methods (algebraic manipulation in one direction, lurking isometry arguments in the other).

1 Unfortunately there is an ω missing in equation (4.7) of [4]. The derivation given there shows that the correct formula is the present one.

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1. Five types of tangent There are certainly nondegenerate tangents δ ∈ T G for which the solution of Car δ is not essentially unique. Consider, for example, δ of the form   δ = (2z, z 2 ), 2c(1, z) for some z ∈ D and nonzero complex c. We call such a tangent royal: it is tangent to the ‘royal variety’ def

R = {(2z, z 2 ) : z ∈ D} in G. By a simple calculation, for any ω ∈ T, Φω (2z, z 2 ) = −z,

Dv Φω (2z, z 2 ) = −c,

where v = 2c(1, z), so that Φω (2z, z 2 ) and Dv Φω (2z, z 2 ) are independent of ω. It follows from Theorem 0.1 that Φω solves Car δ for all ω ∈ T and that |δ|car =

|Dv Φω (2z, z 2 )| |c| = . 1 − |Φω (2z, z 2 )|2 1 − |z|2

(1.1)

Now if ω1 , ω2 are distinct points of T, there is no automorphism m of D such that Φω1 = m ◦ Φω2 ; this is a consequence of the fact that (2¯ ω, ω ¯ 2 ) is the unique singularity of Φω in the closure Γ of G. Hence the solution of Car δ is not essentially unique. Similar conclusions hold for another interesting class of tangents, which we call flat. These are the tangents of the form   ¯ z), c(β, ¯ 1) (λ, v) = (β + βz, for some β ∈ D and c ∈ C \ {0}. It is an entertaining calculation to show that |(λ, v)|car =

|Dv Φω (λ)| |c| = 1 − |Φω (λ)|2 1 − |z|2

(1.2)

for all ω ∈ T. Again, the solution to Car(λ, v) is far from being essentially unique. There are also tangents δ ∈ T G such that Φω solves Car δ for exactly two values of ω in T; we call these purely balanced tangents. They can be described concretely as follows. For any hyperbolic automorphism m of D (that is, one that has two fixed points ω1 and ω2 in T) let hm in G(D) be given by hm (z) = (z + m(z), zm(z)) for z ∈ D. A purely balanced tangent has the form δ = (hm (z), chm (z))

(1.3)

for some hyperbolic automorphism m of D, some z ∈ D and some c ∈ C \ {0}. It is easy to see that, for ω ∈ T, the composition Φω ◦ hm is a rational inner function of degree at most 2 and that the degree reduces to 1 precisely when ¯ 2 . Thus, for these two values of ω (and only these), Φω ◦ hm ω is either ω ¯ 1 or ω is an automorphism of D. It follows that Φω solves Car δ if and only if ω = ω ¯1 or ω ¯2.

Carath´eodory extremal functions

5

A fourth type of tangent, which we call exceptional, is similar to the purely balanced type, but differs in that the hyperbolic automorphism m of D is replaced by a parabolic automorphism, that is, an automorphism m of D which has a single fixed point ω1 in T, which has multiplicity 2. The same argument as in the previous paragraph shows that Φω solves the Carath´eodory problem if and only if ω = ω ¯1. The fifth and final type of tangent is called purely unbalanced. It consists of the tangents δ = (λ, v) ∈ T G such that Φω solves Car δ for a unique value eit0 of ω in T and  d2 |Dv Φeit (λ)|  < 0. (1.4) dt2 1 − |Φeit (λ)|2 t=t0 The last inequality distinguishes purely unbalanced from exceptional tangents – the left-hand side of equation (1.4) is equal to zero for exceptional tangents. The five types of tangent are discussed at length in our paper [2]. We proved [2, Theorem 3.6] a ‘pentachotomy theorem’, which states that every nondegenerate tangent in T G is of exactly one of the above five types. We also give, for a representative tangent of each type, a cartoon showing the unique complex geodesic in G touched by the tangent [2, Appendix B]. It follows trivially from Theorem 0.1 that, for every nondegenerate tangent δ ∈ T G, either (1) there exists a unique ω ∈ T such that Φω solves Car δ, or (2) there exist at least two values of ω in T such that Φω solves Car δ. The above discussion shows that Case (1) obtains for purely unbalanced and exceptional tangents, while Case (2) holds for royal, flat and purely balanced tangents. For the purpose of this paper, the message to be drawn is that Case (1) is generic in the following sense. Consider a point λ ∈ G. Each tangent v in Tλ G has a ‘complex direction’ Cv, which is a one-dimensional subspace of C2 , or in other words, a point of the projective space CP2 . The directions corresponding to the royal (if any) and flat tangents at λ are just single points in CP2 , while, from the constructive nature of the expression (1.3) for a purely balanced tangent, it is easy to show that there is a smooth one-real-parameter curve of purely balanced directions (see [1, Section 1]). It follows that the set of directions Cv ∈ CP2 for which a unique Φω solves Car δ contains a dense open set in CP2 . To summarise: Proposition 1.1. For every λ ∈ G there exists a dense open set Vλ in CP2 such that whenever Cv ∈ Vλ , there exists a unique ω ∈ T such that Φω solves Car(λ, v).

2. Tangents with a unique extremal Φω In Section 1 we discussed extremal functions of the special form Φω , ω ∈ T, for the Carath´eodory problem in G. However, there is no reason to expect that the Φω will be the only extremal functions. For example, if δ = (λ, v) is a nondegenerate tangent and Φω1 , . . . , Φωk all solve Car δ, then one can

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J. Agler, Z.A. Lykova and N.J. Young

generate a large class of other extremal functions as follows. Choose an automorphism mj of D such that mj ◦ Φωj (λ) = 0 and Dv (mj ◦ Φωj )(λj ) > 0 for j = 1, . . . , k. Then each mj ◦ Φωj solves Car δ, and so does any convex combination of them. Nevertheless, if there is a unique ω ∈ T such that Φω is extremal for Car δ, then the solution of Car δ is essentially unique. Theorem 2.1. Let δ be a nondegenerate tangent in G such that Φω solves Car δ for a unique value of ω in T. If ψ solves Car δ, then there exists an automorphism m of D such that ψ = m ◦ Φω . For the proof recall the following model formula [5, Definition 2.1 and Theorem 2.2]. Definition 2.2. A G-model for a function ϕ on G is a triple (M, T, u) where M is a separable Hilbert space, T is a contraction acting on M and u : G → M is an analytic map such that, for all s, t ∈ G, 1 − ϕ(t)ϕ(s) = (1 − t∗T sT )u(s), u(t)M

(2.1)

where, for s ∈ G, sT = (2s2 T − s1 )(2 − s1 T )−1 . A G-model (M, T, u) is unitary if T is a unitary operator on M. def

For any domain Ω we define the Schur class S(Ω) to be the set of holomorphic maps from Ω to the closed unit disc D− . Theorem 2.3. Let ϕ be a function on G. The following three statements are equivalent. (1) ϕ ∈ S(G); (2) ϕ has a G-model; (3) ϕ has a unitary G-model (M, T, u). From a G-model of a function ϕ ∈ S(G) one may easily proceed by means of a standard lurking isometry argument to a realization formula ϕ(s) = A + BsT (1 − DsT )−1 C,

all s ∈ G,

for ϕ, where ABCD is a contractive or unitary colligation on C⊕M. However, for the present purpose it is convenient to work directly from the G-model. We also require a long-established fact about G [4], related to the fact that the Carath´eodory and Kobayashi metrics on T G coincide. Lemma 2.4. If δ is a nondegenerate tangent to G and ϕ solves Car δ, then there exists k in G(D) such that ϕ ◦ k = idD . Moreover, if ψ is any solution of Car δ, then ψ ◦ k is an automorphism of D. We shall need some minor measure-theoretic technicalities. Lemma 2.5. Let Y be a set and let A:T×Y ×Y →C be a map such that

Carath´eodory extremal functions

7

(1) A(·, z, w) is continuous on T for every z, w ∈ Y ; (2) A(η, ·, ·) is a positive kernel on Y for every η ∈ T. Let M be a separable Hilbert space, let T be a unitary operator on M with spectral resolution  T =

η dE(η), T

and let v : Y → M be a mapping. Let  C(z, w) = A(η, z, w) dE(η)v(z), v(w)

(2.2)

T

for all z, w ∈ Y . Then C is a positive kernel on Y . Proof. Consider any finite subset {z1 , . . . , zN } of Y . We must show that the N × N matrix  N C(zi , zj ) i,j=1 is positive. Since A(·, zi , zj ) is continuous on T for each i and j, we may approximate the N × N -matrix-valued function [A(·, zi , zj )] uniformly on T by integrable simple functions of the form  [fij ] = b χ τ  

for some N × N matrices b and Borel sets τ , where χ denotes ‘characteristic function’. Moreover we may do this in such a way that each b is a value [A(η, zi , zj )] for some η ∈ T, hence is positive. Then 

N   N fij (η) dE(η)zi , zj  = b ∗ E(τ )vi , vj  i,j=1 (2.3) 

τ

i,j=1



where ∗ denotes the Schur (or Hadamard) product of matrices. Since the matrix E(τ )vi , vj  is positive and the Schur product of positive matrices is positive, every approximating sum of the form (2.3) is positive, and hence the integral in equation (2.2) is a positive matrix.  Lemma 2.6. For i, j = 1, 2 let aij : T → C be continuous and let each aij have only finitely many zeros in T. Let νij be a complex-valued Borel measure on T such that, for every Borel set τ in T,  2 νij (τ ) i,j=1 ≥ 0. Let X be a Borel subset of T and suppose that  2 aij (η) i,j=1 is positive and of rank 2 for all η ∈ X. Let

 2 C = cij i,j=1 

where cij =

aij (η) dνij (η) X

for i, j = 1, 2.

If rank C ≤ 1, then either c11 = 0 or c22 = 0.

8

J. Agler, Z.A. Lykova and N.J. Young

Proof. By hypothesis the set def

Z =

2 

{η ∈ T : aij (η) = 0}

i,j=1

is finite. Exactly as in the proof of Lemma 2.5, for any Borel set τ in T, 

2 aij dνij ≥ 0. τ

(2.4)

ij=1

Suppose that C has rank at most 1 but c11 and c22 are both nonzero. Then there exists a nonzero 2 × 1 matrix c = [c1 c2 ]T such that C = cc∗ for i, j = 1, 2 and c1 , c2 are nonzero. For any Borel set τ ⊂ X,  





+ a dν aij dνij ≤ aij dνij = C = cc∗ . ij ij = τ

τ

X\τ

X

Consequently there exists a unique μ(τ ) ∈ [0, 1] such that 

aij dνij = μ(τ )C.

(2.5)

τ

It is easily seen that μ is a Borel probability measure on X. Note that if η ∈ Z, say aij (η) = 0, then on taking τ = {η} in equation (2.5), we deduce that μ({η})ci c¯j = 0. Since c1 , c2 are nonzero, it follows that μ({η}) = 0. Hence μ(Z) = 0. Equation (2.5) states that μ is absolutely continuous with respect to νij on X and the Radon–Nikodym derivative is given by dμ ci c¯j = aij dνij for i, j = 1, 2. Hence, on X \ Z, dνij =

ci c¯j dμ, aij

i, j = 1, 2.

(2.6)

Pick a compact subset K of X \ Z such that μ(K) > 0. This is possible, since μ(X \ Z) = 1 and Borel measures on T are automatically regular. By compactness, there exists a point η0 ∈ K such that, for every open neighbourhood U of η0 , μ(U ∩ K) > 0. Notice that, for η ∈ T \ Z,  

ci c¯j 2 |c1 c2 |2 det aij (η) =− < 0. det aij (η) i,j=1 a11 (η)a22 (η)|a12 (η)|2 Thus [ci c¯j aij (η0 )−1 ] has a negative eigenvalue. Therefore there exists a unit vector x ∈ C2 , an ε > 0 and an open neighourhood U of η0 in T such that  

ci c¯j aij (η)−1 x, x < −ε

Carath´eodory extremal functions for all η ∈ U . We then have 





νij (U ∩ K) x, x = 

 U ∩K



= U ∩K

ci c¯j aij (η)

−1



9

 dμ(η)x, x



ci c¯j aij (η)−1 x, x dμ(η)

< −εμ(U ∩ K) < 0.   This contradicts the positivity of the matricial measure νij . Hence either c1 = 0 or c2 = 0.  Proof of Theorem 2.1. Let δ be a nondegenerate tangent to G such that Φω is the unique function from the collection {Φη }η∈T that solves Car δ. Let ψ be a solution of Car δ. We must find an automorphism m of D such that ψ = m ◦ Φω . By Lemma 2.4, there exists k in G(D) such that Φω ◦ k = idD ,

(2.7)

and moreover, the function def

m = ψ◦k

(2.8)

ϕ = m−1 ◦ ψ.

(2.9)

ϕ ◦ k = m−1 ◦ ψ ◦ k = m−1 ◦ m = idD .

(2.10)

is an automorphism of D. Let

Then By Theorem 2.3, there is a unitary G-model (M, T, u) for ϕ. By the spectral theorem for unitary operators, there is a spectral measure E(.) on T with values in B(M) such that  η dE(η). T = T

Thus, for s ∈ G, sT = (2s T − s )(2 − s T ) 2

1

1

−1

 = T

Φη (s) dE(η).

Therefore, for all s, t ∈ G, 1 − ϕ(t)ϕ(s) = (1 − t∗T sT )u(s), u(t)M    = 1 − Φη (t)Φη (s) dE(η)u(s), u(t)M .

(2.11)

T

Consider z, w ∈ D and put s = k(z), t = k(w) in equation (2.11). Invoke equation (2.10) and divide equation (2.11) through by 1 − wz ¯ to obtain, for

10

J. Agler, Z.A. Lykova and N.J. Young

z, w ∈ D,





1=

+ {ω}

T\{ω}

1 − Φη ◦ k(w)Φη ◦ k(z) dE(η)u ◦ k(z), u ◦ k(w) 1 − wz ¯

= I1 + I 2

(2.12)

where I1 (z, w) = E({ω})u ◦ k(z), u ◦ k(w) ,  1 − Φη ◦ k(w)Φη ◦ k(z) I2 (z, w) = dE(η)u ◦ k(z), u ◦ k(w) . (2.13) 1 − wz ¯ T\{ω} The left-hand side 1 of equation (2.12) is a positive kernel of rank one on D, and I1 is also a positive kernel. The integrand in I2 is a positive kernel on D for each η ∈ T, by Pick’s theorem, since Φη ◦ k is in the Schur class. Hence, by Lemma 2.5, I2 is also a positive kernel on D. Since I1 + I2 has rank 1, it follows that I2 has rank at most 1 as a kernel on D. By hypothesis, Φη does not solve Car δ for any η ∈ T \ {ω}. Therefore Φη ◦ k is a Blaschke product of degree 2, and consequently, for any choice of distinct points z1 , z2 in D, the 2 × 2 matrix

2  2 def 1 − Φη ◦ k(zi )Φη ◦ k(zj ) aij (η) i,j=1 = (2.14) 1 − z¯i zj i,j=1 is a positive matrix of rank 2 for every η ∈ T \ {ω}. In particular, a11 (η) > 0 for all η ∈ T \ {ω}. Moreover, each aij has only finitely many zeros in T, as may be seen from the fact that aij is a ratio of trigonometric polynomials in η. To be explicit, if we temporarily write k = (k 1 , k 2 ) : D → G, then equation (2.14) expands to aij (η) = P (η)/Q(η) where     P (η) = 4 1 − k 2 (zi )k 2 (zj ) − 2η k 1 (zj ) − k 1 (zi )k 2 (zj )   − 2¯ η k 1 (zi ) − k 2 (zi )k 1 (zj ) , Q(η) = (1 − z¯i zj )(2 − ηk 1 (zi ))− (2 − ηk 1 (zj )). Let νij = E(·)u ◦ k(zi ), u ◦ k(zj ) . Clearly [νij (τ )] ≥ 0 for every Borel subset τ of T \ {ω}. By definition (2.13),  I2 (zi , zj ) = aij dνij T\{ω}

for i, j = 1, 2. Moreover, by equation (2.12),



1 [I2 (zi , zj )] ≤ [I1 (zi , zj )] + [I2 (zi , zj )] = 1 It follows that





aij dνij = [I2 (zi , zj ] = κ 1 1 T\{ω}

1 . 1

1 1

(2.15)

Carath´eodory extremal functions

11

for some κ ∈ [0, 1]. We may now apply Lemma 2.6 with X = T \ {ω} to deduce that κ = 0 and hence I2 (zi , zj ) = 0. In particular,  0 = I2 (z1 , z1 ) = a11 dν11 . T\{ω}

Since a11 > 0 on T \ {ω}, it follows that ν11 (T \ {ω}) = 0, which is to say that E(T \ {ω})u ◦ k(z1 ) = 0. (2.16) Since z1 , z2 were chosen arbitrarily in T \ {ω}, we have I2 ≡ 0 and therefore, by equation (2.12), 1 = I1 = E({ω})u ◦ k(z), u ◦ k(w)

(2.17)

for all z, w ∈ D. It follows that E({ω})u ◦ k(z) − E({ω})u ◦ k(w)2 = 0 for all z, w, and hence that there exists a unit vector x ∈ M such that E({ω})u ◦ k(z) = x for all z ∈ D. In equation (2.11), choose t = k(w) for some w ∈ D. Since Φω ◦ k = idD , we have for all s ∈ G, 1 − wϕ(s) ¯ = 1 − ϕ ◦ k(w)ϕ(s)     1 − Φη ◦ k(w)Φη (s) dE(η)u(s), u ◦ k(w) = + {ω}

T\{ω}

= (1 − wΦ ¯ ω (s)) u(s), x    + 1 − Φη ◦ k(w)Φη (s) dE(η)u(s), u ◦ k(w) . T\{ω}

In view of equation (2.16), the scalar spectral measure in the second term on the right-hand side is zero on T \ {ω}. Hence the integral is zero, and so, for all s ∈ G and w ∈ D, 1 − wϕ(s) ¯ = (1 − wΦ ¯ ω (s)) u(s), x .

(2.18)

Put w = 0 to deduce that u(s), x = 1 for all s ∈ G, then equate coefficients of w ¯ to obtain ϕ = Φω . Hence, by equation (2.8), ψ = m ◦ ϕ = m ◦ Φω as required.  On combining Theorem 2.1 and Proposition 1.1 we obtain the statement in the abstract. Corollary 2.7. Let λ ∈ G. For a generic direction Cv in CP2 , the solution of the Carath´eodory problem Car(λ, v) is essentially unique.

12

J. Agler, Z.A. Lykova and N.J. Young

It will sometimes be useful in the sequel to distinguish a particular Carath´eodory extremal function from a class of functions that are equivalent up to composition with automorphisms of D. Consider any tangent δ ∈ T G and any solution ϕ of Car δ. The functions m ◦ ϕ, with m an automorphism of D, also solve Car δ, and among them there is exactly one that has the property m ◦ ϕ(λ) = 0 and Dv (m ◦ ϕ)(λ) > 0, or equivalently, (m ◦ ϕ)∗ (δ) = (0, |δ|car ). (2.19) We shall say that ϕ is well aligned at δ if ϕ∗ (δ) = (0, |δ|car ). With this terminology the following is a re-statement of Theorem 2.1. Corollary 2.8. If δ is a nondegenerate tangent in G such that Φω solves Car δ for a unique value of ω in T, then there is a unique well-aligned solution of Car δ. It is expressible as m ◦ Φω for some automorphism m of D.

3. Royal tangents At the opposite extreme from the tangents studied in the last section are the royal tangents to G. Recall that these have the form   (3.1) δ = (2z, z 2 ), 2c(1, z) for some z ∈ D and nonzero complex number c. As we observed in Section 1, |δ|car =

|c| 1 − |z|2

and all Φω , ω ∈ T, solve Car δ. In this section we shall describe all extremal functions for Car δ for royal tangents δ, not just those of the form Φω . Theorem 3.1. Let δ ∈ T G be the royal tangent   δ = (2z, z 2 ), 2c(1, z)

(3.2)

for some z ∈ D and c ∈ C \ {0}. A function ϕ ∈ D(G) solves Car δ if and only if there exists an automorphism m of D and Ψ ∈ S(G) such that, for all s ∈ G,   Ψ(s) 1 2 2 1 1 1 ϕ(s) = m 2 s + 4 ((s ) − 4s ) . (3.3) 1 − 12 s1 Ψ(s) Proof. We shall lift the problem Car δ to a Carath´eodory problem on the bidisc D2 , where we can use the results of [3] on the Nevanlinna–Pick problem on the bidisc. Let π : D2 → G be the ‘symmetrization map’, π(λ1 , λ2 ) = (λ1 + λ2 , λ1 λ2 ) and let k : D → D2 be given by k(ζ) = (ζ, ζ) for ζ ∈ D. Consider the royal tangent δ of equation (3.2) and let δzc = ((z, z), (c, c)) ∈ T D2 .

Carath´eodory extremal functions

k (z,c)

π

(2z,z2)

ϕ ϕ (δ) ∗

δ

δ zc 2

ID

13

ID

G

ID

Figure 1 Observe that π  (λ) = and so



1 λ2

1 , λ1

  π∗ (δzc ) = (π(z, z), π  (z, z)(c, c)) = (2z, z 2 ), 2c(1, z) = δ,

(3.4)

while k∗ ((z, c)) = (k(z), k  (z)c) = ((z, z), (c, c)) = δzc . Consider any ϕ ∈ D(G). Figure 1 illustrates the situation. It is known that every Carath´eodory problem on the bidisc is solved by one of the two co-ordinate functions Fj (λ) = λj for j = 1 or 2 (for a proof see, for example, [2, Theorem 2.3]). Thus 2

|D(c,c) Fj (z, z)| j=1,2 1 − |Fj (z, z)|2 |c| = 1 − |z|2 = |δ|car .

|δzc |D car = max

Here of course the superscript D2 indicates the Carath´eodory extremal problem on the bidisc. Hence, for ϕ ∈ D(G), |c| 1 − |z|2 |c| ⇐⇒ |ϕ∗ ◦ π∗ (δzc )| = 1 − |z|2 |c| ⇐⇒ |ϕ∗ (δ)| = 1 − |z|2 ⇐⇒ ϕ solves Car δ.

ϕ ◦ π solves Car δzc ⇐⇒ |(ϕ ◦ π)∗ (δzc )| =

by the chain rule by equation (3.4) (3.5)

Next observe that a function ψ ∈ D(D2 ) solves Car δzc if and only if ψ ◦ k is an automorphism of D. For if ψ ◦ k is an automorphism of D, then it satisfies |(z, c)| = |(ψ ◦ k)∗ (z, c)| = |ψ∗ ◦ k∗ (z, c)| = |ψ∗ (δzc )|,

14

J. Agler, Z.A. Lykova and N.J. Young

which is to say that ψ solves Car δzc . Conversely, if ψ solves Car δzc , then ψ ◦ k is an analytic self-map of D that preserves the Poincar´e metric of a nondegenerate tangent to D, and is therefore (by the Schwarz–Pick lemma) an automorphism of D. On combining this observation with equivalence (3.5) we deduce that ϕ solves Car δ ⇐⇒ there exists an automorphism m of D such that m−1 ◦ ϕ ◦ π ◦ k = idD .

(3.6)

For a function f ∈ D(D ), it is easy to see that f ◦ k = idD if and only if f solves the Nevanlinna–Pick problem 2

(0, 0) → 0,

( 12 , 12 ) → 12 .

(3.7)

See [3, Subsection 11.5] for the Nevanlinna–Pick problem in the bidisc. Hence ϕ solves Car δ ⇐⇒ there exists an automorphism m of D such that m−1 ◦ ϕ ◦ π solves the Nevanlinna–Pick problem (3.7). (3.8) In [3, Subsection 11.6] Agler and McCarthy use realization theory to show the following. A function f ∈ S(D2 ) satisfies the interpolation conditions f (0, 0) = 0,

f ( 12 , 12 ) =

1 2

(3.9)

if and only if there exist t ∈ [0, 1] and Θ in the Schur class of the bidisc such that, for all λ ∈ D2 , f (λ) = tλ1 + (1 − t)λ2 + t(1 − t)(λ1 − λ2 )2

Θ(λ) . (3.10) 1 − [(1 − t)λ1 + tλ2 ]Θ(λ)

Inspection of the formula (3.10) reveals that f is symmetric if and only if t = 12 and Θ is symmetric. Hence the symmetric functions in S(D2 ) that satisfy the conditions (3.9) are those given by f (λ) = 12 λ1 + 12 λ2 + 14 (λ1 − λ2 )2

Θ(λ) 1 − 12 (λ1 + λ2 )Θ(λ)

(3.11)

for some symmetric Θ ∈ S(D2 ). Such a Θ induces a unique function Ψ ∈ S(G) such that Θ = Ψ ◦ π, and we may write the symmetric solutions f of the problem (3.9) in the form f = f˜ ◦ π where, for all s = (s1 , s2 ) in G, f˜(s) = 12 s1 + 14 ((s1 )2 − 4s2 )

Ψ(s) . 1 − 12 s1 Ψ(s)

(3.12)

Let ϕ solve Car δ. By the equivalence (3.8), there exists an automorphism m of D such that m−1 ◦ ϕ ◦ π solves the Nevanlinna–Pick problem (3.7). Clearly m−1 ◦ ϕ ◦ π is symmetric. Hence there exists Ψ ∈ S(G) such that, for all λ ∈ D2 , m−1 ◦ ϕ(s) = 12 s1 + 14 ((s1 )2 − 4s2 ) Thus ϕ is indeed given by the formula (3.3).

Ψ(s) . 1 − 12 s1 Ψ(s)

(3.13)

Carath´eodory extremal functions

15

Conversely, suppose that for some automorphism m of D and Ψ ∈ S(G), a function ϕ is defined by equation (3.3). Let f = m−1 ◦ϕ◦π. Then f is given by the formula (3.11), where Θ = Ψ ◦ π. Hence f is a symmetric function that satisfies the interpolation conditions (3.9). By the equivalence (3.8), ϕ solves Car δ. 

4. Flat tangents In this section we shall give a description of a large class of Carath´eodory extremals for a flat tangent. Recall that a flat tangent has the form   ¯ z), c(β, ¯ 1) δ = (β + βz, (4.1) for some z ∈ D and c = 0, where β ∈ D. Such a tangent touches the ‘flat geodesic’ def ¯ w) : w ∈ D}. Fβ = {(β + βw, The description depends on a remarkable property of sets of the form R ∪ Fβ , β ∈ D: they have the norm-preserving extension property in G [2, Theorem 10.1]. That is, if g is any bounded analytic function on the variety R∪Fβ , then there exists an analytic function g˜ on G such that g = g˜|R ∪ Fβ and the supremum norms of g and g˜ coincide. Indeed, the proof of [2, Theorem 10.1] gives an explicit formula for one such g˜ in terms of a Herglotz-type integral. Let us call the norm-preserving extension g˜ of g constructed in [2, Chapter 10] the special extension of g to G. It is a simple calculation to show that R and Fβ have a single point in common. By equation (1.2), for δ in equation (4.1) |δ|car =

|c| . 1 − |z|2

Theorem 4.1. Let δ be the flat tangent   ¯ z), c(β, ¯ 1) δ = (β + βz,

(4.2)

to G, where β ∈ D and c ∈ C \ {0}. Let ζ, η be the points in D such that ¯ η) ∈ R ∩ Fβ (2ζ, ζ 2 ) = (β + βη, and let m be the unique automorphism of D such that m∗ ((z, c)) = (0, |δ|car ). For every function h ∈ S(D) such that h(ζ) = m(η) the special extension g˜ to G of the function g : R ∪ Fβ → D,

(2w, w2 ) → h(w),

¯ w) → m(w) (β + βw,

for w ∈ D is a well-aligned Carath´eodory extremal function for δ.

(4.3)

16

J. Agler, Z.A. Lykova and N.J. Young

Proof. First observe that there is indeed a unique automorphism m of D such that m∗ ((z, c)) = (0, |δ|car ), by the Schwarz–Pick lemma. Let ¯ w) k(w) = (β + βw,

for w ∈ D,

so that Fβ = k(D) and k∗ ((z, c)) = δ. By the definition (4.3) of g, g ◦ k = m. Consider any function h ∈ S(D) such that h(ζ) = m(η). By [2, Lemma 10.5], the function g defined by equations (4.3) is analytic on R ∪ Fβ . We claim that the special extension g˜ of g to G is a well-aligned Carath´eodory extremal function for δ. By [2, Theorem 10.1], g˜ ∈ D(G). Moreover, (˜ g )∗ (δ) = (˜ g )∗ ◦ k∗ ((z, c)) = (˜ g ◦ k)∗ ((z, c)) = (g ◦ k)∗ ((z, c)) = m∗ ((z, c)) = (0, |δ|car ) as required. Thus the Poincar´e metric of (˜ g )∗ (δ) on T D is |(˜ g )∗ (δ)| = |(0, |δ|car )| = |δ|car . Therefore g˜ is a well-aligned Carath´eodory extremal function for δ.



Clearly the map g → g˜ is injective, and so this procedure yields a large class of Carath´eodory extremals for δ, parametrized by the Schur class. Remark 4.2. In the converse direction, if ϕ is any well-aligned Carath´eodory extremal for δ, then ϕ is a norm-preserving extension of its restriction to R∪Fβ , which is a function of the type (4.3). Thus the class of all well-aligned Carath´eodory extremal functions for δ is given by the set of norm-preserving analytic extensions to G of g in equation (4.3), as h ranges over functions in the Schur class taking the value m(η) at ζ. Typically there will be many such extensions of g, as can be seen from the proof of [2, Theorem 10.1]. An extension is obtained as the Cayley transform of a function defined by a Herglotz-type integral with respect to a probability measure μ on T2 . In the proof of [2, Lemma 10.8], μ is chosen to be the product of two measures μR and μF on T; examination of the proof shows that one can equally well choose any measure μ on T2 such that μ(A × T) = μR (A),

μ(T × A) = μF (A)

for all Borel sets A in T.

Thus each choice of h ∈ S(D) satisfying h(ζ) = m(η) can be expected to give rise to many well-aligned Carath´eodory extremals for δ.

5. Purely balanced tangents In this section we find a large class of Carath´eodory extremals for purely balanced tangents in G by exploiting an embedding of G into the bidisc.

Carath´eodory extremal functions

17

Lemma 5.1. Let Φ = (Φω1 , Φω2 ) : G → D2 where ω1 , ω2 are distinct points in T. Then Φ is an injective map from G to D2 . Proof. Suppose Φ is not injective. Then there exist distinct points (s1 , s2 ), (t1 , t2 ) ∈ G such that Φωj (s1 , s2 ) = Φωj (t1 , t2 ) for j = 1, 2. On expanding and simplifying this relation we deduce that s1 − t1 − 2ωj (s2 − t2 ) − ωj2 (s1 t2 − t1 s2 ) = 0. A little manipulation demonstrates that both (s1 , s2 ) and (t1 , t2 ) lie on the complex line def

 = {(s1 , s2 ) ∈ C2 : (ω1 + ω2 )s1 − 2ω1 ω2 s2 = 2}. However,  does not meet G. For suppose that (s1 , s2 ) ∈  ∩ G. Then there exists β ∈ D such that ¯ 2, s1 = β + βs ¯ 2 ) − 2. 2ω1 ω2 s2 = (ω1 + ω2 )s1 − 2 = (ω1 + ω2 )(β + βs On solving the last equation for s2 we find that s2 = −¯ ω1 ω ¯2

2 − (ω1 + ω2 )β , 2 − (¯ ω1 + ω ¯ 2 )β¯

whence |s2 | = 1, contrary to the hypothesis that (s1 , s2 ) ∈ G. Hence Φ is injective on G.  Remark 5.2. Φ has an analytic extension to the set Γ \ {(2¯ ω1 , ω ¯ 12 ), (2¯ ω2 , ω ¯ 22 )}, 2 where Γ is the closure of G in C . However this extension is not injective: it takes the constant value (−¯ ω2 , −¯ ω1 ) on a curve lying in ∂G. Theorem 5.3. Let δ = (λ, v) be a purely balanced tangent to G and let Φω solve Car δ for the two distinct points ω1 , ω2 ∈ T. Let mj be the automorphism of D such that mj ◦ Φωj is well aligned at δ for j = 1, 2 and let Φ = (Φ1 , Φ2 ) = (m1 ◦ Φω1 , m2 ◦ Φω2 ) : G → D2 .

(5.1)

For every t ∈ [0, 1] and every function Θ in the Schur class of the bidisc the function F = tΦ1 + (1 − t)Φ2 + t(1 − t)(Φ1 − Φ2 )2

Θ◦Φ 1 − [(1 − t)Φ1 + tΦ2 ]Θ ◦ Φ

is a well-aligned Carath´eodory extremal function for δ. Proof. By Lemma 5.1, Φ maps G injectively into D2 . By choice of mj , (mj ◦ Φωj )∗ (δ) = (0, |δ|car ). Hence Φ∗ (δ) = ((0, 0), |δ|car (1, 1)) ,

(5.2)

18

J. Agler, Z.A. Lykova and N.J. Young

which is tangent to the diagonal {(w, w) : w ∈ D} of the bidisc. Since the diagonal is a complex geodesic in D2 , we have |Φ∗ (δ)|car = (0, |δ|car ). As in Section 3, we appeal to [3, Subsection 11.6] to assert that, for every t ∈ [0, 1] and every function Θ in the Schur class of the bidisc, the function f ∈ C(D2 ) given by f (λ) = tλ1 + (1 − t)λ2 + t(1 − t)(λ1 − λ2 )2

Θ(λ) (5.3) 1 − [(1 − t)λ1 + tλ2 ]Θ(λ) def

solves Car(Φ∗ (δ)). For every such f the function F = f ◦ Φ : G → D satisfies F∗ (δ) = (f ◦ Φ)∗ (δ) = f∗ (Φ∗ (δ)) = (0, |δ|car ). Thus F is a well-aligned Carath´eodory extremal for δ. On writing out F using equation (5.3) we obtain equation (5.2).  Remark 5.4. The range of Φ is a subset of D2 containing (0, 0) and is necessarily nonconvex, by virtue of a result of Costara [8] to the effect that G is not isomorphic to any convex domain. Φ(G) is open in D2 , since the Jacobian determinant of (Φω1 , Φω2 ) at (s1 , s2 ) is 4(ω1 − ω2 )(1 − ω1 ω2 s2 ) (2 − ω1 s1 )2 (2 − ω2 s1 )2 which has no zero in G. Carath´eodory extremals F given by equation (5.3) have the property that the map F ◦ Φ−1 on Φ(G) extends analytically to a map in D(D2 ). There may be other Carath´eodory extremals ϕ for δ for which ϕ ◦ Φ−1 does not so extend. Accordingly we do not claim that the Carath´eodory extremals described in Theorem 5.3 constitute all extremals for a purely balanced tangent.

6. Relation to a result of L. Kosi´ nski and W. Zwonek Our main result in Section 2, on the essential uniqueness of solutions of Car δ for purely unbalanced and exceptional tangents, can be deduced from [15, Theorem 5.3] and some known facts about the geometry of G. However, the terminology and methods of Kosi´ nski and Zwonek are quite different from ours, and we feel it is worth explaining their statement in our terminology. Kosi´ nski and Zwonek speak of left inverses of complex geodesics where we speak of Carath´eodory extremal functions for nondegenerate tangents. These are essentially equivalent notions. By a complex geodesic in G they mean a holomorphic map from D to G which has a holomorphic left inverse. Two complex geodesics h and k are equivalent if there is an automorphism m of D such that h = k ◦ m, or, what is the same, if h(D) = k(D). It is known (for example [4, Theorem A.10]) that, for every nondegenerate tangent δ to G, there is a unique complex geodesic k of G up to equivalence such that δ is tangent to k(D). A function ϕ ∈ D(G) solves Car δ if and only if ϕ ◦ k is an automorphism of D. Hence, for any complex geodesic k and any

Carath´eodory extremal functions

19

nondegenerate tangent δ to k(D), to say that k has a unique left inverse up to equivalence is the same as to say that Car δ has an essentially unique solution. Kosi´ nski and Zwonek also use a different classification of types of complex geodesics (or equivalently tangent vectors) in G, taken from [16]. There it is shown that every complex geodesic k in G, up to composition with automorphisms of D on the right and of G on the left, is of one of the following types. (1)

√ √ √ √ k(z) = (B( z) + B(− z), B( z)B(− z) where B is a non-constant Blaschke product of degree 1 or 2 satisfying B(0) = 0;

(2) k(z) = (z + m(z), zm(z)) where m is an automorphism of D having no fixed point in D. These types correspond to our terminology from [2] (or from Section 1) in the following way. Recall that an automorphism of D is either the identity, elliptic, parabolic or hyperbolic, meaning that the set {z ∈ D− : m(z) = z} consists of either all of D− , a single point of D, a single point of T or two points in T. (1a) If B has degree 1, so that B(z) = cz for some c ∈ T, then, up to equivalence, k(z) = (0, −c2 z). These we call the flat geodesics. The general tangents to flat  geodesics are the  flat tangents described in Section 1, ¯ z), c(β, ¯ 1) for some β ∈ D, z ∈ D and nonzero that is δ = (β + βz, c ∈ C. (1b) If B(z) = cz 2 for some c ∈ T, then k(z) = (2cz, c2 z 2 ). Thus k(D) is the royal variety R, and the tangents to k(D) are the royal tangents. (1c) If B has degree 2 but is not of the form (1b), say B(z) = cz(z − α)/(1 − α ¯ z) where c ∈ T and α ∈ D \ {0}, then   2c(1 − |α|2 )z, c2 z(z − α2 ) k(z) = . 1−α ¯2z Here k(D) is not R but it meets R (at the point (0, 0)). It follows that k(D) is a purely unbalanced geodesic and the tangents to k(D) are the purely unbalanced tangents. (2a) If m is a hyperbolic automorphism of D, then k(D) is a purely balanced geodesic and its tangents are purely balanced tangents. (2b) If m is a parabolic automorphism of D, then k(D) is an exceptional geodesic, and its tangents are exceptional tangents. With this description, Theorem 5.3 of [15] can be paraphrased as stating that a complex geodesic k of G has a unique left inverse (up to equivalence) if and only if k is of one of the forms (1c) or (2b). These are precisely the purely unbalanced and exceptional cases in our terminology, that is, the cases of tangents δ for which there is a unique ω ∈ T such that Φω solves Car δ, in agreement with our Theorem 2.1.

20

J. Agler, Z.A. Lykova and N.J. Young

The authors prove their theorem with the aid of a result of Agler and McCarthy on the uniqueness of solutions of 3-point Nevanlinna–Pick problems on the bidisc [3, Theorem 12.13]. They also use the same example from Subsection 11.6 of [3] which we use for different purposes in Sections 3 and 5.

References [1] J. Agler, Z.A. Lykova and N.J. Young, A geometric characterization of the symmetrized bidisc, preprint. [2] J. Agler, Z.A. Lykova and N.J. Young, Geodesics, retracts, and the extension property in the symmetrized bidisc, 106 pp., to appear in Memoirs of the American Mathematical Society, arXiv:1603.04030 . [3] J. Agler and J.E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics 44, American Mathematical Society, Providence, RI, 2002. [4] J. Agler and N.J. Young, The hyperbolic geometry of the symmetrised bidisc, J. Geometric Analysis 14 (2004), 375–403. [5] J. Agler and N.J. Young, Realization of functions on the symmetrized bidisc, J. Math. Anal. Applic. 453 no. 1 (2017), 227– 240. [6] H. Bart, I.C. Gohberg and M.A. Kaashoek, Minimal factorization of matrix and operator functions, Birkh¨ auser Verlag, Basel, 1979, 277 pp. [7] T. Bhattacharyya, S. Pal and S. Shyam Roy, Dilations of Γ-contractions by solving operator equations, Advances in Mathematics 230 (2012), 577–606. [8] C. Costara, The symmetrized bidisc and Lempert’s theorem, Bull. Lond. Math. Soc. 36 (2004), 656–662. [9] A. Edigarian and W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), 364–374. [10] A.E. Frazho, S. ter Horst and M.A. Kaashoek, State space formulas for stable rational matrix solutions of a Leech problem, Indagationes Math. 25 (2014), 250–274. [11] M. Jarnicki and P. Pflug, On automorphisms of the symmetrised bidisc, Arch. Math. (Basel) 83 (2004), 264–266. [12] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, 2nd extended edition, De Gruyter, Berlin, 2013. [13] M.A. Kaashoek and F. van Schagen, The inverse problem for Ellis-Gohberg orthogonal matrix functions, Integral Equ. Oper. Theory 80 (2014), 527–555. [14] S. Kobayashi, Hyperbolic Complex Spaces, Grundlehren der mathematischen Wissenschaften 318, Springer Verlag, 1998. [15] L. Kosi´ nski and W. Zwonek, Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock, J. Geom. Analysis 26 (2016), 1863–1890. [16] P. Pflug and W. Zwonek, Description of all complex geodesics in the symmetrized bidisc, Bull. London Math. Soc. 37 (2005), 575–584. [17] J. Sarkar, Operator theory on symmetrized bidisc, Indiana Univ. Math. J. 64 (2015), 847–873.

Carath´eodory extremal functions

21

[18] M. Trybula, Invariant metrics on the symmetrized bidisc, Complex Variables and Elliptic Equations 60 no. 4 (2015), 559–565. Jim Agler Department of Mathematics, University of California at San Diego, CA 92103 USA Zinaida A. Lykova School of Mathematics, Statistics and Physics, Newcastle University Newcastle upon Tyne NE1 7RU U.K. e-mail: [email protected] N.J. Young School of Mathematics, Statistics and Physics, Newcastle University Newcastle upon Tyne NE1 7RU U.K. and School of Mathematics, Leeds University Leeds LS2 9JT U.K. e-mail: [email protected]

Standard versus strict Bounded Real Lemma with infinite-dimensional state space III: The dichotomous and bicausal cases J.A. Ball, G.J. Groenewald and S. ter Horst Dedicated to Rien Kaashoek on the occasion of his 80th birthday.

Abstract. This is the third installment in a series of papers concerning the Bounded Real Lemma for infinite-dimensional discrete-time linear input/state/output systems. In this setting, under appropriate conditions, the lemma characterizes when the transfer function associated with the system has contractive values on the unit circle, expressed in terms of a linear matrix inequality, often referred to as the Kalman– Yakubovich–Popov (KYP) inequality. Whereas the first two installments focussed on causal systems with the transfer functions extending to an analytic function on the disk, in the present paper the system is still causal but the state operator is allowed to have nontrivial dichotomy (the unit circle is not contained in its spectrum), implying that the transfer function is analytic in a neighborhood of zero and on a neighborhood of the unit circle rather than on the unit disk. More generally, we consider bicausal systems, for which the transfer function need not be analytic in a neighborhood of zero. For both types of systems, by a variation on Willems’ storage-function approach, we prove variations on the standard and strict Bounded Real Lemma. We also specialize the results to nonstationary discrete-time systems with a dichotomy, thereby recovering a Bounded Real Lemma due to Ben-Artzi–Gohberg–Kaashoek for such systems. Mathematics Subject Classification (2010). Primary 47A63; Secondary 47A48, 93B20, 93C55, 47A56. Keywords. KYP inequality, storage function, bounded real lemma, infinite dimensional linear systems, dichotomous systems, bicausal systems.

This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 93039, 90670, and 93406).

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_2

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24

J.A. Ball, G.J. Groenewald and S. ter Horst

1. Introduction This is the third installment in a series of papers on the bounded real lemma for infinite-dimensional discrete-time linear systems and the related Kalman– Yakubovich–Popov (KYP) inequality. We consider discrete-time input-stateoutput linear systems determined by the following equations  x(n + 1) = Ax(n) + Bu(n), Σ := (n ∈ Z), (1.1) y(n) = Cx(n) + Du(n), where A : X → X , B : U → X , C : X → Y and D : U → Y are bounded linear Hilbert space operators i.e., X , U and Y are Hilbert spaces and the system matrix associated with Σ takes the form





A B X X M= : → . (1.2) C D U Y Associated with the system Σ is the transfer function FΣ (z) = D + zC(I − zA)−1 B

(1.3)

which necessarily defines an analytic function on some neighborhood of the origin in the complex plane with values in the space L(U , Y) of bounded linear operators from U to Y. The bounded real lemma is concerned with the question of characterizing (in terms of A, B, C, D) when FΣ has analytic continuation to the whole unit disk D such that the supremum norm of FΣ over the unit disk FΣ ∞,D := sup{FΣ (z) : z ∈ D} satisfies either (i) FΣ ∞,D ≤ 1 (standard version), or (ii) FΣ ∞,D < 1 (strict version). We first note the following terminology which we shall use. Given a selfadjoint operator H on a Hilbert space X , we say that (i) H is strictly positive-definite (H  0) if there is a δ > 0 so that Hx, x ≥ δx2 for all x ∈ X . (ii) H is positive-definite if Hx, x > 0 for all 0 = x ∈ X . (iii) H is positive-semidefinite (H  0) if Hx, x ≥ 0 for all x ∈ X . Given two selfadjoint operators H, K on X , we write H  K or K ≺ H if H − K  0 and similarly for H  K or K  H. Note that if X is finitedimensional, then strictly positive-definite and positive-definite are equivalent. Then the standard and strict bounded real lemmas for the finite-dimensional setting (where X , U , Y are all finite dimensional and one can view A, B, C, D as finite matrices) are as follows. A B ], and F as Theorem 1.1. Suppose that we are given X , U , Y, M = [ C Σ D in (1.1), (1.2), (1.3), with X , U , Y being finite-dimensional Hilbert spaces. Then:

(1) Standard Bounded Real Lemma (see [1]): Assume that (A, B) is controllable (i.e., spank≥0 {Im Ak B} = X ) and (C, A) is observable (i.e.,  k k≥0 ker CA = {0}). Then FΣ ∞,D ≤ 1 if and only if there exists a positive-definite matrix H satisfying the Kalman–Yakubovich–Popov

Infinite-Dimensional Bounded Real Lemma III (KYP) inequality:

∗ H A B C D 0

0 IY



A C

B D





H 0

0 IU

25

.

(1.4)

(2) Strict Bounded Real Lemma (see [18]): Assume that all eigenvalues of A are in the unit disk D. Then FΣ ∞,D < 1 if and only if there exists a positive-definite matrix H so that

∗





A B H 0 A B H 0 ≺ . (1.5) C D 0 IY C D 0 IU Infinite-dimensional versions of the standard bounded real lemma have been studied by Arov–Kaashoek–Pik [2] and the authors [6, 7], while infinitedimensional versions of the strict bounded real lemma have been analyzed by Yakubovich [23, 24], Opmeer–Staffans [17] and the authors [6, 7]. In this paper we wish to study the following variation of the bounded real lemma, which we shall call the dichotomous bounded real lemma. Given the A B ] and associated transfer function F as system with system matrix M = [ C Σ D in (1.1), (1.2), (1.3), we now assume that the operator A admits dichotomy, i.e., we assume that A has no spectrum on the unit circle T. Under this assumption it follows that the transfer function FΣ in (1.3) can be viewed as an analytic L(U , Y)-valued function on a neighborhood of the unit circle T. The dichotomous bounded real lemma is concerned with the question of characterizing in terms of A, B, C, D when it is the case that FΣ ∞,T := sup{FΣ (z) : z ∈ T} satisfies either FΣ ∞,T ≤ 1 (standard version) or (ii) FΣ ∞,T < 1 (strict version). For the finite-dimensional case we have the following result. A B ] and F Theorem 1.2. Suppose that we are given X , U , Y and M = [ C Σ D as in (1.1), (1.2), (1.3), with X , U , Y being finite-dimensional Hilbert spaces and with A having no eigenvalues on the unit circle T. Then:

(1) Finite-dimensional standard dichotomous bounded real lemma: Assume that Σ is minimal ((A, B) is controllable and (C, A) is observable). Then the inequality FΣ ∞,T ≤ 1 holds if and only if there exists an invertible selfadjoint matrix H which satisfies the KYP inequality (1.4). Moreover, the dimension of the spectral subspace of A over the unit disk is equal to the number of positive eigenvalues (counting multiplicities) of H and the dimension of the spectral subspace of A over the exterior of the closed unit disk is equal to the number of negative eigenvalues (counting multiplicities) of H. (2) Finite-dimensional strict dichotomous bounded real lemma: The strict inequality FΣ ∞,T < 1 holds if and only if there exists an invertible selfadjoint matrix H which satisfies the strict KYP inequality (1.5). Moreover, the inertia of A (the dimensions of the spectral subspace of A for the disk and for the exterior of the closed unit disk) is related to the inertia of H (dimension of negative and positive eigenspaces) as in the standard dichotomous bounded real lemma (item (1) above).

26

J.A. Ball, G.J. Groenewald and S. ter Horst

We note that Theorem 1.2 (2) appears as Corollary 1.2 in [10] as a corollary of more general considerations concerning input-output operators for nonstationary linear systems with an indefinite metric; to make the connection between the result there and the strict KYP inequality (1.5), one should observe that a standard Schur-complement computation converts the strict inequality (1.5) to the pair of strict inequalities I − B ∗ HB − D∗ D  0, H − A∗ HA − C ∗ C − (A∗ HB + C ∗ D)Z −1 (B ∗ HA + D∗ C)  0 where Z = I − B ∗ HB − D∗ D. We have not located an explicit statement of Theorem 1.2 (1) in the literature; this will be a corollary of the infinite-dimensional standard dichotomous bounded real lemma which we present in this paper (Theorem 7.1 below). Note that if F = FΣ is a transfer function of the form (1.3), then necessarily F is analytic at the origin. One approach to remove this restriction is to designate some other point z0 where F is to be analytic and adapt the formula (1.3) to a realization “centered at z0 ” (see [5, page 141] for details): e.g., for the case z0 = ∞, one can use F (z) = D + C(zI − A)−1 B. To get a single chart to handle an arbitrary location of poles, one can use the bicausal realizations used in [4] (see [8] for the nonrational operator-valued case); for the setting here, where we are interested in rational matrix functions analytic on a neighborhood of the unit circle T, we suppose that 



 





+ B + − B − A A X+ X+ X− X− M+ =  : → , M = : → (1.6) −  − 0 U Y U Y C+ D C + , σ(A + ), and of A − , σ(A − ), are two system matrices with spectrum of A contained in the unit disk D and that F (z) is given by  + zC + (I − z A + )−1 B + + C − (I − z −1 A − )−1 B − . F (z) = D

(1.7)

We shall give an interpretation of (1.7) as the transfer function of a bicausal exponentially stable system in Section 3 below. In any case we can now pose the question for the rational case where all spaces X± , U , Y in (1.6) are finite dimensional: characterize in terms of M+ and M− when it is the case that F ∞,T ≤ 1 (standard case) or F ∞,T < 1 (strict case). To describe the result we need to introduce the bicausal KYP inequality to be satisfied by a selfadjoint operator H=

H− H0∗

H0 H+



on X =

X− X+

(1.8)

Infinite-Dimensional Bounded Real Lemma III given by



∗ C ∗ 0 A − − ∗ ∗ 0 A C + + ∗ B ∗ C ∗  ∗ 0 B + − − +D

I



 

H− H0 0 H0∗ H+ 0 0 0 I

∗ 0 0 A − 0 I 0 ∗ 0 I B −





I 0 0 + + 0 A B − A − C + C − B − +D  C

H− H0 0 H0∗ H+ 0 0 0 I



− 0 B − A 0 I 0 0 0 I

27



,

(1.9)

as well as the strict bicausal KYP inequality: for some  > 0 we have    ∗ 

I

∗ C ∗ I 0 A  A  ∗ B  0 0 − − A A H− H0 0 − − 0 − − 2   ∗ ∗ 0 A+ B+ 0 A C 0 I 0 + H0∗ H+ 0 + + ∗ B ∗ C ∗  ∗ 0 B + − − +D





0

∗ 0 0 A − 0 I 0 ∗ 0 I B −

0 I



∗ A  ∗  B − − 0 B− B− +I

− A − C − C − B − +D  C

H− H0 0 H0∗ H+ 0 0 0 I



− 0 B − A 0 I 0 0 0 I

.

(1.10)

+ , B + ) We say that the system matrix-pair (M+ , M− ) is controllable if both (A    and (A− , A− B− ) are controllable, and that (M+ , M− ) is observable if both + , A + ) and (C − A − , A − ) are observable. We then have the following result. (C Theorem 1.3. Suppose that we are given X+ , X− , U , Y, M+ , M− as in (1.6) + and A − with X+ , X− , U , Y finite-dimensional Hilbert spaces and both A having spectrum inside the unit disk D. Further, suppose that F is the rational matrix function with no poles on the unit circle T given by (1.7). Then: (1) Finite-dimensional standard bicausal bounded real lemma: Assume that (M+ , M− ) is controllable and observable. Then we have F ∞,T ≤ 1 if and only there exists an invertible selfadjoint solution H as in (1.8) of the bicausal KYP inequality (1.9). Moreover H+  0 and H− ≺ 0. (2) Finite-dimensional strict bicausal bounded real lemma: The strict inequality F ∞,T < 1 holds if and only if there exists an invertible selfadjoint solution H as in (1.8) of the strict bicausal KYP inequality (1.10). Moreover, in this case H+  0 and H− ≺ 0. We have not located an explicit statement of these results in the literature; they also are corollaries of the infinite-dimensional results which we develop in this paper (Theorem 7.3 below). The goal of this paper is to explore infinite-dimensional analogues of Theorems 1.2 and 1.3 (both standard and strict versions). For the case of trivial dichotomy (the stable case where σ(A) ⊂ D in Theorem 1.2), we have recently obtained such results via two distinct approaches: (i) the state-spacesimilarity theorem approach (see [6]), and (ii) the storage-function approach (see [7]) based on the work of Willems [21, 22]. Both approaches in general involve additional complications in the infinite-dimensional setting. In the first approach (i), one must deal with possibly unbounded pseudo-similarity rather than true similarity transformations, as explained in the penetrating paper of Arov–Kaashoek–Pik [3]; one of the contributions of [6] was to identify additional hypotheses (exact or 2 -exact controllability and observability) which

28

J.A. Ball, G.J. Groenewald and S. ter Horst

guarantee that the pseudo-similarities guaranteed by the Arov–Kaashoek– Pik theory can in fact be taken to be bounded and boundedly invertible. In the second approach (ii), no continuity properties of a storage function are guaranteed a priori and in general one must allow a storage function to take the value +∞; nevertheless, as shown in [7], it is possible to show that the Willems available storage function Sa and a regularized version of the Willems required supply Sr (at least when suitably restricted) have a quadratic form coming from a possibly unbounded positive-definite operator (Ha and Hr respectively) which leads to a solution (in an adjusted generalized sense required for the formulation of what it should mean for an unbounded operator to be a solution) of the KYP inequality. Again, if the system satisfies an exact or 2 -exact controllability/observability hypothesis, then we get finite-valued quadratic storage functions and associated bounded and boundedly invertible solutions of the KYP inequality. It seems that the first approach (i) (involving the state-space-similarity theorem with pseudo-similarities) does not adapt well in the dichotomous setting, so we here focus on the second approach (ii) (computation of extremal storage functions). For the dichotomous setting, there again is a notion of storage function but now the storage functions S can take values on the whole real line rather than just positive values, and quadratic storage functions should have the form S(x) = Hx, x (at least for x restricted to some appropriate domain) with H (possibly unbounded) selfadjoint rather than just positive-definite. Due to the less than satisfactory connection between closed forms and closed operators for forms not defined on the whole space and not necessarily semi-bounded (see, e.g., [20, 15]), it is difficult to make sense of quadratic storage functions in the infinite-dimensional setting unless the storage function is finite-valued and the associated self-adjoint operator is bounded. Therefore, for the dichotomous setting here we deal only with the case where 2 -exact controllability/observability assumptions are imposed at the start, and we are able to consider only storage functions S which are finite real-valued with the associated selfadjoint operators in an quadratic representation equal to bounded operators. Consequently, our results require either the strict inequality condition F ∞,T < 1 on the transfer function F , or an 2 -exact or exact controllability/observability assumption on the operators in the system matrices. Consequently, unlike what is done in [6, 7] for the causal trivial-dichotomy setting, the present paper has nothing in the direction of a bounded real lemma for a dichotomous or exponentially dichotomous system under only (approximate) controllability and observability assumptions for the case where F ∞,T = 1. The paper is organized as follows. Apart from the current introduction, the paper consists of seven sections. In Sections 2 and 3 we introduce the dichotomous systems and bicausal systems, respectively, studied in this paper and derive various basic results used in the sequel. Next, in Section 4 we introduce the notion of a storage function for discrete-time dichotomous linear systems as well as the available storage Sa and required supply Sr

Infinite-Dimensional Bounded Real Lemma III

29

storage functions in this context and show that they indeed are storage functions (pending the proof of a continuity condition which is obtained later from Theorem 5.2). In Section 5 we show, under certain conditions, that Sa and Sr are quadratic storage functions by explicit computation of the corresponding invertible selfadjoint operators Ha and Hr . The results of Sections 4 and 5 are extended to bicausal systems in Section 6. The main results of the present paper, i.e., the infinite-dimensional versions of Theorems 1.2 and 1.3, are proven in Section 7. In the final section, Section 8, we apply our dichotomous bounded real lemma to discrete-time, nonstationary, dichotomous linear systems and recover a result of Ben-Artzi–Gohberg–Kaashoek [10].

2. Dichotomous system theory We assume that we are given a system Σ as in (1.1) with system matrix A B ] and associated transfer function F as in (1.3) with A having M = [C Σ D dichotomy. As a neighborhood of the unit circle T is in the resolvent set of A, by definition of A having a dichotomy, we see that FΣ (z) is analytic and uniformly bounded in z on a neighborhood of T. One way to make this explicit is to decompose FΣ in the form FΣ = FΣ,+ + FΣ,− where FΣ,+ (z) is analytic and uniformly bounded on a neighborhood of the closed unit disk D and where FΣ,− (z) is analytic and uniformly bounded on a neighborhood of the closed exterior unit disk De as follows. The fact that A admits a dichotomy implies there is a direct sum (not ˙ − , so necessarily orthogonal) decomposition of the state space X = X+ +X that with respect to this decomposition A has a block diagonal matrix decomposition of the form





A− 0 X− X− A= : → (2.1) 0 A+ X+ X+ where A+ := A|X+ ∈ L(X+ ) has spectrum inside the unit disk D and A− := A|X− ∈ L(X− ) has spectrum in the exterior of the closed unit disk De = C\D. It follows that A+ is exponentially stable, rspec (A+ ) < 1, and A− is invertible with inverse A−1 − exponentially stable. Occasionally we will view A+ and A− as operators acting on X and, with some abuse of notation, write A−1 − for what is really a generalized inverse of A− :

−1



X− X− A− 0 −1 ∼ : A− = → , X+ X+ 0 0 i.e., the Moore–Penrose generalized inverse of A− in case the decomposition ˙ + is orthogonal – the meaning will be clear from the context. Now X− +X decompose B and C accordingly:





  X− X− B− : U→ and C = C− C+ : → Y. (2.2) B= B+ X+ X+

30

J.A. Ball, G.J. Groenewald and S. ter Horst

We may then write FΣ (z) = D + zC(I − zA)−1 B   I − zA− = D + z C− C+ 0

0 I − zA+

−1

B− B+



= D + zC− (I − zA− )−1 B− + zC+ (I − zA+ )−1 B+ −1 −1 −1 = −C− A−1 A− ) B− + D + zC+ (I − zA+ )−1 B+ − (I − z

= FΣ,− (z) + FΣ,+ (z), where −1 −1 −1 FΣ,− (z) = −C− A−1 A− ) B− = − − (I − z

∞ 

n+1 C− (A−1 B− z −n (2.3) − )

n=0

is analytic on a neighborhood of De , with the series converging in operator norm on De due to the exponential stability of A−1 − , and where FΣ,+ (z) = D + zC+ (I − zA+ )−1 B+ = D +

∞ 

n C+ An−1 + B+ z

(2.4)

n=1

is analytic on a neighborhood of D, with the series converging in operator norm on D due to the exponential stability of A+ . Furthermore, from the convergent series expansions for FΣ,+ in (2.4) and for FΣ,− in (2.3) we read off that FΣ has the convergent Laurent expansion on the unit circle T ∞ 

FΣ (z) =

Fn z n

n=−∞

with Laurent coefficients Fn given by ⎧ −1 ⎪ ⎨D − C− A− B− n−1 Fn = C+ A+ B+ ⎪ ⎩ −C− An−1 − B−

if n = 0, if n > 0, if n < 0.

(2.5)

As FΣ ∞,T := sup{FΣ (z) : z ∈ T} < ∞, it follows that FΣ defines a bounded multiplication operator: MFΣ : L2U (T) → L2Y (T),

MFΣ : f (z) → FΣ (z)f (z)

with MFΣ  = FΣ ∞,T . If we write this operator as a block matrix MFΣ = [MFΣ ]ij (−∞ < i, j < ∞) with respect to the orthogonal decompositions L2U (T) =

∞ 

znU ,

L2Y (T) =

n=−∞

∞ 

znY

n=−∞

for the input and output spaces for MFΣ , it is a standard calculation to verify that [MFΣ ]ij = Fi−j , i.e., the resulting bi-infinite matrix [MFΣ ]ij is the Laurent matrix LFΣ associated with FΣ given by LFΣ = [Fi−j ]∞ i,j=−∞

(2.6)

Infinite-Dimensional Bounded Real Lemma III

31

where Fn is as in (2.5). Another expression of this identity is the fact that MFΣ : L2U (T) → L2Y (T) is just the frequency-domain expression of the time n  (z) = ∞ domain operator LFΣ : 2U (Z) → 2Y (Z), i.e., if we let u n=−∞ u(n)z 2 2  (z) = in LU (T) be the bilateral Z-transform of u in U (Z) and similarly let y  ∞ n 2 2 y(n)z in L (T) be the bilateral Z-transform of y in  (Z), then Y Y n=−∞ we have the relationship y = LF Σ u

 (z) = FΣ (z) · u  (z) for all z ∈ T. y

⇐⇒

(2.7)

We now return to analyzing the system-theoretic properties of the dichotomous system (1.1). Associated with the system operators A, B, C, D are the diagonal operators A, B, C, D acting between the appropriate 2 -spaces indexed by Z: A = diagk∈Z [A] : 2X (Z) → 2X (Z),

B = diagk∈Z [B] : 2U (Z) → 2X (Z),

C = diagk∈Z [C] : 2X (Z) → 2Y (Z),

D = diagk∈Z [D] : 2U (Z) → 2Y (Z).

(2.8)

We also introduce the bilateral shift operator S : 2X (Z) → 2X (Z),

S : {x(k)}k∈Z → {x(k − 1)}k∈Z

and its inverse S −1 = S ∗ : {x(k)}k∈Z → {x(k + 1)}k∈Z . We can then rewrite the system equations (1.1) in aggregate form  −1 S x = Ax + Bu, Σ := y = Cx + Du,

(2.9)

We shall say that a system trajectory (u, x, y) = {(u(n), x(n), y(n)}n∈Z is 2 -admissible if all of u, x, and y are in 2 : u = {u(n)}n∈Z ∈ 2U (Z), x = {x(n)}n∈Z ∈ 2X (Z), y = {y(n)}n∈Z ∈ 2Y (Z). Note that the constant-diagonal structure of A, B, C, D implies that each of these operators intertwines the bilateral shift operator on the appropriate 2 (Z)-space: AS = SA,

BS = SB,

CS = SC,

DS = SD,

(2.10)

where S is the bilateral shift operator on 2W where W is any one of U , X , Y depending on the context. It is well known (see, e.g., [9, Theorem 2.2]) that the operator A admitting a dichotomy is equivalent to S −1 − A being invertible as an operator on 2X (Z). Hence the dichotomy hypothesis enables us to solve uniquely for x ∈ 2X (Z) and y ∈ 2Y (Z) for any given u ∈ 2U (Z): x = (S −1 − A)−1 Bu = (I − SA)−1 SBu =: TΣ, is u, y = (D + C(S −1 − A)−1 B)u = (D + C(I − SA)−1 SB)u =: TΣ u,

(2.11)

where TΣ, is = (S −1 − A)−1 B : 2U (Z) → 2X (Z), TΣ = D + C(S

−1

− A)

−1

B:

2U (Z)

→

2Y (Z),

(2.12) (2.13)

32

J.A. Ball, G.J. Groenewald and S. ter Horst

are the respective input-state and input-output maps. In general the inputoutput map TΣ in (2.13) is not causal. Given an 2U (Z)-input signal u, rather than specification of an initialization condition on the state x(0), as in standard linear system theory for systems running on Z+ , in order to specify a uniquely determined state trajectory x for a given input trajectory u, the extra information required to solve uniquely for the state trajectory x in the dichotomous system (1.1) or (2.9) is the specification that x ∈ 2X (Z), i.e., that the resulting trajectory (u, x, y) be 2 -admissible. Next we express various operators explicitly in terms of A± , B± , C± and D. The following lemma provides the basis for the formulas derived in the remainder of the section. In fact this lemma amounts to the easy direction of the result of Ben-Artzi–Gohberg [9, Theorem 2.2] mentioned above. Lemma 2.1. Let Σ be the dichotomous system (1.1) with A decomposing as in (2.1) with A as in (2.8). Then (S −1 − A)−1 = (I − SA)−1 S acting on 2X (Z) is given explicitly as the following block matrix with rows and columns indexed by Z [(S −1 − A)−1 ]ij =

Ai−j−1 + −Ai−j−1 −

for i > j, , for i ≤ j

with A0+ = PX+ .

(2.14)

˙ + , we can identify 2X (Z) with Proof. Via the decomposition X = X− +X 2 2 ˙ X (Z). Write S+ for the bilateral shift operator and A+ for the X− (Z)+ + block diagonal operator with A+ diagonal entries, both acting on 2X+ (Z), and write S− for the bilateral shift operator and A− for the block diagonal operator with A− diagonal entries, both acting on 2X− (Z). Then with respect to the above decomposition of 2X (Z) we have

−1 0 I − S − A− −1 (I − SA) = 0 I − S + A+

(I − S− A− )−1 0 = . 0 (I − S+ A+ )−1 Since A+ has its spectrum in D, so do A+ and S+ A+ , and thus ∞ ∞   k k (I − S+ A+ )−1 = (S+ A+ )k = S+ A+ k=0

k=0

where we make use of observation (2.10) to arrive at the final infinite-series −1 −1 −1 expression. Similarly, A−1 − has spectrum in D implies that A− and A− S− have spectrum in D, and hence (I − S− A− )−1 = (S− A− )−1 ((S− A− )−1 − I)−1 −1 −1 −1 −1 = −A−1 ) = −A−1 − S− (I − (S− A− ) − S−

∞  k=0

−1 = −A−1 − S−

∞  k=0

−k A−k − S− = −

∞  k=1

−k A−k − S− .

(S− A− )−k

Infinite-Dimensional Bounded Real Lemma III

33

Inserting the formulas for (I − S+ A+ )−1 and (I − S− A− )−1 in the formula for (I −SA)−1 , multiplying by S on the right and writing out in block matrix form, we obtain the desired formula for (I − SA)−1 S = (S −1 − A)−1 .  We now compute the input-output map TΣ and input-state map TΣ, is explicitly. Proposition 2.2. Let Σ be the dichotomous system (1.1) with A decomposing as in (2.1) and B and C as in (2.2). The input-output map TΣ : 2U (Z) → 2Y (Z) and input-state map TΣ, is : 2U (Z) → 2X (Z) of Σ are then given by the following block matrix, with rows and columns indexed over Z: ⎧ i−j−1 ⎪ B+ if i > j, ⎨C+ A+ Ai−j−1 B+ if i > j, + [TΣ ]ij = D − C− A−1 [T ] = B if i = j, Σ, is ij − i−j−1 − ⎪ −A B if i ≤ j. − ⎩ − −C− Ai−j−1 B− if i < j, − In particular, TΣ is equal to the Laurent operator LFΣ of the transfer function FΣ given in (2.6), and for u ∈ 2U (Z) and y ∈ 2Y (Z) with bilateral Z-transform notation as in (2.7), y = TΣ u

⇐⇒

 (z) = FΣ (z) · u  (z) for almost all z ∈ T. y

(2.15)

Proof. Recall that TΣ can be written as TΣ = D + C(I − SA)−1 SB = D + C(S −1 − A)−1 B. The block matrix formula for TΣ now follows directly from the block matrix formula for (S −1 −A)−1 obtained in Lemma 2.1. Comparison of the formula for [TΣ ]ij with the formula (2.5) for the Laurent coefficients {Fn }n∈Z of FΣ shows that TΣ = LFΣ as operators from L2U (T) to L2Y (T). Finally the identity (2.15) follows upon combining the identity LFΣ = TΣ with the general identity (2.7).  It is convenient to also view TΣ = LFΣ as a block 2 × 2 matrix with respect to the decomposition 2U (Z) = 2U (Z− ) ⊕ 2U (Z+ ) for the input-signal space and 2Y (Z) = 2Y (Z− )⊕2Y (Z+ ) for the output-signal space. We can then write

2

2

F F U (Z− ) Y (Z− ) H T Σ Σ LFΣ = : → (2.16) 2U (Z+ ) 2Y (Z+ ) HFΣ TFΣ where ⎧ i−j−1 ⎪ B+ for 0 > i > j, ⎨C+ A+ −1  [TFΣ ]ij : i 0 are arbitrary, we conclude that (A−1 , A−1 B) is controllable.  The following 2 -admissible-trajectory interpolation result will be useful in the sequel. Proposition 2.7. Suppose that Σ is a dichotomous linear system as in (1.1), (2.1), (2.2), and that we are given a vector u ∈ U and x ∈ X . Assume that Σ is dichotomously 2 -exactly controllable. Then there exists an 2 -admissible system trajectory (u, x, y) for Σ such that u(0) = u,

x(0) = x.

Proof. As Σ is  -exactly controllable, we know that Wc− and Wc+ are surjective. Write x = x+ + x− with x± ∈ X± . Choose u− ∈ 2U (Z− ) so that −1  Wc+ u− = x+ . Next solve for x− so that x− = A−1 − x− − A− B− u, i.e., set 2

x− := A− x− + A− B− u.

(2.26)

Use the surjectivity of the controllability operator Wc− to find u+ ∈ 2U (Z+ ) so that Wc− u+ = x− . We now define a new input signal u by ⎧ ⎪ if n < 0, ⎨u− (n) u(n) = u if n = 0, ⎪ ⎩ u+ (n − 1) if n > 0. Since u+ and u− are 2 -sequences, we obtain that u ∈ 2U (Z). Now let (u, x, y) be the 2 -admissible system trajectory determined by the input sequence u. Clearly u(0) = u. So it remains to show that x(0) = x. To see this, note that   x(0) = Wc u = Wc+ u− + Wc− ( u 0 · · · + Su+ ) −1 −1 −1  − = x+ − A−1 − B− u + A− Wc u+ = x+ − A− B− u + A− x− = x+ + x− = x.



3. Bicausal systems Even for the setting of rational matrix functions, it is not the case that a rational matrix function F which is analytic on a neighborhood of the unit circle T necessarily has a realization of the form (1.3), as such a realization for F implies that F must be analytic at the origin. What is required instead

38

J.A. Ball, G.J. Groenewald and S. ter Horst

is a slightly more general notion of a system, which we will refer to as a bicausal system, defined as follows. A bicausal system Σ consists of a pair of input-state-output linear systems Σ+ and Σ− with Σ+ running in forward time and Σ− running in backward time − x− (n + 1) + B − u(n), x− (n) = A Σ− : (n ∈ Z), (3.1)  y− (n) = C− x− (n), Σ+ :

+ x+ (n) + B + u(n), x+ (n + 1) = A + x+ (n) + Du(n),  y+ (n) = C

(n ∈ Z),

(3.2)

− on X− exponentially with Σ− having state space X− and state operator A  + on X+ stable (i.e., σ(A− ) ⊂ D) and Σ+ having state space X+ and A + ) ⊂ D). A system trajectory consists of a triple exponentially stable (σ(A {u(n), x(n), y(n)}n∈Z such that  x  $ X− % u(n) ∈ U, x(n) = x− ∈ X+ , y(n) = y− (n) + y+ (n) with y± (n) ∈ Y + such that (u, x− , y− ) is a system trajectory of Σ− and (u, x+ , y+ ) is a system  x trajectory of Σ2+ . We say that the system trajectory 2 (u, x, y) = (u, x− , y− + y+ ) is  -admissible if all system signals are in  : + u ∈ 2U (Z),

x+ ∈ 2X+ (Z),

x− ∈ 2X− (Z),

y± ∈ 2Y (Z).

+ , given u ∈ 2 (Z), there is Due to the assumed exponential stability of A U a uniquely determined x+ ∈ 2X+ (Z) and y+ ∈ 2Y (Z) so that (u, x+ , y+ ) is − due to the an 2 -admissible system trajectory for Σ+ and similarly for A − . The result is as follows. assumed exponential stability of A + Proposition 3.1. Suppose that Σ = (Σ+ , Σ− ) is a bicausal system, with A  exponentially stable as an operator on X+ and A− exponentially stable as an operator on X− . Then: 1. Given any u ∈ 2U (Z), there is a unique x+ ∈ 2X+ (Z) satisfying the first system equation in (3.2), with the resulting input-state map TΣ+ ,is mapping 2U (Z) to 2X+ (Z) given by the block matrix [TΣ+ ,is ]ij =

i−j−1 B + A + 0

for i > j, for i ≤ j.

(3.3)

The unique output signal y+ ∈ 2Y (Z) resulting from the system equations (3.2) with given input u ∈ 2U (Z) and resulting uniquely determined state trajectory x+ in 2X+ (Z) is then given by y+ = TΣ+ u with TΣ+ : 2U (Z) → 2Y (Z) having block matrix representation given by ⎧ + for i > j,  i−j−1 B ⎪ ⎨C+ A+  (3.4) [TΣ+ ]ij = D for i = j, ⎪ ⎩ 0 for i < j.

Infinite-Dimensional Bounded Real Lemma III

39

Thus TΣ+ ,is and TΣ+ are block lower-triangular (causal) Toeplitz operators. 2. Given any u ∈ 2U (Z), there is a unique x− ∈ 2X− (Z) satisfying the first system equation in (3.1), with resulting input-state map TΣ− ,is : 2U (Z) → 2X− (Z) having block matrix representation given by [TΣ− ,is ]ij =

0 j−i B − A −

for i > j, for i ≤ j.

(3.5)

The unique output signal y− ∈ 2Y (Z) resulting from the system equations (3.1) with given input u ∈ 2U (Z) and resulting uniquely determined state trajectory x− in 2X− (Z) is then given by y− = TΣ− u with TΣ− : 2U (Z) → 2Y (Z) having block matrix representation given by 0 − A j−i B − C

[TΣ− ]ij =

−

for i > j, for i ≤ j.

(3.6)

Thus TΣ− ,is and TΣ− are upper-triangular (anticausal) Toeplitz operators. 3. The input-state map for the combined bicausal system Σ = (Σ+ , Σ− ) is then given by

2

X− (Z) TΣ− ,is TΣ,is = : 2U (Z) → 2X (Z) = TΣ+ ,is 2X+ (Z) with block entries  0 matrix   X(with  notation using the natural identifications − ∼ X+ ∼ and X ) = X+ − = 0 [TΣ,is ]ij =

i−j−1 B + A + j−i  B − A −

for i > j, for i ≤ j.

(3.7)

Moreover, the input-output map TΣ : 2U (Z) → 2Y (Z) of Σ is given by TΣ = TΣ+ + TΣ− : 2U (Z) → 2Y (Z), having block matrix decomposition given ⎧  i−j−1 B + ⎪ ⎨C+ A+    [TΣ ]ij = D + C− B− ⎪ ⎩  j−i  C − A B−

by

−

4. For u ∈ Z-transforms

and y ∈

2Y (Z),

u(n)z ∈

L2U (T),

2U (Z)

 (z) = u

∞ 

n

n=−∞

for i > j, for i = j, for i < j.

 and y  be the respective bilateral let u  (z) = y

∞ 

y(n)z n ∈ L2Y (T).

n=−∞

Then y = TΣ u

⇐⇒

(3.8)

 (z) = FΣ (z) · u  (z) for all z ∈ T y

40

J.A. Ball, G.J. Groenewald and S. ter Horst where FΣ (z) is the transfer function of the bicausal system Σ given by − (I − z −1 A − )−1 B − + D  + zC + (I − z A + )−1 B + FΣ (z) = C =

∞ 

− A n B  −n + (D  +C − B − ) + C − −z

n=1

∞ 

+ A n−1 B + z n . C +

(3.9)

n=1

Furthermore, the Laurent operator LFΣ : 2U (Z) → 2Y (Z) associated with the function FΣ ∈ L∞ L(U ,Y) (T) as in (2.6) is identical to the input-output operator TΣ for the bicausal system Σ LFΣ = TΣ ,

(3.10)

and hence also, for u ∈ 2U (Z) and y ∈ 2Y (Z) and notation as in (2.7), y = TΣ u

⇐⇒

 (z) = FΣ (z) · u  (z) for all z ∈ T. y

(3.11)

Proof. We first consider item (1). Let us rewrite the system equations (3.2) in aggregate form Σ+ :

S −1 x+ = A+ x+ + B+ u  y+ = C+ x+ + Du

(3.12)

where + ] ∈ L(2X ), A+ = diagk∈Z [A + + ] ∈ L(2U (Z), 2X (Z)), B+ = diagk∈Z [B + + ] ∈ L(2X (Z), 2Y (Z)), C+ = diagk∈Z [C +

(3.13)

 = diagk∈Z [D] ∈ L(2 (Z), 2 (Z)). D U Y + has trivial ex+ implies that A The exponential stability assumption on A ponential dichotomy (with state-space X− = {0}). As previously observed + implies that we can solve the first (see [9]), the exponential dichotomy of A system equation (3.12) of Σ+ uniquely for x+ ∈ 2X+ (Z): x+ = (S −1 − A+ )−1 B+ u =: TΣ+ ,is u

(3.14)

and item (1) follows. From the general formula (2.14) for (S −1 − A)−1 in (2.14), we see that for our case here the formula for the input-state map TΣ+ ,is for the system Σ+ is given by (3.3). From the aggregate form of the system equations (3.12) we see that the resulting input-output map TΣ+ : 2U (Z) → 2Y (Z) is then given by  + C+ (S −1 − A+ )−1 B+ = D  + C+ TΣ ,is . T Σ+ = D + The block matrix decomposition (3.4) for the input-output map TΣ+ now follows directly from plugging in the matrix decomposition (3.3) for TΣ+ ,is into this last formula.

Infinite-Dimensional Bounded Real Lemma III

41

The analysis for item (2) proceeds in a similar way. Introduce operators − ] ∈ L(2 ), A− = diagk∈Z [A X− − ] ∈ L(2U (Z), 2X (Z)), B− = diagk∈Z [B −

(3.15)

− ] ∈ L(2X (Z), 2Y (Z)), C− = diagk∈Z [C − Write the system (3.1) in the aggregate form Σ− :

x− = A− S −1 x− + B− u y− = C− x− .

(3.16)

− is exponentially stable, so also is A− and we may compute (I − As A − S −1 )−1 via the geometric series, using also that A − and S −1 commute as A observed in (2.10), ∞  − S −1 )−1 = k− S −k (I − A A k=0

from which we deduce the block matrix representation [(I − A− S −1 )−1 ]ij =

0 j−i A −

for i > j, for i ≤ j.

We next note that we can solve the first system equation in (3.16) for x− in terms of u:  =: TΣ ,is u. x− = (I − A− S −1 )−1 Bu − Combining this with the previous formula for the block-matrix entries for (I − A− S −1 )−1 leads to the formula (3.5) for the matrix entries of TΣ− ,is . From the second equation for the system (3.16) we see that then y− is uniquely determined via the formula y− = C− x− = C− TΣ ,is u. −

Plugging in the formula (3.5) for the block matrix entries of TΣ− ,is then leads to the formula (3.6) for the block matrix entries of the input-output map TΣ− for the system Σ− . Item (3) now follows by definition of the input-output map TΣ of the bicausal system Σ as the sum TΣ = TΣ,− + TΣ,+ of the input-output maps for the anticausal system Σ− and the causal system Σ+ along with the formulas for TΣ,± obtained in items (1) and (2). We now analyze item (4). Define FΣ by either of the equivalent formulas + and A − , we see that FΣ is a in (3.9). Due to the exponential stability of A continuous L(U , Y)-valued function on the unit circle T, and hence the multiplication operator MFΣ : f (z) → FΣ (z) · f (z) is a bounded operator from L2U (T) into L2Y (T). From the second formula for FΣ (z) in (3.9) combined with the formula (3.8) for the  block matrix entries of TΣ , we see that the ∞ Laurent expansion for F (z) = n=−∞ Fn z n on T is given by Fn = [TΣ ]n,0 and that the Laurent matrix [LFΣ ]ij = Fi−j is the same as the matrix for the

42

J.A. Ball, G.J. Groenewald and S. ter Horst

input-output operator [TΣ ]ij . We now see the identity (3.10) as an immediate consequence of the general identity (2.7). Finally, the transfer-function property (3.11) follows immediately from (3.10) combined with the general identity (2.7).  Remark 3.2. From the form of the input-output operator TΣ and transfer function FΣ of the dichotomous system Σ in (1.1)–(1.2) that were obtained in Proposition 2.2 with respect to the decompositions of A in (2.1) and of B and C in (2.2) it follows that a dichotomous system can be represented as a bicausal system (3.1)–(3.2) with + , A + , B + , D)  = (C+ , A+ , B+ , D), (C (3.17) − , A − , B − ) = (C− , A−1 , −A−1 B− ). (C − − The extra feature that a bicausal system coming from a dichotomous system − is invertible. In fact, if the operator A − in a bicausal system has is that A (3.1)-(3.2) is invertible, it can be represented as a dichotomous system (1.1) as well, by reversing the above transformation. Indeed, one easily verifies that − invertible, if Σ = (Σ+ , Σ− ) is a bicausal system given by (3.1)–(3.2) with A then the system (1.1) with     $ % + + A B 0    A= , B = , C = , D=D C C + − −1 −1 B − 0 A −A − − is a dichotomous system whose input-output operator and transfer function are equal to the input-output operator and transfer function from the original bicausal system. To a large extent, the theory of dichotomous systems presented in Section 2 carries over to bicausal systems, with proofs that can be directly obtained from the translation between the two systems given above. We describe here the main features. The Laurent operator LFΣ = TΣ can again be decomposed as in (2.16)  F and TF are given by where now the Toeplitz operators T Σ Σ ⎧ j−i    ⎪ for i < j < 0, ⎨C− A− B−  F ]ij : i 0, (4.8)

Then (u, x, y) is again an 2 -admissible system trajectory. Proof. We must verify that (u, x, y) satisfy the system equations (1.1) for all n ∈ Z. For n < 0 this is clear since (u , x , y ) is a system trajectory. Since x (0) = x (0), we see that this holds for n = 0. That it holds for n > 0 follows easily from the fact that (u , x , y ) is a system trajectory. Finally note that (u , x , y ) and (u , x , y ) both being 2 -admissible implies that (u, x, y) is 2 -admissible.  Our next goal is to show that Sa and Sr are storage functions for Σ, and among all storage functions they are the minimum and maximum ones. We postpone the proof of Step 4 in the proof of items (1) and (2) in the following proposition to Section 5 below. Proposition 4.4. Let Σ be a dichotomous linear system as in (1.1) such that (4.3) holds. Then: (1) Sa is a storage function for Σ. (2) Sr is a storage function for Σ. (3) If S is any other storage function for Σ, then  0 ) ≤ Sr (x0 ) for all x0 ∈ X . Sa (x0 ) ≤ S(x Proof of (1) and (2). The proof proceeds in several steps. Step 1: Sa and Sr are finite-valued on X . Let x0 ∈ X = Im Wc . By the 2 exact controllability assumption, there is a u0 ∈ 2U (Z) so that x0 = Wc u0 .

Infinite-Dimensional Bounded Real Lemma III

47

Let (u0 , x0 , y0 ) be the unique 2 -admissible system trajectory of Σ defined by the input u0 . Then ∞  Sa (x0 ) ≥ y0 (n)2 − u0 (n)2 ≥ −u0 2 > −∞ n=0

and similarly Sr (x0 ) ≤ u0 2 < ∞. It remains to show Sa (x0 ) < ∞ and Sr (x0 ) > −∞. Let (u, x, y) be any 2 -admissible system trajectory of Σ with x(0) = x0 . Let (u0 , x0 , y0 ) be the particular 2 -admissible system trajectory with x(0) = x0 as chosen above. Then by Lemma 4.3 we may piece together these two trajectories to form a new 2 -admissible system trajectory (u , x , y ) of Σ defined as follows: u (n) =

u0 (n) u(n)

if n < 0, if n ≥ 0,

y (n) =

y0 (n) y(n)

if n < 0, if n ≥ 0.

x (n) =

x0 (n) x(n)

if n ≤ 0, if n > 0,

Since TΣ  ≤ 1 and (u , x , y ) is a system trajectory, we know that +∞ 

y (n)2 = y 2 = TΣ u 2 ≤ u 2 =

n=−∞

+∞ 

u (n)2 .

n=−∞

Let us rewrite this last inequality in the form ∞ 

y(n)2 −

n=0

∞ 

u(n)2 ≤

−1  n=−∞

n=0

−1 

u0 (n)2 −

y0 (n)2 < ∞.

n=−∞

It follows that the supremum of the left-hand side over all 2 -admissible trajectories (u, x, y) of Σ with x(0) = x0 is finite, i.e., Sa (x0 ) < ∞. A similar argument shows that Sr (x0 ) > −∞ as follows. Given an arbitrary 2 -admissible system trajectory (u, x, y) with x(0) = x0 , Lemma 4.3 enables us to form the composite 2 -admissible system trajectory (u , x , y ) of Σ defined by u(n) u0 (n)

if n < 0, if n ≥ 0,

y(n) y0 (n) +∞

if n < 0, if n ≥ 0.

u (n) = y (n) = Then the fact that −1  n=−∞

u(n)2 −

n=−∞ −1  n=−∞

x (n) =

y (n)2 ≤

y(n)2 ≥

∞

∞  n=0

n=−∞

x(n) x0 (n)

if n ≤ 0, if n > 0,

u (n)2 gives us that

y0 (n)2 −

∞ 

u0 (n)2 > −∞,

n=0

and it follows from the definition (4.7) that Sr (x0 ) > −∞. By putting all these pieces together we see that both Sa and Sr are finite-valued on X = Im Wc .

48

J.A. Ball, G.J. Groenewald and S. ter Horst

Step 2: Sa (0) = Sr (0) = 0. This fact follows from the explicit quadratic form for Sa and Sr obtained in Theorem 5.2 below, but we include here an alternative more conceptual proof to illustrate the ideas. By noting that (0, 0, 0) is an 2 -admissible system trajectory, we see from the definitions of Sa in (4.6) and Sr in (4.7) that Sa (0) ≥ 0 and Sr (0) ≤ 0. Now let (u, x, y) be any 2 -admissible system trajectory such that x(0) = 0. Another application of Lemma 4.3 then implies that (u , x , y ) given by (u (n), x (n), y (n)) =

(0, 0, 0) (u(n), x(n), y(n))

if n < 0, if n ≥ 0

is also an 2 -admissible system trajectory. From the assumption that TΣ  ≤ 1 we get that 0≤

∞ 

(u (n)2U − y (n)2Y ) =

n=−∞

so

∞ 

(u(n)2U − y(n)2Y ),

n=0 ∞ 

(y(n)2Y − u(n)2U ) ≤ 0

(4.9)

n=0 2

whenever (u, x, y) is an  -admissible system trajectory with x(0) = 0. From the definition in (4.6) we see that Sa (0) is the supremum over all such expressions on the left hand side of (4.9), and we conclude that Sa (0) ≤ 0. Putting this together with the first piece above gives Sa (0) = 0. Similarly, note that if (u, x, y) is an 2 -admissible trajectory with x(0) = Wc u = 0, then again by Lemma 4.3 (u (n), x (n), y (n)) =

(u(n), x(n), y(n)) (0, 0, 0)

if n < 0, if n ≥ 0

is also an 2 -admissible system trajectory. Since TΣ  ≤ 1 we get 0≤

∞ 

(u (n)2U − y (n)2Y ) =

n=−∞

−1 

(u(n)2U − y(n)2Y ).

n=−∞

From the definition (4.7) of Sr (0) it follows that Sr (0) ≥ 0. Putting all these pieces together, we arrive at Sa (0) = Sr (0) = 0. Step 3: Both Sa and Sr satisfy the energy balance inequality (4.1). For x0 ∈ X , set → − (0) = Wc u  = x0 }, U x0 = { u ∈ 2U (Z) : x Also, for any Hilbert space W let P+ on 2W (Z) be the orthogonal projection on 2W (Z+ ) and P− = I − P+ . Then we can write Sa (x0 ) and Sr (x0 ) as  2 −P+ u  2 Sa (x0 ) = sup P+ y → −  ∈ U x0 u

and

Sr (x0 ) =

 2 −P− y  2 , inf P− u → −

 ∈ U x0 u

 is the output of Σ defined by the input u  ∈ 2U (Z). In  = TΣ u where y 2  ∈ U (Z) is an input trajectory, then the corresponding uniquely general, if u

Infinite-Dimensional Bounded Real Lemma III

49

 := determined 2 -admissible state and output trajectories are denoted by x  and y  := TΣ u . TΣ,is u Now let (u, x, y) be an arbitrary fixed system trajectory for the dichotomous system Σ and fix n ∈ Z. Set → − (0) = x(n), u  (0) = u(n)}. U ∗ = { u ∈ 2U (Z) : x → − Note that U ∗ is nonempty by simply quoting Proposition 2.7. Observe that → − → − → − , y  ) with u  ∈ U∗ U ∗ ⊂ U x(n) . For an 2 -admissible system trajectory ( u, x  (0) = y(n) and x (1) = x(n + 1). Furthermore, (S ∗ u , S ∗x , S ∗ y  ) is we have y also an 2 -admissible system trajectory of Σ and )(0) = x (1) = x(n + 1). (S ∗ x Hence

→ − → − → − : u  ∈ U ∗ } ⊂ U x(n+1) . S ∗ U ∗ = {S ∗ u

 (0) = y(n) and u  (0) = u(n) we have Next, since y  2 − P+ u  2 = P+ S ∗ y  2 − P+ S ∗ u  2 + y(n)2 − u(n)2 P+ y and  2 − P− S ∗ y  2 = P− u  2 − P− y  2 + u(n)2 − y(n)2 . P− S ∗ u We thus obtain that Sa (x(n)) =

sup

→ −  ∈ U x(n) u

 2 − P+ u   ≥ sup P+ y  2 − P+ u  2 P+ y → − ∈ U ∗ u

 2 − P+ S ∗ u  2 = y(n)2 − u(n)2 + sup P+ S ∗ y → − ∈ U ∗ u

= y(n)2 − u(n)2 +

sup

→ −  ∈S ∗ U ∗ u

 2 − P+ u  2 , P+ y

and similarly for Sr we have Sr (x(n + 1)) = = ≤

inf

→ −  ∈ U x(n+1) u

inf

 2 − P− y  2 P− u

→ −  ∈ U x(n+1) S∗u

inf

→ −  ∈S ∗ U ∗ S∗u

 2 − P− S ∗ y  2 P− S ∗ u

 2 − P− S ∗ y  2 P− S ∗ u

 2 − P− S ∗ y  2 = inf P− S ∗ u → − ∈ U ∗ u

 2 − P− y  2 . = u(n)2 − y(n)2 + inf P− u → − ∈ U ∗ u

To complete the proof of this step it remains to show that Sa (x(n + 1)) =

sup

→ −  ∈S ∗ U ∗ u

 2 − P+ u  2 =: sa , P+ y

 2 − P− y  2 =: sr . Sr (x(n)) = inf P− u → − ∈ U ∗ u

(4.10)

50

J.A. Ball, G.J. Groenewald and S. ter Horst → − → − We start with Sa . Since S ∗ U ∗ ⊂ U x(n+1) we see that sa ≤ Sa (x(n + 1)).

, y  ) be an 2 -admissible system To show that also sa ≥ Sa (x(n + 1)), let ( u, x → −  ∈ U x(n+1) . The problem is to show trajectory with u ∞ 

( y(n)2 −  u(n)2 ) ≤ sa .

(4.11)

n=0

→ − , y  ) be any 2 -admissible trajectory with u  ∈ S ∗ U ∗. Toward this goal, let ( u, x We then patch the two system trajectories together by setting   (k) = u

 (k) u  (k) u

if k < 0, if k ≥ 0,

  (k) = y

 (k) y  (k) y

if k < 0, if k ≥ 0.

 (k) = x

(k) x (k) x

if k ≤ 0, if k > 0,

Clearly the input, state and output trajectories are all 2 -sequences. Note , y  ) and ( , y  ) are both 2 -admissible system trajectories. Since that ( u, x u, x  (−1) = u(n), x (−1) = x(n), y  (−1) = y(n) and x (0) = x(n + 1), we see u that (0) = A (0). x x(−1) + B u(−1) = Ax(n) + Bu(n) = x(n + 1) = x  , y   ) is We can now apply once again Lemma 4.3 to conclude that ( u , x → − 2   ∈ S ∗ U ∗, also an  -admissible trajectory for Σ. Furthermore, we have u  = P+ y   and P+ u  = P+ u   . Thus P+ y  2 − P+ u  2 = P+ y   2 − P+ u   2 ≤ P+ y

sup

→ −  ∈S ∗ U ∗ u

 2 − P+ u  2 =: sa . P+ y

Taking the supremum on the left-hand side over all 2 -admissible system → −  ∈ U x(n+1) then yields Sa (x(n + 1)) ≤ sa , and the first trajectories with u equality in (4.10) holds as required. To prove the second equality in (4.10) we follow a similar strategy, which → − → − we will only sketch here. The inclusion U ∗ ⊂ U x(n) shows sr is an upper → −  ∈ U x(n) , y  ) be 2 -admissible system trajectory with u bound. Any ( u, x can be patched together with an 2 -admissible system trajectory with input → −  , y  ) sequence from U ∗ to form a new 2 -admissible system trajectory ( u , x → −  = P− u   and P− y  = P− y   , so that   in U ∗ , P− u with u  2 − P− y  2 = P− u   2 − P− y   2 ≥ inf  2 − P− y  2 =: sr P− u P− u → − ∈ U ∗ u

which then yields that sr is also a lower bound for Sr (x(n)) as required. Step 4: Both Sa and Sr are continuous at 0. This is a consequence of the explicit quadratic form obtained for Sa and Sr in Theorem 5.2 below.

Infinite-Dimensional Bounded Real Lemma III

51

Proof of (3). Let S be any storage function for Σ. Let x0 ∈ Im Wc and let (u, x, y) be any 2 -admissible dichotomous system trajectory for Σ with x(0) = x0 . Then S satisfies the energy balance relation   S(x(n + 1)) − S(x(n)) ≤ u(n)2U − y(n)2Y .

(4.12)

Summing from n = 0 to n = N then gives   0 ) = S(x(N   S(x(N + 1)) − S(x + 1)) − S(x(0)) ≤

N  

 u(n)2U − y(n)2Y .

(4.13)

n=0

As x ∈ 2X (Z) and S as part of being a storage function is continuous at 0    with S(0) = 0, we see from x(N + 1) → 0 that S(x(N + 1)) → S(0) = 0 as N → ∞. Hence letting N → ∞ in (4.13) gives  0) ≤ −S(x

∞  

 u(n)2U − y(n)2Y .

n=0

But by definition, the infimum of the right-hand side of this last expression over all system trajectories (u, x, y) of Σ such that x(0) = x0 is exactly  0 ) ≤ −Sa (x0 ), and thus Sa (x0 ) ≤ S(x  0 ) for −Sa (x0 ). We conclude that −S(x any x0 ∈ Im Wc . Similarly, if we sum up (4.12) from n = −N to n = −1 we get  0 ) − S(x(−N    S(x )) = S(x(0)) − S(x(−N )) ≤

−1 

(u(n)2U − y(n)2Y ).

n=−N

Letting N → ∞ in this expression then gives  0) ≤ S(x

−1 

(u(n)2U − y(n)2Y ).

n=−∞

But by definition the infimum of the right-hand side of this last inequality over all 2 -admissible system trajectories (u, x, y) with x(0) = x0 is exactly  0 ) ≤ Sr (x0 ). This completes the proof equal to Sr (x0 ). We conclude that S(x of part (3) of Proposition 4.4.  Quadratic storage functions and spatial KYP inequalities: the dichotomous setting. Let us say that a function S : X → R is quadratic if there exists a bounded selfadjoint operator H on X such that S(x) = SH (x) := Hx, x for all x ∈ X . Trivially any function S = SH of this form satisfies conditions (1) and (3) in the definition of storage function (see the discussion around (4.1)). To characterize which bounded selfadjoint operators H give rise to S = SH being a storage function, as we are assuming that our blanket assumption (4.3) is in force, we may quote the result of Remark 4.2 to substitute the local version (4.4) ((4.5)) of the energy-balance condition in place of the

52

J.A. Ball, G.J. Groenewald and S. ter Horst

original version (4.1) (respectively (4.2) for the strict case). Condition (4.4) applied to SH leads us to the condition H(Ax + Bu), Ax + Bu − Hx, x ≤ u2 − Cx + Du2 , or, equivalently, H(Ax+Bu), Ax+Bu + Cx+Du, Cx+Du ≤ Hx, x + u, u . holding for all x ∈ X and u ∈ U . In a more matricial form, we may write instead 





 H 0 x x , 0 I u u 









 H 0 A B x A B x , ≥0 (4.14) − 0 I C D u C D u for all x ∈ X and u ∈ U . Hence H satisfies the spatial version (4.14) of the KYP inequality (1.4). By elementary Hilbert space theory, namely, that a selfadjoint operator X on a complex Hilbert space is uniquely determined by its associated quadratic form x → Xx, x, it follows that H solves the KYP inequality (1.4), but now for an infinite-dimensional setup. If we start with the strict version (4.5) of the local energy-balance condition, we arrive at the following criterion for the quadratic function SH to be a strict storage function for the system Σ, namely the spatial version of the strict KYP inequality: 





 H 0 x x , 0 I u u ! 











! ! x !2 H 0 A B x A B x ! − , ≥ ! ! u ! , (4.15) 0 I C D u C D u and hence also the strict KYP inequality in operator form (1.5), again now for the infinite-dimensional setting. Following the above computations in reversed order shows that the spatial KYP inequality (4.14) and strict spatial KYP inequality (4.15) imply that SH is a storage function and strict storage function, respectively. Proposition 4.5. Let Σ be a dichotomous linear system as in (1.1). Let H be a bounded, self adjoint operator on X . Then SH is a quadratic storage function for Σ if and only if H is a solution of the KYP inequality (1.4). Moreover, SH is a strict quadratic storage function if and only if H is a solution of the strict KYP inequality (1.5).

5. The available storage and required supply We assume throughout this section that the dichotomous linear system Σ satisfies the standing assumption (4.3). Under these conditions we shall show that the available storage Sa and required supply Sr are quadratic storage

Infinite-Dimensional Bounded Real Lemma III

53

functions and we shall obtain explicit formulas for the associated selfadjoint operators Ha and Hr satisfying the KYP inequality (1.4). The assumption that FΣ ∞,T ≤ 1 implies that the associated Laurent operator LFΣ in (2.6) is a contraction, so that the Toeplitz operators TFΣ  F (2.17) are also contractions. Thus I − LF L∗ and I − L∗ LF are and T Σ Σ Σ FΣ FΣ both positive operators. Writing out these operators in terms of the operator matrix decomposition (2.16) we obtain ⎡ ⎤ 2  ∗  F H∗ − H  F T∗ DT −T Σ Σ FΣ FΣ  ∗ − H FΣ HF Σ FΣ ⎦, I − LFΣ L∗FΣ = ⎣ 2  ∗ − TF H ∗ −HFΣ T DT − HFΣ H∗FΣ ∗ Σ FΣ FΣ FΣ (5.1)   2 ∗ ∗ ∗ H F D − H H −H T − T F F Σ Σ Σ F F F F Σ Σ Σ T Σ . I − L∗FΣ LFΣ = ∗ T F ∗ H F −T∗ HF − H D2 − H Σ

FΣ

TFΣ

Σ

FΣ

− LFΣ L∗FΣ

FΣ

Σ

− L∗FΣ LFΣ

In particular, from I and I being positive operators we read off that 2 2  ∗ DT  HFΣ H∗FΣ , DT ∗  ∗  HF Σ HF Σ , FΣ FΣ (5.2) 2 2 ∗ H F . DT  H∗FΣ HFΣ , DT  H F Σ  Σ F Σ

FΣ

Applying Douglas’ lemma [11] along with the factorizations in (2.21) enables us to prove the following result. Lemma 5.1. Assume the dichotomous system Σ in (1.1) satisfies (4.3). Then there exist unique injective bounded linear operators Xo,+ , Xo,− , Xc,+ and Xc,− such that Xo,+ : X+ → 2U (Z+ ),

Wo+ = DT∗F Xo,+ ,

Im Xo,+ ⊂ Im DT∗F ,

(5.3)

Xo,− : X− → 2U (Z− ),

Wo− = DT  ∗ Xo,− ,

Im Xo,− ⊂ Im DT ∗ ,

(5.4)

Xc,+ : X+ → 2Y (Z+ ), Xc,− : X− →

2Y (Z− ),

Σ

FΣ

Σ

FΣ

(Wc+ )∗ = DT  F Xc,+ , Σ

(Wc− )∗

Im Xc,+ ⊂ Im DT  F , (5.5) Σ

Im Xc,− ⊂ Im DTFΣ . (5.6)

= DTFΣ Xc,− ,

Proof. We give the details of the proof only for Xo,+ as the other cases are similar. The argument is very much like the proof of Lemma 4.8 in [7] where the argument is more complicated due the unbounded-operator setting there. 2 Since DT  HFΣ HFΣ , the Douglas factorization lemma [11] implies ∗ FΣ

the existence of a unique contraction operator Yo,+ : 2U (Z− ) → 2U (Z+ ) with Wo+ Wc+ = HFΣ = DT∗F Yo,+ Σ

and

Im Yo,+ ⊂ Im DT∗F . Σ

As Im Wc = X+ , the open mapping theorem guarantees that Wc+ has a bounded right inverse Wc+† := Wc+∗ (Wc+ Wc+∗ )−1 . Moreover, u = Wc+† x is the least norm solution of the equation Wc+ u = x: Wc+† (x) = arg min {u22 (Z− ) : u ∈ D(Wc+ ), x = Wc+ u} U

We now define Xo,+ by Xo,+ = Yo,+ Wc+† .

(x ∈ Im Wc+ ).

54

J.A. Ball, G.J. Groenewald and S. ter Horst

We then observe DT∗F Xo,+ = DT∗F Yo,+ Wc+† = HFΣ Wc+† = Wo+ Wc+ Wc+† = Wo+ Σ

Σ

giving the factorization (5.3) as wanted. Moreover, the factorization Xo,+ = Yo,+ Wc+† implies that Im Xo,+ ⊂ Im Yo,+ ⊂ Im DT∗F ; this property comΣ bined with the factorization (5.3) makes the choice of Xo,+ unique. Moreover, the containment Im Xo,+ ⊂ Im DT∗F combined with the injectivity of Wo+ Σ forces the injectivity of Xo,+ .  We are now ready to analyze both the available storage function Sa and the required supply function Sr for a system meeting hypotheses (4.3). Theorem 5.2. Suppose that Σ is a dichotomous discrete-time linear system as in (1.1) with transfer function FΣ so that the hypotheses (4.3) are satisfied. Then Sa = SHa and Sr = SHr are quadratic storage functions with associated selfadjoint operators Ha and Hr bounded and boundedly invertible on X , and Sa and Sr are given by ! !2 ! ! Sa (x0 ) = Xo,+ (x0 )+ 2 − !Pa T∗FΣ Xo,+ (x0 )+ − Pa DTFΣ Wc−† (x0 )− ! , (5.7) ! !2 ! ∗ ! +† 2 Sr (x0 ) = !Pr T  F Wc (x0 )+ ! − Xo,− (x0 )−  , (5.8) FΣ Xo,− (x0 )− − Pr DT Σ

with x0 = (x0 )+ ⊕ (x0 )− the decomposition of x0 with respect to the direct ˙ − , the operators Xo,+ and Xo,− as in Lemma 5.1, and sum X = X+ +X Wc−† = Wc−∗ (Wc− Wc−∗ )−1 , Wc+† = Wc+∗ (Wc+ Wc+∗ )−1 , Pa = P(DT KerWc− )⊥ , Pr = P(D  KerWc+ )⊥ . TF

FΣ

Σ

In particular, Sa and Sr are continuous. ˙ + inducing the decomIf we assume that the decomposition X = X− +X positions (2.1) and (2.2) is actually orthogonal, which can always be arranged via an invertible similarity-transformation change of coordinates in X if necessary, then with respect to the orthogonal decomposition X = X− ⊕ X+ , Ha and Hr are given explicitly by ∗

∗ Xo,+ (I − TFΣ Pa T∗FΣ )Xo,+ Xo,+ TFΣ Pa DTFΣ Wc−† Ha = , (5.9) Wc−†∗ DTFΣ Pa T∗FΣ Xo,+ −Wc−†∗ DTFΣ Pa DTFΣ Wc−†  +†∗  +† ∗ Wc DT −Wc+†∗ DT  F Pr D T  F Wc  F Pr TFΣ Xo,− Σ Σ Σ Hr =  F Pr D  W∗† −X ∗ (I − T  F Pr T  ∗ )Xo,− . (5.10) −X ∗ T o,−

Σ

TFΣ

c

o,−

Σ

FΣ

Furthermore, the dimension of the spectral subspace of A over the unit disk agrees with the dimension of the spectral subspace of Ha and Hr over the positive real line (= dim X+ ), and the dimension of the spectral subspace of A over the exterior of the closed unit disk agrees with the dimension of the spectral subspace of Ha and Hr over the negative real line (= dim X− ).

Infinite-Dimensional Bounded Real Lemma III

55

Proof. To simplify notation, in this proof we write simply F rather than FΣ . We start with the formula for Sa . Fix x0 ∈ Im Wc . Let (u, x, y) be any system trajectory of Σ such that x0 = Wc u. The first step in the calculation of Sa is to reformulate the formula from the definition (4.6) in operator-theoretic form: Sa (x0 ) =

sup

u : Wc u=x0

(LF u)|Z+ 22 (Z+ ) − u|Z+ 22 (Z+ ) . Y

U

(5.11)

From the formulas (2.16), (2.17), and (2.18) for LF , in greater detail we have Sa (x0 ) =

sup u+ , u− :

Wc+ u− =(x0 )+ , Wc− u+ =(x0 )−

HF u− + TF u+ 2 − u+ 2 .

where u− ∈ 2U (Z− ) and u+ ∈ 2U (Z+ ) and where x0 = (x0 )− + (x0 )+ is the decomposition of x0 into X− and X+ components. Recalling the factorization HF = Wo+ Wc+ from (2.18) as well as the constraint on u− , we rewrite the objective function in the formula for Sa (x0 ) as HF u− + TF u+ 2 − u+ 2 = Wo+ (x0 )+ + TF u+ 2 − u+ 2 . Furthermore, by assumption Wc+ is surjective, so there is always a u− ∈ 2U (Z− ) which achieves the constraint Wc+ u− = (x0 )+ . In this way we have eliminated the parameter u− and the formula for Sa (x0 ) becomes Sa (x0 ) =

sup u+ ∈2U (Z+ ) :

Wc− u+ =(x0 )−

Wo+ (x0 )+ + TF u+ 2 − u+ 2 .

(5.12)

By Lemma 5.1 there is a uniquely determined injective linear operator Xo,+ from X+ to Im DT∗F so that Wo+ = DT∗F Xo,+ . Then the objective function in (5.12) becomes Wo+ (x0 )+ + TF u+ 2 − u+ 2 = DT∗F Xo,+ (x0 )+ + TF u+ 2 − TF u+ 2 − DTF u+ 2 = DT∗F Xo,+ (x0 )+ 2 + 2ReDT∗F Xo,+ (x0 )+ , TF u+  − DTF u+ 2 = DT∗F Xo,+ (x0 )+ 2 + 2ReXo,+ (x0 )+ , TF DTF u+  − DTF u+ 2 = DT∗F Xo,+ (x0 )+ 2 + 2ReT∗F Xo,+ (x0 )+ , DTF u+  − DTF u+ 2 = DT∗F Xo,+ (x0 )+ 2 + T∗F Xo,+ (x0 )+ 2 − T∗F Xo,+ (x0 )+ − DTF u+ 2 = Xo,+ (x0 )+ 2 − T∗F Xo,+ (x0 )+ − DTF u+ 2 . In this way we arrive at the decoupled formula for Sa (x0 ): Sa (x0 ) = Xo,+ (x0 )+ 2 −

inf

T∗F Xo,+ (x0 )+ − DTF u+ 2 . (5.13)

u+ : Wc− u+ =(x0 )−

By assumption Wc− is surjective and hence Wc− is right invertible with right inverse equal to Wc−∗ (Wc− Wc−∗ )−1 . In particular, the minimal-norm solution u0+ of Wc− u+ = (x0 )− is given by u0+ = Wc−∗ (Wc− Wc−∗ )−1 (x0 )− = Wc−† (x0 )− , and then any other solution has the form u+ = u0+ + v+ where v+ ∈ Ker Wc− .

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J.A. Ball, G.J. Groenewald and S. ter Horst

By standard Hilbert space theory, it then follows that inf

u+ : Wc− u+ =(x0 )−

T∗F Xo,+ (x0 )+ − DTF u+ 2

!  ! ! !2 = !P(DT KerWc− )⊥ T∗F Xo,+ (x0 )+ − DTF u0+ ! F !  ∗ !2 ! = Pa TF Xo,+ (x0 )+ − DTF Wc−† (x0 )− !

and we arrive at the formulas (5.7) for Sa . A few more notational manip˙ + is an ulation leads to the explicit formula (5.9) for Ha when X = X− +X orthogonal decomposition. In a similar vein, the formula (4.7) for Sr can be written in operator form as Sr (x0 ) = inf u− 2 − P2Y (Z− ) LF u2 . u : Wc u=x0

Then the objective function can be written as  F u+ + T  F u− 2 u− 2 − P2Y (Z− ) LF u2 = u− 2 − H  F u− 2 = u− 2 − W− (x0 )− + T  F u− 2 = u− 2 − Wo− Wc− u+ + T o 2  = u− 2 − DT  ∗ Xo,− (x0 )− + TF u−  F

2 2   = u− 2 − DT  ∗ Xo,− (x0 )−  − 2ReDT  ∗ Xo,− (x0 )− , TF u−  − TF u−  F

F

2 2  = −DT  ∗ Xo,− (x0 )−  − 2ReXo,− (x0 )− , DT  ∗ TF u−  + DT  F u−  F

F

2 2 ∗ = −DT  ∗ Xo,− (x0 )−  − 2ReTF Xo,− (x0 )− , DT  F u−  + DT  F u−  F

2 − 2 2 ∗ ∗ = −DT  ∗ Xo,− (x0 )−  − TF Xo,− (x0 )  + TF Xo,− (x0 )− − DT  F u−  F

 ∗ Xo,− (x0 )− − D  u− 2 = −Xo,− (x0 )− 2 + T F TF where now u+ is eliminated and the constraint on the free parameter u− is Wc+ u− = (x0 )+ . Thus Sr (x0 ) = −Xo,− (x0 )− 2 +

inf

u− : Wc+ u− =(x0 )+

 ∗ Xo,− (x0 )− − D  u− 2 . T F TF

We note that all possible solutions u− of the constraint Wc+ u− = (x0 )+ are given by u− = Wc+∗ (Wc+ Wc+∗ )−1 (x0 )+ +v− = Wc+† (x0 )+ +v− where v− ∈ KerWc+ . Then standard Hilbert space theory leads to the formulas (5.8) for Sr ; a little more careful manipulation leads to the explicit form (5.10) for Hr . We next wish to verify that Ha and Hr are invertible. This follows as an application of results referred to as inertial theorems; these results are well known for the finite-dimensional setting (indeed it is these results which are behind the proof of the inertial statements in Theorems 1.1 and 1.2—see e.g. [16]). As these results are not so well known for the infinite-dimensional setting, we go through the results in some detail here.

Infinite-Dimensional Bounded Real Lemma III

57

As a consequence of Proposition 4.5, we know that Ha is a solution of the KYP inequality (1.4). From the (1,1)-entry of (1.4) we see, in particular, that Ha − A∗ Ha A − C ∗ C  0. Write Ha as a block operator matrix with respect to the direct sum decom˙ − as position X = X+ +X



Ha− Ha0 X− Ha = on . (5.14) ∗ Ha0 Ha+ X+ We can then rewrite the above inequality as

∗



A− 0 Ha− Ha0 A− Ha− Ha0 − ∗ ∗ 0 A∗+ Ha0 0 Ha0 Ha+ Ha+

∗

0 C−  C−  ∗ A+ C+

 C+ .

From the diagonal entries of this block-operator inequality we get −1 ∗−1 ∗ −1 ∗ ∗ −Ha− + A∗−1 − Ha− A−  A− C− C− A− , Ha+ − A+ Ha+ H+ A+  C+ C+ .

An inductive argument then gives −Ha− 

N 

∗−N ∗ A∗−n C− C− A−n Ha− A−N − − A− − ,

n=1

Ha+ 

N 

∗N +1 +1 ∗ n A∗n Ha+ AN . + C+ C+ A+ + A+ +

n=0

A−1 −

As both and A+ are exponentially stable, we may take the limit as N → ∞ in both of the above expressions to get −Ha−  (Wo− )∗ Wo− ,

Ha+  (Wo+ )∗ Wo+ .

By the dichotomous 2 -exactly observable assumption, both operators (Wo− )∗ and (Wo+ )∗ are surjective, and hence (Wo− )∗ Wo− and (Wo+ )∗ Wo+ are also surjective. Thus we can invoke the open mapping theorem to get that both (Wo− )∗ Wo− and (Wo+ )∗ Wo+ are bounded below. We conclude that both Ha+ and −Ha− are strictly positive-definite, i.e., there is an  > 0 so that Ha+  I and Ha−  −I. In particular, both Ha+ and Ha− are invertible. It remains to put all this together to see that Ha and Hr are invertible. We do the details for Ha as the proof for Hr is exactly the same. By Schur complement theory (see, e.g., [12]), applied to the block matrix decomposition of Ha in (5.14), given that the operator Ha+ is invertible (as we have already verified), then Ha is also invertible if and only if the Schur comple−1 ∗ ment S(Ha ; Ha+ ) := Ha− − Ha0 Ha+ Ha0 is invertible. But we have already verified that both Ha− and −Ha+ are strictly negative-definite. Hence the Schur complement is the sum of a strictly negative-definite operator and an at worst negative-semidefinite operator, and hence is itself strictly negativedefinite and therefore also invertible. We next note the block diagonalization of Ha associated with the Schur-complement computation:





−1 I 0 Ha− Ha0 S(Ha ; Ha+ ) 0 I Ha0 Ha+ . = −1 ∗ ∗ 0 Ha+ Ha+ Ha0 Ha+ 0 I Ha0 I

58

J.A. Ball, G.J. Groenewald and S. ter Horst $

Thus Ha is congruent with

S(Ha ;Ha+ ) 0 0 Ha+

S(Ha ; Ha+ ) ≺ 0 on X− ,

% where we have seen that Ha+  0 on X+ .

In this way we arrive at the (infinite-dimensional) inertial relations between H and A: the dimension of the spectral subspace of A over the unit disk is the same as the dimension of the spectral subspace of H over the positive real axis, namely dim X+ , and the dimension of the spectral subspace of A over the exterior of the unit disk is the same as the dimension of the spectral subspace of H over the negative real axis, namely dim X− .  Remark 5.3. Rather than the full force of assumption (4.3), let us now only assume that FΣ ∞,T ≤ 1. A careful analysis of the proof shows that Ha and Hr each being bounded requires only the dichotomous 2 -exact controllability assumption (surjectivity of Wc ). The invertibility of each of Ha and Hr requires in addition the dichotomous 2 -exact observability assumption (surjectivity of Wo∗ ). Moreover, if the 2 -exact observability condition is weakened   to observability (i.e., n≥0 KerC+ An+ = {0} and n≥0 KerC− A−n−1 = {0}), − then one still gets that Ha and Hr are injective but their respective inverses may not be bounded. Remark 5.4. If F ∞,T < 1 (where we are setting F = FΣ ), then DT∗F and DT  ∗ are invertible, and we can solve uniquely for the operators X0,+ and F X0,− in Lemma 5.1: −1 + X0,+ = DT ∗ Wo , F

− X0,− = D−1  ∗ Wo . TF

We may then plug in these expressions for X0,+ and X0,− into the formulas (5.7), (5.8), (5.9), (5.10) to get even more explicit formulas for Sa , Sr , Ha and Hr .

6. Storage functions for bicausal systems We now consider how the analysis in Sections 4 and 5, concerning storage functions S : X → R, available storage Sa and required supply Sr , quadratic storage function SH , etc., can be adapted to the setting of a bicausal system Σ = (Σ− , Σ+ ) with subsystems (3.1) and (3.2), where now 2 -admissible trajectories refer to signals of the form (u, x− ⊕x+ , y) such that y = y− +y+ with (u, x− , y− ) an 2 -admissible system trajectory of Σ− and (u, x+ , y+ ) an 2 -admissible system trajectory of Σ+ . We define S : X := X− ⊕X+ → R to be a storage function for Σ exactly as was done in Section 4 for the dichotomous case, i.e., we demand that 1. S is continuous at 0, 2. S satisfies the energy balance relation (4.1) along all 2 -admissible system trajectories of the bicausal system Σ = (Σ− , Σ+ ), and 3. S(0) = 0.

Infinite-Dimensional Bounded Real Lemma III

59

We again say that S is a strict storage function for Σ if the strict energybalance relation (4.2) holds over all 2 -admissible system trajectories (u, x, y) for the bicausal system Σ = (Σ− , Σ+ ). By following the proof of Proposition 4.1 verbatim, but now interpreted for the more general setting of a bicausal system Σ = (Σ− , Σ+ ), we arrive at the following result. Proposition 6.1. Suppose that S is a storage function for the bicausal system ± exponentially stable. Then the inputΣ = (Σ− , Σ+ ) in (3.1)–(3.2), with A output map TΣ is contractive (TΣ  ≤ 1). In case S is a strict storage function for Σ, the input-output map is a strict contraction (TΣ  < 1). To get further results for bicausal systems, we impose the condition (4.3), interpreted properly for the bicausal setting as explained in Section 3. In particular, with the bicausal 2 -exact controllability assumption in place, we get the following analogue of Remark 4.2. Remark 6.2. We argue that the second condition (4.1) (respectively, (4.2) for the strict case) in the definition of a storage function for a bicausal system Σ = (Σ− , Σ+ ) (assumed to be 2 -exactly controllable) can be replaced by the local condition + x+ + B + u)) − S((A − x− + B − u) ⊕ x+ ) S(x− ⊕ (A − A − x− + C + x+ + (C − B − + D)u2 ≤ u2 − C

(6.1)

for the standard case, and by its strict version + x+ + B + u)) − S((A − x− + B − u) ⊕ x+ ) S(x− ⊕ (A − x− + B − u2 + x+ 2 + u2 ) + 2 (A − A − x− + C + x+ + (C − B − + D)u2 ≤ u2 − C

(6.2)

for the strict case. Indeed, by translation invariance of the system equations, it suffices to check the bicausal energy-balance condition (4.1) only at n = 0 for any 2 -admissible trajectory (u, x, y). In terms of x− := x− (n + 1),

x+ := x+ (n),

u := u(n),

(6.3)

we can solve for the other quantities appearing in (4.1) for the case n = 0: + x+ + B + u, x+ (1) = A − x− + B − u, x− (0) = A − A − x− + C+ x+ + (C − B − + D)u.  y(0) = C Then the energy-balance condition (4.1) for the bicausal system Σ reduces to (6.1), so (6.1) is a sufficient condition for S to be a storage function (assuming conditions (1) and (3) in the definition of a storage function also hold). Conversely, given any x− ∈ X− , x+ ∈ X+ , u ∈ U, the trajectory-interpolation result Proposition 3.3 assures us that we can always embed the vectors x− , x+ , u into an 2 -admissible trajectory so that (6.3) holds. We then see that condition (6.1) holding for all x− , x+ , u is also necessary for S to be a storage

60

J.A. Ball, G.J. Groenewald and S. ter Horst

function. The strict version works out in a similar way, again by making use of the interpolation result Proposition 3.3 . We next define functions Sa : X → R and Sr : X → R via the formulas (4.6) and (4.7) but with Wc taken to be the controllability operator as in (3.21) for a bicausal system. One can also check that the following bicausal version of Proposition 4.4 holds, but again with the verification of the continuity property for Sa and Sr postponed until more detailed information concerning Sa and Sr is developed below. Proposition 6.3. Let Σ = (Σ− , Σ+ ) be a bicausal system as in (3.1)–(3.2), ± exponentially stable. Assume that (4.3) holds. Then: with A 1. Sa is a storage function for Σ. 2. Sr is a storage function for Σ. 3. If S is any storage function for Σ, then  0 ) ≤ Sr (x0 ) for all x0 ∈ X . Sa (x0 ) ≤ S(x Proof. The proof of Proposition 4.4 for the causal dichotomous setting extends verbatim to the bicausal setting once we verify that the patching technique of Lemma 4.3 holds in exactly the same form for the bicausal setting. We therefore suppose that (u , x , y ),

(u , x , y )

(6.4) 

are  -admissible trajectories for the bicausal system Σ such that x (0) = x (0). In more detail, this means that there are two 2 -admissible system   trajectories of the form (u , x− , y− ) and (u , x− , y− ) for the anticausal system Σ− such that x− (0) = x− (0) and two 2 -admissible system trajectories   ) and (u , x+ , y+ ) for the causal system Σ+ with of the form (u , x+ , y+ x+ (0) = x+ (0) such that we recover the state and output components of the original trajectories for the bicausal system (6.4) via 2

x (n) = x− (n) ⊕ x+ (n),

x (n) = x− (n) ⊕ x+ (n),

  y (n) = y− (n) + y+ (n),

  y (n) = y− (n) + y+ (n).

Let us define a composite input trajectory by u(n) =

u (n) u (n)

if n < 0, if n ≥ 0.

We apply the causal patching lemma to the system Σ+ (having trivial dichotomy) to see that the composite trajectory (u, x+ , y+ ) with state and output given by x+ (n) =

x+ (n) x+ (n)

if n ≤ 0, if n > 0,

y+ (n) =

 y+ (n)  y+ (n)

if n < 0, if n ≥ 0,

is an 2 -admissible trajectory for the causal system Σ+ . Similarly, we apply a reversed-orientation version of the patching given by Lemma 4.3 to see that

Infinite-Dimensional Bounded Real Lemma III

61

the trajectory (u, x− , y− ) with state and output given by x− (n) x− (n)

x− (n) =

if n < 0, if n ≥ 0,

y− (n) =

 y− (n)  y− (n)

if n < 0, if n ≥ 0

is an 2 -admissible system trajectory for the anticausal system Σ− . It then follows from the definitions that the composite trajectory (u, x, y) given by (4.8) is an 2 -admissible system trajectory for the bicausal system Σ as wanted.  Quadratic storage functions and spatial KYP inequalities: the bicausal setting. We define a quadratic function S : X → R as was done at the end of Section 4 above: S(x) = Hx, x where H is a bounded selfadjoint operator on X . For the bicausal setting, we wish to make explicit that X has a decomposition as $X =%X− ⊕ X+ which we now wish to write as a column deX composition X = X− . After a change of coordinates which we choose not to + go through explicitly, we may assume that this decomposition is orthogonal. Then any selfadjoint operator H on X has a 2 × 2 matrix representation



H − H0 X− H= on X = (6.5) H0∗ H+ X+ with associated quadratic function SH now given by 

 H− H 0 x − x SH (x− ⊕ x+ ) = H(x− ⊕ x+ ), x− ⊕ x+  = , − . H0∗ H+ x+ x+ If we apply the local criterion for a given function S to be a storage function in the bicausal setting as given by Remark 6.2, we arrive at the following criterion for SH to be a storage function for the bicausal system Σ: *$ %$ % $ %+ x− x− H− H0 + x+ +B + u , A + x+ +B + u A H0∗ H+ % $ %+ *$ %$ H− H 0 − x− +B − u − x− +B − u A A − , ∗ x+ x+ H H+ 0

2 − A − x− + C + x+ + (C − B − + D)u  ≤ u − C . 2

which amounts to the spatial version of the bicausal KYP inequality (1.9). Similarly, SH is a strict storage function exactly when there is an  > 0 so that *$ %$ % $ %+ x− x− H− H0 + x+ +B + u , A + x+ +B + u A H0∗ H+ *$ %$ % $ %+ H− H 0 − x− +B − u − x− +B − u A A − , ∗ x+ x+ H H+ 0

− x− + B − u2 + x+ 2 + u2 ) +  (A 2

2 − A − x− + C + x+ + (C − B − + D)u  ≤ u2 − C .

One can check that this is just the spatial version of the strict bicausal KYP inequality (1.10). One can now check that the assertion of Proposition 4.5 goes through as stated with the dichotomous linear systems in (1.1) replaced by a + and A − both exponentially stable), and bicausal system (3.1)–(3.2) (with A with KYP inequality (respectively strict KYP inequality (1.5)) replaced with

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J.A. Ball, G.J. Groenewald and S. ter Horst

bicausal KYP inequality (1.9) (respectively strict bicausal KYP inequality (1.10)). We have thus arrived at the following extension of Proposition 4.5 to the bicausal setting. Proposition 6.4. Suppose that Σ = (Σ− , Σ+ ) is a bicausal system (3.1)–(3.2), ± exponentially stable. Let H be a selfadjoint operator as in (6.5), with A where we assume that coordinates are chosen so that the decomposition X = X− ⊕ X+ is orthogonal. Then SH is a quadratic storage function for Σ if and only if H is a solution of the bicausal KYP inequality (1.9). Moreover, SH is a strict storage function for Σ if and only if H is a solution of the strict bicausal KYP inequality (1.10). Furthermore, as noted in Section 3, the Hankel factorizations (3.22) also hold in the bicausal setting. Hence Lemma 5.1 goes through as stated, the only modification being the adjustment of the formulas for the operators Wo± , Wc± to those in (3.20) (rather than (2.19), (2.20)). It then follows that Theorem 5.2 holds with exactly the same formulas (5.7), (5.8), (5.9), (5.10) for Sa , Sr , Ha and Hr , again with the adjusted formulas for the operators Wo± and Wc± . As Sa = SHa and Sr = SHr with Ha and Hr bounded and boundedly invertible selfadjoint operators on X− ⊕ X+ , it follows that Sa and Sr are continuous, completing the missing piece in the proof of Proposition 6.3 above. We have arrived at the following extension of Theorem 5.2 to the bicausal setting. Theorem 6.5. Suppose that Σ = (Σ− , Σ+ ) is a bicausal system (3.1)–(3.2), ± exponentially stable, satisfying the standing hypothesis (4.3). Define with A the available storage Sa and the required supply Sr as in (4.6)–(4.7) (properly interpreted for the bicausal rather than dichotomous setting). Then Sa and Sr are continuous. In detail, Sa and Sr are given by the formulas (5.7)–(5.8), or equivalently, Sa = SHa and Sr = SHr where Ha and Hr are given explicitly as in (5.9) and (5.10). Remark 6.6. A nice exercise is to check that the bicausal KYP inequality − is (1.9) collapses to the standard KYP inequality (1.4) in the case that A  invertible so that the bicausal system Σ can be converted to a dichotomous  is a bicausal system as in system as in Remark 3.2. Let us assume that Σ  (3.1) and (3.2). We assume that A is invertible and we make the substitution (3.17) to convert to a dichotomous linear system as in (1.1), (2.1), (2.2). The resulting bicausal KYP inequality then becomes   ∗

I

I 0 A−1∗ C− 0 0 −

0 A∗ +

∗ C+

∗ ∗ ∗ C− +D ∗ 0 B+ −B− A∗−1 −





A−1∗ 0 0 − 0 I 0 ∗ −B− A∗−1 0 I −



H− H 0 0 H0∗ H+ 0 0 0 I

H− H0 0 H0∗ H+ 0 0 0 I



0 A+ B+ −1 C− A−1 − C+ −C− A− B− +D

−1 A−1 − 0 −A− B− 0 I 0 0 0 I

.

(6.6)

However the spatial version of the bicausal KYP inequality (1.9) corresponds to the quadratic form based at the vector

x− (1) x+ (0) u(0)

while the spatial version of

Infinite-Dimensional Bounded Real Lemma III

63

the dichotomous KYP inequality (1.4) is the quadratic form based

x (causal) − (0) at the vector x+ (0) , where the conversion from the latter to the former is u(0)

given by

x

− (0)

x+ (0) u(0)

=

$ A−

0 B− 0 I 0 0 0 I

% x− (1)

x+ (0) u(0)

.

To recover the dichotomous KYP inequality (1.4)$ from (6.6), % it therefore still remains to conjugate both sides of (6.6) by T = ∗

A− 0 B − 0 I 0 0 0 I

the right by T and on the left by T ). Note next that I

$ % A− 0 0 A− 0 B− 0 A+ B+ = 0 0 I 0 −1 C− A−1 0 0 I C− − C+ −C− A− B− +D −1

$ % $I 0 0% A 0 B A− 0 −A−1 B − − − − = 0I0 . 0 I 0 0 I 0 0

0

0 0 I

I

0 B− A+ B+ C+ D

0

0 I

,

0 0 I

Hence conjugation of both sides of (6.6) by T results in  ∗ 

A 0 B ∗ A− 0 C − H− H0 0 H− − − ∗ ∗ ∗ 0 A+ C + 0 A+ B+  H0∗ H0 H + 0 ∗ ∗ B− B+ D∗

(i.e., multiply on

C− C + D

0

H0 0 H+ 0 0 I



which is just the dichotomous KYP inequality (1.4) written out when the matrices are expressed in the decomposed form (2.1), (2.2), (6.5). The connection between the strict KYP inequalities for the bicausal setting (1.10) and the dichotomous setting (1.5) works out similarly. In fact all the results presented here for dichotomous systems follow from the corresponding result for the bicausal setting by restricting to the associated − = A−1 invertible. bicausal system having A −

7. Dichotomous and bicausal bounded real lemmas In this section we derive infinite-dimensional versions of the finite-dimensional Bounded Real Lemmas stated in the introduction. Combining the results of Propositions 4.1, 4.4, 4.5 and Theorem 5.2 leads us to the following infinite-dimensional version of the standard dichotomous bounded real lemma; this result contains Theorem 1.2 (1), as stated in the introduction, as a corollary. Theorem 7.1. Standard dichotomous bounded real lemma: Assume that the linear system Σ in (1.1) has a dichotomy and is dichotomously 2 -exactly controllable and observable (both Wc and Wo∗ are surjective). Then the following are equivalent: 1. FΣ ∞,T := supz∈T FΣ (z) ≤ 1. 2. There is a bounded and boundedly invertible selfadjoint operator H on X which satisfies the KYP inequality (1.4). Moreover, the dimension of the spectral subspace of A over the unit disk (respectively, exterior of the closed unit disk) agrees with the dimension of the spectral subspace of H over the positive real line (respectively, over the negative real line).

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J.A. Ball, G.J. Groenewald and S. ter Horst

We shall next show how the infinite-dimensional version of the strict dichotomous bounded real lemma (Theorem 1.2 (2)) can be reduced to the standard version (Theorem 7.1) by the same technique used for the stable (non-dichotomous) case (see [18, 6, 7]). The result is as follows; the reader can check that specializing the result to the case where all signal spaces U , X , Y are finite-dimensional results in Theorem 1.2 (2) from the introduction as a corollary. Note that, as in the non-dichotomous case (see [6, Theorem 1.6]), there is no controllability or observability condition required here. Theorem 7.2. Strict dichotomous bounded real lemma: Assume that the linear system Σ in (1.1) has a dichotomy. Then the following are equivalent: 1. FΣ ∞,T := supz∈T FΣ (z) < 1. 2. There is a bounded and boundedly invertible selfadjoint operator H on X which satisfies the strict KYP inequality (1.5). Moreover the inertia of A partitioned by the unit circle lines up with the inertia of H (partitioned by the point 0 on the real line) as in the standard dichotomous bounded real lemma (Theorem 7.1 above). Proof. The proof of (2) ⇒ (1) is a consequence of Propositions 4.1, 4.4, and 4.5, so it suffices to prove (1) ⇒ (2). To simplify the notation, we again write F rather than FΣ throughout this proof. We therefore assume that F ∞,T < 1. For  > 0, we let Σ be the discrete-time linear system (1.1) with system matrix M given by M =

A C

B D



⎡

A ⎢ C := ⎢ ⎣ IX 0

B D 0 IU

⎤ IX 0 ⎥ ⎥ 0 ⎦ 0

(7.1)

with associated transfer function ⎡

⎤ ⎤ ⎡ 0 C   0⎦ + z ⎣IX ⎦ (I − zA)−1 B IX 0 0 ⎤ F (z) zC(I − zA)−1 = ⎣z(I − zA)−1 B 2 z(I − zA)−1 ⎦ . IU 0

D F (z) = ⎣ 0 IU ⎡

(7.2)

As M and M have the same state-dynamics operator A, the system Σ inherits the dichotomy property from Σ. As the resolvent expression z(I − zA)−1 is uniformly bounded in norm on T, the fact that F ∞,T < 1 implies that F ∞,T < 1 as long as  > 0 is chosen sufficiently small. Moreover,

Infinite-Dimensional Bounded Real Lemma III

65

when we decompose B and C according to (2.2), we get ⎡ ⎤



U B− IX− 0 X− B = : ⎣ X− ⎦ → , B+ 0 IX+ X+ X+ ⎡ ⎡ ⎤ ⎤ C− C+ Y

⎢IX− ⎢ ⎥ ⎥ X 0 X − −⎥ ⎥: C = ⎢ →⎢ ⎣ 0 ⎣ X+ ⎦ , IX+ ⎦ X+ 0 0 U or specifically

 B− = B− C−

 IX− 0 , ⎤ ⎡ C ⎢IX− ⎥ ⎥ =⎢ ⎣ 0 ⎦, 0

  B+ = B+ 0 IX+ , ⎡ ⎤ C+ ⎢ 0 ⎥ ⎥ C+ = ⎢ ⎣IX+ ⎦ . 0

Hence we see that (A+ , B+ ) is exactly controllable in one step and hence −1 2 is 2 -exactly controllable. Similarly, (A−1 − , A− B− ) is  -exactly controllable −1 −1 2 and both (C+ , A+ ) and (C− A− , A− ) are  -exactly observable. As we also have F ∞,T < 1, in particular F ∞,T ≤ 1, so Theorem 7.1 applies to the system Σ . We conclude that there is bounded, boundedly invertible, selfadjoint operator H on X so that the KYP inequality holds with respect to the system Σ : ∗







A C∗ H 0 A B H 0  . B∗ D∗ 0 IY⊕X ⊕U C D 0 IU ⊕X Spelling this out gives ⎡ ∗ ⎤ ⎡ ⎤ A H A + C ∗ C +  2 I X A∗ H B + C ∗ D A∗ H H 0 0 ⎣ B ∗ H A + D ∗ C B ∗ H B + D∗ D + 2 IU B ∗ H ⎦  ⎣ 0 IU 0 ⎦ . H A H B  2 H 0 0 IX By crossing off the third row and third column, we get the inequality ∗



A∗ H B + C ∗ D H 0 A H A + C ∗ C +  2 I X  B ∗ H A + D ∗ C B ∗ H B + D ∗ D +  2 I U 0 IU or ∗









C ∗ H 0 A B 0 H 0 A 2 IX +  . B ∗ D∗ 0 IY C D 0 IU 0 IU We conclude that H serves as a solution to the strict KYP inequality (1.5) for the original system Σ as wanted.  The results in Section 6 for bicausal systems lead to the following extensions of Theorems 7.1 and 7.2 to the bicausal setting; note that Theorem 1.3 in the introduction follows as a corollary of this result. Theorem 7.3. Suppose that Σ = (Σ+ , Σ− ) is a bicausal linear system with subsystems Σ+ and Σ− as in (3.1) and (3.2), respectively, with both A+ and A− exponentially stable and with associated transfer function FΣ as in (3.9).

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J.A. Ball, G.J. Groenewald and S. ter Horst

1. Assume that Σ is 2 -exactly minimal, i.e., the operators Wc+ , Wc− , (Wo+ )∗ , (Wo− )∗ given by (3.20) are all surjective. Then FΣ ∞,T ≤ 1 if and only if there exists $ % a bounded and boundedly invertible selfadjoint H− H0 solution H = H ∗ H+ of the bicausal KYP inequality (1.9). 0

2. Furthermore, FΣ ∞,T < 1 holds if and only $if there %is a bounded and H H boundedly invertible selfadjoint solution H = H−∗ H+0 of the strict bi0 causal KYP inequality (1.10).

Proof. To verify item (1), simply combine the results of Propositions 6.1, 6.3, 6.4 and Theorem 6.5. As for item (2), note that sufficiency follows already from the stream of Propositions 6.1, 6.3 and 6.4. As for necessity, let us verify that the same -augmented-system technique as used in the proof of Theorem 7.2 can be used to reduce the strict case of the result (item (2)) to the standard case (item (1)). Let us rewrite the bicausal KYP inequality (1.9) as   ∗ ∗    ∗ ∗   ∗ ∗  H− +A− C− C− A− H0 A+ +A− C− C+ ∗ H ∗ +C ∗ C   ∗  ∗  A + 0 + − A− A+ H+ A+ +C+ C+ ∗ ∗ ∗  ∗      ∗C + B H +D C− A− B H+ A+ +D +

0



 where we set

H0 B+ +A− C− D ∗ H+ B + +C ∗ D  A + + ∗ ∗     B H+ B+ +D D

+

+

∗ H− A − A ∗ H0 A ∗ H− B − A − − − ∗  ∗  H 0 A− H+ H 0 B−  ∗ H− B − +I  ∗ H− A − B  ∗ H0 B B − − −



(7.3)

 =C − B − + D.  D

Let us now consider the -augmented system matrices   ⎡ A+ | B+ IX+ ⎤

  X+   X+ A+ B+,   C | D 0 + ⎣ ⎦: U → Y , M+, = = X+ +, D  IX+ | 0 0 X+ C U 

M−,

 A = − C−,



−, B = 0

⎡

0

− A ⎣ C− IX− 0

| IU

0

− IX |B − | 0 | 0 | 0

0 0 0

⎤

⎦:



X− U X−



→

X− Y X− U

 .

Then the system matrix-pair (M,+ , M,− ) defines a bicausal system Σ = (Σ,+ , Σ,− ) where the subsystem Σ,+ is associated the system matrix M,+ and the subsystem Σ,− is associated the system matrix M,− . Note that the + and A − of Σ are exponentially stable and has state-dynamics operators A transfer function F given by



C+  0 D + )−1 [ B+ IX+ ] F (z) = 0 0 + z IX+ (I − z A IU 0 0 

C− − )−1 [ B− IX− ] + IX− (I − z −1 A =

0 + (I−z A + )−1 +C − (I−z A − )−1 F (z) z C + )−1 B + )−1 +2 (I−z −1 A + +(I−z −1 A − )−1 B − 2 z(I−z A − )−1 z(I−z A IU 0

.

Infinite-Dimensional Bounded Real Lemma III

67

Since by assumption the transfer function F associated with the original + and A − bicausal system Σ = (Σ+ , Σ− ) has norm F ∞,T < 1 and both A are exponentially stable, it is clear that we can maintain F ∞,T < 1 with  > 0 as long as we choose  sufficiently small. Due to the presence of the identity matrices in the input and output operators for M+, and M−, , it is clear that the bicausal system Σ is 2 -exactly controllable and 2 -exactly observable in the bicausal sense. Then statement (1) of the present theorem (already verified) applies and we are $ assured % that there is a bounded and boundedly invertible solution H =

H+ H0 H0∗ H−

of the bicausal KYP inequality ± , C ± and D  in (7.3) by (7.3) associated with Σ = (Σ+, , Σ−, ). Replacing B    B±, , C±, and D and noting that   := C −, B −, + D  = D



  D C − 2 IX B − IU 0



eventually leads to the -augmented version of the bicausal KYP inequality: ⎡ ⎤ 2 ∗  ∗ ∗    ∗ ∗   ∗ ∗  2 ∗  H− +A− C− C− A− + A− A− H0 A+ +A− C− C+! ∗ H+ A + +C ∗ C + +2 I A

∗ H ∗ +C ∗ C − A − A ⎢ ⎣ B∗ H ∗ ++D ∗0C A+ +2 B∗ A −

−

− 0 2  ∗ C   H0∗ +C − − A− + A−

+

−

⎡



+

+

 ∗ H+ A + +D  ∗C + B +   + H+ A+ +C ∗ C −

H0 B+ +A− C− D+ A− B− + H+ B + +C ∗ D  A

+ 2  ∗ H+ B + +D  ∗ D+  2B ∗ B  B + − − + IU ∗  2   C− D+ B−

∗ H− A − A ∗ H0 A ∗ H− B − A ∗ H− A − − − − − ⎢ H0∗ A− H+ H0∗ B H0∗ ⎣ B∗ H A B∗ H B∗ H B +I B∗ H − − − − 0 − − − − − − − H− A H0 H− B 2 H− +I

where 

X14 X24 X34 X44



⎤ ⎥ ⎦

⎡

2 ∗ ∗  ∗ C H0 +A − − C− + A− ∗ ∗ ⎢ A H+ +C C−

=⎣

+

+

 ∗C − +2 B ∗ D − 2 ∗  2  C C− + IX −

X14

X24 X34

⎥ ⎦

X44

(7.4)

⎤ ⎥ ⎦.

−

The (4 × matrices appearing in (7.4) are to be understood as opera  4)-block X− $ % X . tors on XU+ where the last component X further decomposes as X = X− + X

Note that the operators in the fourth row a priori are operators with range in X− or X+ ; to get the proper interpretation of these operators as mapping in X , one must compose each $ of%these operators on the left with the canonical X injection of X± into X = X− . Similarly the operators in the fourth columns + a priori are defined only on X− or X+ ; each of these should be composed on the right with the canonical projection of X onto X± . On the other hand the identity operator I appearing in the (4, of the matrix on the right is $ 4)-entry % X− the identity on the whole space X = X+ . With this understanding of the interpretation for the fourth row and fourth column of the matrices in (7.4) in place, the next step is to simply

68

J.A. Ball, G.J. Groenewald and S. ter Horst

cross out the last row and last column in (7.4) to arrive at the reduced block(3 × 3) inequality   ∗ ∗    ∗ ∗   ∗ ∗  H− +A− C− C− A− H0 A+ +A− C− C+ H0 B+ +A− C− D ∗ H ∗ +C ∗ C   ∗  ∗    ∗  A + 0 + − A− A+ H+ A+ +C+ C+ A+ H+ B+ +C+ D  ∗ H ∗ +D  ∗C − A − B  ∗ H+ A + +D  ∗C + B  ∗ H+ B + +D  ∗D  B + + 0 +



+

2

∗ A  ∗ B  A 0 A − − − − 0 IX + 0 − A − 0 B ∗ B  B − − +IU







∗ H− A − A ∗ H0 ∗ H− B − A A − − − − − H0∗ A H+ H0∗ B  ∗ H− A − B  ∗ H0 B  ∗ H− B − +IU B − − −

 .

This last inequality amounts to the spelling out of the strict version of the bicausal KYP inequality (1.9), i.e., to (1.10). 

8. Bounded real lemma for nonstationary systems with dichotomy In this section we show how the main result of Ben Artzi–Gohberg–Kaashoek in [10] (see also [14, Chapter 3] for closely related results) follows from Theorem 7.2 by the technique of embedding a nonstationary discrete-time linear system into an infinite-dimensional stationary (time-invariant) linear system (see [13, Chapter X]) and applying the corresponding result for stationary linear systems. We note that Ben Artzi–Gohberg–Kaashoek took the reverse path: they obtain the result for the stationary case as a corollary of the result for the non-stationary case. In this section we replace the stationary linear system (1.1) with a nonstationary (or time-varying) linear system of the form  x(n + 1) = An x(n) + Bn u(n), Σnon-stat : = (n ∈ Z) (8.1) y(n) = Cn x(n) + Dn u(n), where {An }n∈Z is a bilateral sequence of state space operators (An ∈ L(X )), {Bn }n∈Z is a bilateral sequence of input operators (Bn ∈ L(U, X )), {Cn }n∈Z is a bilateral sequence of output operators (Cn ∈ L(X , Y)), and {Dn }n∈Z is a bilateral sequence of feedthrough operators (Dn ∈ L(U , Y)). We assume that all the operator sequences {An }n∈Z , {Bn }n∈Z , {Cn }n∈Z , {Dn }n∈Z are uniformly bounded in operator norm. The system Σnon−stat is said to have dichotomy if there is a bounded sequence of projection operators {Rn }n∈Z on X such that 1. Rank Rn is constant and the equalities An Rn = Rn+1 An hold for all n ∈ Z, 2. there are constants a and b with a < 1 so that An+j−1 · · · An x ≤ baj x for all x ∈ Im Rn , 1 An+j−1 · · · An y ≥ j y for all y ∈ Ker Rn . ba

(8.2) (8.3)

Infinite-Dimensional Bounded Real Lemma III

69

Let us introduce spaces →

U := 2U (Z),

→

X := 2X (Z),

→

Y := 2Y (Z).

→

→ →

→ →

and define bounded operators A ∈ L(X ), B ∈ L( U , X ), C ∈ L(X , Y ), D ∈ → →

L( U , Y ) by A = diagn∈Z [An ], B = diagn∈Z [Bn ], C = diagn∈Z [Cn ], D = diagn∈Z [Dn ]. →

Define the shift operator S on X by S = [δi,j+1 IX ]i,j∈Z with inverse S

−1

given by S−1 = [δi+1,j IX ]i,j∈Z ,

Then the importance of the dichotomy condition is that S−1 − A is invertible on 2X (Z), and conversely, S−1 − A invertible implies that the system (8.1) has dichotomy (see [10, Theorem 2.2]). In this case, given any 2 -sequence →

x ∈ X , the equation

S−1 x = Ax + x (8.4) −1 −1  2 admits a unique solution x = (S − A) x ∈ X (Z). Write out x as x = {x(n)}n∈Z . Then the aggregate equation (8.4) amounts to the system of equations x(n + 1) = An x(n) + x (n). (8.5) In particular we may take x (n) to be of the form x (n) = Bn u(n) where →

u = {u(n)}n∈Z ∈ 2U (Z). Then we may uniquely solve for x = {x(n)}n∈Z ∈ X so that x(n + 1) = An x(n) + Bu(n). We may then use the output equation in (8.1) to arrive at an output sequence →

y = {y(n)}n∈Z ∈ Y by y(n) = Cn x(n) + Dn u(n). Thus there is a well-defined map TΣ which maps the sequence u = {u(n)}n∈Z →

→

in U to the sequence y = {y(n)}n∈Z in Y . Roughly speaking, here the initial condition is replaced by boundary conditions at ±∞: Rx(−∞) = 0 and (I − R)x(+∞) = 0; we shall use the more precise albeit more implicit operator-theoretic formulation of the input-output map (compare also to the discussion around (3.1) and (3.2) for this formulation in the stationary set→

→

→

ting): TΣ : U → Y is defined as: given a u = {u(n)}n∈Z ∈ U , TΣ u is the →

→

unique y = {y(n)}n∈Z ∈ Y for which there is a x = {x(n)}n∈Z ∈ X so that the system equations (8.1) hold for all n ∈ Z, or in aggregate operator-form, by TΣ = D + C(S−1 − A)−1 B = D + C(I − SA)−1 SB. The main theorem from [10] can be stated as follows.

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J.A. Ball, G.J. Groenewald and S. ter Horst

Theorem 8.1. (See [10, Theorem 1.1].) Given a nonstationary input-stateoutput linear system (8.1), the following conditions are equivalent. 1. The system (8.1) is dichotomous and the associated input-output operator TΣ has operator norm strictly less than 1 (TΣ  < 1). 2. There exists a sequence of constant-inertia invertible selfadjoint operators {Hn }n∈Z ∈ L(X ) with both Hn  and Hn−1  uniformly bounded in n ∈ Z such that ∗







An Cn∗ Hn+1 0 An Bn Hn 0 ≺ (8.6) 0 IY C n D n 0 IU Bn∗ Dn∗ for all n ∈ Z. Proof. One can check that the nonstationary dichotomy condition (8.2)–(8.3) on the operator sequence {An }n∈Z translates to the stationary dichotomy →

→

condition on A with X + = Im R, X − = Ker R where R = diagn∈Z [Rn ]. Then →

→ A B X X U= : → → → C D U Y is the system matrix for a big stationary dichotomous linear system →

Σ :=

→

→

x(n + 1) = A x(n) + B u(n), → → y (n + 1) = C x(n) + D u(n),

(8.7)

→

where →

→

→

→

u = { u(n)}n∈Z ∈ 2→ (Z), U

→

→

x = { x(n)}n∈Z ∈ 2→ (Z), X

y = { y (n)}n∈Z ∈ 2→ (Z). Y

To apply Theorem 7.2 to this enlarged stationary dichotomous system Σ, we first need to check that the input-output map TΣ is strictly contractive. Toward this end, for each k ∈ Z introduce a linear operator →

σk,U : U = 2U (Z) → 2→ (Z) U

defined by →(k)

u = {u(n)}n∈Z → σk u = { u

(n)}n∈Z

where we set →(k)

u

→

(n) = {δm,n+k u(n)}m∈Z ∈ 2U (Z) = U .

In the same way we define σk,X and σk,Y , changing only U to X and Y, respectively, in the definition: →

σk,X : X = 2X (Z) → 2→ (Z), X

→

σk,Y : Y = 2Y (Z) → 2→ (Z). Y

Infinite-Dimensional Bounded Real Lemma III

71

Then one can check that σk,U , σk,X and σk,Y are all isometries. Furthermore, the operators   σ→ = · · · σ−1,U σ0,U σ1,U · · · : 2→ (Z) → 2→ (Z), U U  2U  2 → · · · σ σ σ · · · σ = : → (Z) → → (Z), −1,X 0,X 1,X X X   2X σ→ = · · · σ−1,Y σ0,Y σ1,Y · · · : → (Z) → 2→ (Z), Y

Y

Y

are all unitary. The relationship between the input-output map TΣ for the stationary system Σ and the input-output map TΣ for the original nonstationary system is encoded in the identity TΣ =

∞ 

∗ ∗ σk,Y SY∗k TΣ SUk σk,U = σ→ diag k∈Z [TΣ ] σ→ Y

k=−∞

U

where SU is the bilateral shift on 2U (Z) and SY is the bilateral shift on 2Y (Z), i.e., the input-output map TΣ is unitarily equivalent to the infinite inflation TΣ ⊗ I2 (Z) = diag k∈Z [TΣ ] of TΣ . In particular, it follows that TΣ  = TΣ , and hence the hypothesis that TΣ  < 1 implies that also TΣ  < 1. We may now apply Theorem 7.2 to conclude that there is a bounded →

invertible selfadjoint operator H on X such that  ∗





H 0 A C∗ H 0 A B 0 I→ C D − 0 I→ ≺ 0. B∗ D ∗ Y U Conjugate this identity with the isometry σ0 :  ∗

∗





A B σ0 0 A C∗ H 0 σ0 0 0 I→ C D 0 σ0 0 σ0∗ B∗ D∗ Y ∗





H 0 σ0 0 σ0 0 ≺ 0. − 0 σ0∗ 0 I→ 0 σ0 U

(8.8)

(8.9)

If we define Hn ∈ L(X ) by Hn = ι∗n σ0∗ H σ0 ιn where ιn is the embedding of X into the n-th coordinate subspace of 2X (Z), then one can check that the identity (8.8) collapses to the identity (8.6) holding for all n ∈ Z. This completes the proof of Theorem 8.1.  Acknowledgement. This work is based on the research supported in part by the National Research Foundation of South Africa. Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard.

References [1] B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory Approach, Prentice-Hall, Englewood Cliffs, 1973.

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[2] D.Z. Arov, M.A. Kaashoek and D.R. Pik, Minimal representations of a contractive operator as a product of two bounded operators, Acta Sci. Math. (Szeged) 71 (2005), 313–336. [3] D.Z. Arov, M.A. Kaashoek and D.R. Pik, The Kalman-Yakubovich-Popov inequality for discrete time systems of infinite dimension, J. Operator Theory 55 (2006), 393–438. [4] J.A. Ball, N. Cohen and A.C.M. Ran, Inverse spectral problems for regular improper rational matrix functions, in: Topics in Interpolation Theory and Rational Matrix-Valued Functions (Ed. I. Gohberg), pp. 123–173, Oper. Th. Adv. Appl. 33, Birkh¨ auser Verlag, Basel, 1988. [5] J.A. Ball, I. Gohberg and L. Rodman, Interpolation of Rational Matrix Functions, Oper. Th. Adv. Appl. 45, Birkh¨ auser Verlag, Basel, 1990. [6] J.A. Ball, G.J. Groenewald and S. ter Horst, Standard versus strict Bounded Real Lemma with infinite-dimensional state space I: the State-Space Similarity approach, J. Operator Theory 80 no. 1 (2018), 225–253. [7] J.A. Ball, G.J. Groenewald and S. ter Horst, Standard versus Strict Bounded Real Lemma with infinite-dimensional state space II: the storage function approach, in: The Diversity and Beauty of Applied Operator Theory (Chemnitz 2017), pp. 1–50, Oper. Th. Adv. Appl. 268, Birkh¨ auser/Springer, Cham, 2018. [8] J.A. Ball and M.W. Raney, Discrete-time dichotomous well-posed linear systems and generalized Schur-Nevanlinna-Pick interpolation, Complex Anal. Oper. Theory 1 no. 1 (2007), 1–54. [9] A. Ben-Artzi and I. Gohberg, Band matrices and dichotomy, in: Topics in Matrix and Operator Theory (Rotterdam, 1989), pp. 137–170, Oper. Theory Adv. Appl. 50, Birkh¨ auser, Basel, 1991. [10] A. Ben-Artzi, I. Gohberg and M.A. Kaashoek, Discrete nonstationary bounded real lemma in indefinite metrics, the strict contractive case, in: Operator Theory and Boundary Eigenvalue Problems (Vienna, 1993), pp. 49–78, Oper. Theory Adv. Appl. 80, Birkh¨ auser, Basel, 1995. [11] R.G. Douglas, On majorization, factorization, and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. [12] G.E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach, Texts in Applied Mathematics Vol. 36, Springer-Verlag, New York, 2000. [13] C. Foias, A.E. Frazho, I. Gohberg and M.A. Kaashoek, Metric Constrained Interpolation, Commutant Lifting and Systems, Oper. Theory Adv. Appl. 100, Birkh¨ auser, Basel, 1998. [14] A. Halanay and V. Ionescu, Time-varying Discrete linear Systems: Input-output Operators, Riccati Equations, Disturbance Attenuation, Oper. Th. Adv. Appl. 68, Birkh¨ auser Verlag, Basel, 1994. [15] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer-Verlag, Berlin–Heidelberg, 1980. [16] P. Lancaster and M. Tismenetsky, The Theory of Matrices with Applications, 2nd ed., Academic Press, 1985. [17] M.R. Opmeer and O.J. Staffans, Optimal input-output stabilization of infinitedimensional discrete time-invariant linear systems by output injection, SIAM J. Control Optim. 48 (2010), 5084–5107.

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[18] I.R. Petersen, B.D.O. Anderson and E.A. Jonckheere, A first principles solution to the non-singular H ∞ control problem, Internat. J. Robust Nonlinear Control 1 (1991), 171–185. [19] A. Rantzer, On the Kalman–Yakubovich–Popov lemma, Systems & Control Letters 28 (1996), 7–10. [20] M. Reed and B. Simon, Methods of Mathematical Physics I: Functional Analysis, Academic Press, San Diego, 1980. [21] J.C. Willems, Dissipative dynamical systems Part I: General theory, Arch. Rational Mech. Anal. 45 (1972), 321–351. [22] J.C. Willems, Dissipative dynamical systems Part II: Linear systems with quadratic supply rates, Arch. Rational Mech. Anal. 45 (1972), 352–393. [23] V.A. Yakubovich, A frequency theorem for the case in which the state and control spaces are Hilbert spaces, with an application to some problems in the ˇ 15 no. 3 (1974), 639–668, synthesis of optimal controls. I., Sibirsk. Mat. Z. translation in Sib. Math. J. 15 (1974), 457–476 (1975). [24] V.A. Yakubovich, A frequency theorem for the case in which the state and control spaces are Hilbert spaces, with an application to some problems in the ˇ 16 no. 5 (1975), 1081–1102, synthesis of optimal controls. II., Sibirsk. Mat. Z. translation in Sib. Math. J. 16 (1974), 828–845 (1976). J.A. Ball Department of Mathematics, Virginia Tech Blacksburg, VA 24061-0123 USA e-mail: [email protected] G.J. Groenewald and S. ter Horst Department of Mathematics, Unit for BMI, North-West University Potchefstroom 2531 South Africa e-mail: [email protected] [email protected]

L-free directed bipartite graphs and echelon-type canonical forms Harm Bart, Torsten Ehrhardt and Bernd Silbermann Dedicated to our colleague and friend Marinus A. (Rien) Kaashoek, on the occasion of his eightieth birthday.

Abstract. It is common knowledge that matrices can be brought in echelon form by Gaussian elimination and that the reduced echelon form of a matrix is canonical in the sense that it is unique. In [4], working within the context of the algebra Cn×n upper of upper triangular n × n matrices, certain new canonical forms of echelon-type have been introduced. Subalgebras of Cn×n upper determined by a pattern of zeros have been considered too. The issue there is whether or not those subalgebras are echelon compatible in the sense that the new canonical forms belong to the subalgebras in question. In general they don’t, but affirmative answers were obtained under certain conditions on the given zero pattern. In the present paper these conditions are weakened. Even to the extent that a further relaxation is not possible because the conditions involved are not only sufficient but also necessary. The results are used to study equivalence classes in Cm×n associated with zero patterns. The analysis of the pattern of zeros referred to above is done in terms of graph theoretical notions. Mathematics Subject Classification (2010). Primary 15A21, 05C50; Secondary 05C20. Keywords. Echelon (canonical) form, zero pattern matrices, directed (bipartite) graph, partial order, L-free graph, N-free graph, in/out-ultra transitive graph, equivalence classes of matrices.

1. Introduction It is well known that a matrix can be brought in echelon form by Gaussian elimination. More precisely, a row echelon form, respectively a column echelon form, of a matrix is brought about by applying Gaussian elimination on The second author (T.E.) was supported by the Simons Foundation Collaboration Grant # 525111.

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_3

75

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the rows, respectively the columns, of the matrix in question. Such echelon forms are generally not unique. Uniqueness does hold, however, for the socalled reduced echelon form. The reduced row echelon form, respectively the reduced column echelon form, of a given matrix M will be denoted by MrEch ,  

respectively McEch . Note that McEch = E EM E)rEch E, where E is the n×n reverse-identity matrix and the symbol signals the operation of taking the transpose. Let M ∈ Cm×n with m and n positive integers. Then there exist invertible matrices L ∈ Cm×m and R ∈ Cn×n such that LM = MrEch and M R = McEch . How is the situation when we restrict ourselves to working with matrices coming from a given linear subspace of Cm×n , say A? Of course one cannot expect the matrices MrEch and McEch to belong to A. This issue was taken up in [4] for m = n. In the first instance, only upper triangular matrices were considered. Working within that context of n×n Cupper , the algebra of upper triangular n×n matrices, new canonical forms of echelon type were introduced. In the second instance, subalgebras of Cn×n upper determined by a pattern of zeros were considered too. Under certain graph theoretical conditions on the pattern it could be proved that the new canonical forms (or some of them) indeed belong to the given subalgebras. In other words, sufficient conditions for echelon compatibility of the subalgebras, or rather the underlying partial orders determining the zero pattern in question, could be established. The present paper expands on what was done in [4] in two directions. First, by dropping the restriction of working completely within the framework of upper triangular matrices (which presupposes taking m = n). Second, by weakening the assumptions on the zero patterns under consideration to such an extent that a necessary and sufficient condition for echelon compatibility as meant above can be established. The results obtained this way are used to investigate equivalence classes in Cm×n associated with certain zero pattern linear subspaces of Cm×n . The zero patterns in questions are determined by the property of being L-free which generalizes graph theoretical concepts that have appeared in the literature before. Namely, the property of being N-free (cf. [9]) and the notions of in-ultra and out-ultra transitivity (cf. [3] and [4]). Apart from the introduction (Section 1), this paper consists of seven sections. Here is a brief indication of what they are about. In Section 2 we expand on definitions and results as can be found in Section 3 of [4] for upper triangular matrices. The basic results on the more general echelon-type canonical forms coming about in this way are efficiently proved via a trick which enables a reduction to the upper triangular case dealt with in [4]. Further we indicate an algorithm for constructing (one typical instance of) the canonical forms meant above. This algorithm plays a crucial role in the main result of the paper. Section 3 contains graph theoretical preparations needed for working with linear subspaces of Cm×n determined by zero patterns. Generalizing the notions of in-ultra transitivity and outultra transitivity that were employed in [3] and [1], the new concept of an

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L-free directed bipartite graph (formally a binary relation) is introduced. It bears a certain analogy to (but is different from and considerably more general than) the property of being N-free as can be found in the literature (cf. [9]). Also preparations are made to collect information about the left or right equivalence matrices that will appear in the next section (Section 4) where echelon compatibility for L-free directed bipartite graphs is the central topic. The special situation dealt with in [4] where the bipartite graph actually is a digraph of upper triangular type is revisited in Section 5. It turns out that the results in [4] can be easily recovered from the more general ones obtained here. Section 6 contains an application concerning the counting of the number of certain equivalence classes in Cm×n . The conclusions in question complement those in [4], Section 9. The short final section of the present paper is concerned with comments on characterizing L-free directed bipartite graphs and suggestions for further generalizations and research. Acknowledgement. The authors thank Henry Martyn Mulder for stimulating discussions about the graph theoretical aspects of this paper.

2. Canonical echelon-type forms In this section we generalize the results of [4], Section 4. Graph theoretical considerations do not come into play yet. 2.1. Terminology, notation and statement of basic results Here we expand on material from [4], Section 3. Throughout m and n stands for positive integers. The linear space of (complex) m × n matrices is denoted by Cm×n . Let A ∈ Cm×n . If a row of A is nonzero, its leading entry is its first nonzero entry. We say that A is in, or has, upward echelon form if the leading entries of any two different nonzero rows of A are in different columns of A while, moreover, when A has a leading entry in a (nonzero) row of A, the entries above it in the column in question are all zero. If, in addition, all leading entries in A are 1 it is said that A is in, or has, monic upward echelon form. In the above, the focus was on row structure. Zooming in on the column structure of the matrices involved we have the following analogues. Let B ∈ Cm×n . If a column of B is nonzero, its trailing entry is its last nonzero entry. We say that B is in, or has, starboard echelon form if the trailing entries of any two different nonzero columns of B are in different rows of B while, moreover, when B has a trailing entry in a (nonzero) column of B, the entries to the right of it in the row in question are all zero. If, in addition, all trailing entries in B are 1 it is said that B is in, or has, monic starboard echelon form. Next we turn to frames. We say that F ∈ Cm×n is an echelon frame when all rows and columns of F contain at most one nonzero entry. In case all these nonzero entries are 1, the echelon frame is said to be monic. (Square

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monic echelon frames are also called rook matrices or partial permutation matrices; cf. [10].) Here is a characterization in terms of upward and in starboard echelon forms. Theorem 2.1. Let F ∈ Cn×n . Then F is a an echelon frame if and only if F is both in upward and in starboard echelon form. Also, F is a monic echelon frame if and only if F is both in monic upward and in monic starboard echelon form. In the case when m = n and F is upper triangular, the (simple, but not completely trivial) proof of the first part of the theorem is given in [4], Section 3. By the way, there the term generalized diagonal matrix is used. With slight adaptations the argument in question goes through for the more general situation considered here. The second part of the theorem follows immediately from the first. Next we formulate a couple of theorems and corollaries generalizing results from [4], Sections 4 and 5. For their verification we refer to the next subsection. The algebra of upper triangular n × n matrices is denoted by Cn×n upper . A matrix will be called monic if it is upper triangular and has only ones on its diagonal. Such a matrix is of course invertible and its inverse is monic again. Theorem 2.2. Let M ∈ Cm×n . Then there is precisely one U ∈ Cm×n such that U is in upward echelon form and U = LM for some monic matrix L ∈ Cm×m upper . The matrix U is said to be the canonical upward echelon form of M ↑ ↑ – written MrEch . Also, a monic matrix L such that MrEch = LM will be ↑ called a left equivalence matrix corresponding to MrEch . Such an equivalence matrix is generally not unique. Mutatis mutandis the same can be said for the (other) equivalence matrices coming up below. Theorem 2.3. Let M ∈ Cm×n . Then there is precisely one S ∈ Cm×n such that S is in starboard echelon form and S = M R for some monic matrix R ∈ Cn×n upper . The matrix S is said to be the canonical starboard echelon form of M – → → written McEch . Also, a monic matrix R such that McEch = M R will be called → a right equivalence matrix corresponding to McEch . Theorem 2.4. Let M ∈ Cm×n . Then there is precisely one echelon frame F ∈ Cm×n such that F = LM R for some monic matrices L ∈ Cm×m upper and R ∈ Cn×n . upper The matrix G is said to be the canonical echelon frame of M – written MEchfr . Also, monic matrices L and R satisfying MEchfr = LM R will be designated left and right equivalence matrices corresponding to MEchfr , respectively. At the expense of sacrificing the monicity requirement on the left and right equivalence matrices, in favor of plain invertibility, monic versions of the canonical upward echelon form and starboard echelon form are obtained.

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 ∈ Cm×n such Theorem 2.5. Let M ∈ Cm×n . Then there exists precisely one U    that U is in monic upward echelon form and U = LM for some invertible  ∈ Cm×m L upper .  is said to be the canonical monic upward echelon form of The matrix U ↑ M – written MmrEch . Also, an invertible upper triangular matrix matrix L ↑ such that MmrEch = LM will be called a left equivalence matrix corresponding ↑ to MmrEch . Theorem 2.6. Let M ∈ Cm×n . Then there exists precisely one S ∈ Cm×n such that S is in monic starboard echelon form and S = M R for some invertible  ∈ Cn×n . R upper The matrix S is said to be the canonical monic starboard echelon form → of M – written MmcEch . Also, an invertible upper triangular matrix matrix → R such that MmrEch = M R will be called a right equivalence matrix corre→ sponding to MmcEch . Theorem 2.7. Let M ∈ Cm×n . Then there is precisely one monic eche R  for some invertible matrices lon frame F ∈ Cm×n such that F = LM  ∈ Cm×m and R  ∈ Cn×n . L upper upper  is said to be the canonical monic echelon frame or, for The matrix G brevity, just the canonical frame of M . It is denoted by Mfr . Invertible upper  R  will be designated left  and R  satisfying Mfr = LM triangular matrices L and right equivalence matrices corresponding to Mfr , respectively. There are several relationships between the canonical forms indicated above. On a superficial level we have  

→ McEch = Em (En M Em )↑rEch En ,  

→ MmcEch = Em (En M Em )↑mrEch En ,  

↑ MrEch = Em (En M Em )→ En , cEch  

↑ MmrEch = Em (En M Em )→ En . mcEch Here Em is the m×m reverse-identity matrix, En is the n×n reverse-identity matrix, and the symbol signals the operation of taking the transpose. To present some more sophisticated connections we first need to extend our conceptual apparatus a bit. Let A ∈ Cm×n . The matrix obtained from A by leaving the leading entries in A untouched but replacing all other entries by zeros will be called the leading entry reduction of A. Similarly, we will use the term trailing entry reduction of A for the matrix obtained from A by leaving the trailing entries in A untouched and replacing all other entries by zeros.

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Theorem 2.8. Let M ∈ Cn×n . Then ↑ ↑ ↑ → → MEchfr = (MrEch )Echfr = (MrEch )→ cEch = (McEch )Echfr = (McEch )rEch . ↑ Also, the matrix MEchfr is the leading entry reduction of MrEch and it is the → trailing entry reduction of McEch . ↑ In particular, the leading entry reduction of MrEch and the trailing entry → reduction of McEch coincide. Informally speaking, the leading entry structure ↑ → of MrEch and the trailing entry structure of McEch are identical, both with regard to positions and entry values.

Theorem 2.9. Let M ∈ Cn×n . Then ↑ ↑ ↑ → → Mfr = (MmrEch )Echfr = (MmrEch )→ cEch = (MmcEch )Echfr = (MmcEch )rEch ↑ ↑ ↑ → → = (MrEch )fr = (MrEch )→ mcEch = (McEch )fr = (McEch )mrEch ↑ ↑ ↑ → → = (MmrEch )fr = (MmrEch )→ mcEch = (MmcEch )fr = (MmcEch )mrEch . ↑ Also, the matrix Mfr is the leading entry reduction of MmrEch as well as the → trailing entry reduction of MmcEch . ↑ In particular, the leading entry reduction of MmrEch and the trailing en→ try reduction of MmcEch coincide. Loosely speaking, the leading entry struc↑ → ture of MmrEch and the trailing entry structure of MmcEch are identical, both with regard to positions and (but this is trivial now) entry values.

2.2. Verification via reduction to the upper triangular case The results stated in Subsection 2.1 can be proved directly. We will follow a different path, however, and take advantage of the work concerning the (square) upper triangular case done already in [4]. For this, we need some preparation. The k × k identity matrix, i.e., the unit element in the algebra Ck×k , is denoted by Ik . Let M ∈ Cm×n and put s = m + n. With M we associate the s × s . by stipulating that matrix M   0 M . M= . (2.1) 0 0 Here, by slight abuse of notation, the zeros stand for the zero matrices of . ∈ Cs×s the appropriate sizes. Clearly M upper . Thus the material on canonical echelon-type forms developed in [4] for upper triangular matrices applies. We shall now indicate how this can be used to deduce the results indicated in the previous subsection. From [4], Theorem 4.2 we know that there is precisely one U  ∈ Cs×s upper . for some monic matrix such that U  is in upward echelon form and U  = L M L ∈ Cs×s upper . In line with (2.1), write

L-free bipartite graphs and echelon-type canonical forms  

L =

L

L0

0

L+

81

 .

Then L is a monic m × m matrix (and L+ is a monic n × n matrix). Also      0 LM L L0 0 M . = U . = = L M 0 0 0 L+ 0 0 From this its is clear that U = LM is in upward echelon form. Suppose now that U1 ∈ Cm×n is a matrix in upward echelon form such that U1 = L1 M for some monic m × m matrix L1 . Then 

L1

0

0

In



 .= M

L1

0

0

In



0

M

0

0



 =

0

L1 M

0

0



 =

0

U1

0

0



is in upward echelon form too. The first factor in the (extreme) left-hand side of the above expression is monic. But then the matrices     0 LM 0 L1 M and 0 0 0 0 must be the same. Hence U1 = L1 M = LM = U . This proves Theorem 2.2. In the same way the other results in Subsection 2.1 can be related to the corresponding ones in [4], Sections 4 and 5. As a byproduct one obtains  ↑ .↑ = M M rEch rEch ,

 ↑ .↑ M mrEch = MmrEch ,

→ .→ = M  M cEch cEch ,

→ .→  M mcEch = MmcEch ,

.Echfr = M  M Echfr ,

.fr = M / M fr .

In combination with the results in Section 6 of [4], these identities elucidate Theorems 2.8 and 2.9. 2.3. An algorithm for constructing canonical forms In view of the proof Theorem 4.1 below it is necessary to give a description ↑ of how MrEch can be obtained from M . The algorithm consists (generally) of several steps. Using the extension trick employed in the previous subsection, these steps can be read off from the proof of Theorem 7.1 in [4] and represent the actions to be taken in order to get the standard reduced row echelon form. Some of the steps are possibly ‘empty’ in the sense that no action is needed: entries that a priori should be turned into zeros are already zero. n Let the m × n matrix M = [mi,j ]m i=1 j=1 be given. Suppose its rank is r. Then the procedure consists of precisely r steps, each of which is possible to carry out concretely. The case r = 0 is trivial (no action necessary), so we assume r to be positive.

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Step 1. As M is not the m × n zero matrix, there is a first column of M containing a nonzero entry. Suppose it is the j(1)-th column. Now let i(1) be the largest integer i among 1, . . . , m for which mi,j(1)   does not vanish. Then M has a leading entry at the position i(1), j(1) . When there are other nonzero entries in positions above this leading entry, that is in positions (i, j(1)) with i < j(1), we change them into zeros by applying appropriate row operations. This comes down to multiplying M from the left with a monic (upper triangular) m × m matrix L1 , resulting in the matrix M1 = L1 M . When r = 1 we are ready. So assume r ≥ 2. Step 2. Consider the matrix M1 . Note that the first j(1)−1 columns of both M and M1 vanish. Identify the second (non-vanishing) column of M1 containing a leading entry (the first such column of M1 is the j(1)-th, of course). Suppose it is the j(2)-th column of M1 . Let i(2) be the largestinteger i among 1, . . . , m such that M1 has a leading entry at the position i, j(2) . In other words, find the largest i among {1, . . . , m} \ {i(1)} such that the entry at position  i, j(2) is nonzero. Then i(2) = i(1) and j(1) < j(2). Also, the matrix M1 has leading entries at the positions (i(1), j(1)) and (i(2), j(2)). When there are nonzero entries in positions above the latter leading entry, that is in positions (i, j(2)) with i < i(2), we turn them into zeros by applying appropriate row operations. This comes down to multiplying M1 from the left with a monic m × m matrix L2 , resulting in the matrix M2 = L2 M1 = L2 L1 M . Carrying this out does not change the first j(2) − 1 columns of M1 . When r = 2 we are ready. So assume r ≥ 3. Step 3. Now look at M2 = L2 M1 (= L2 L1 M ). As just noted, the first j(2) − 1 columns of M1 and M2 are the  same.  Also M  1 and M2 have leading entries at the positions i(1), j(1) and i(2), j(2) . Next find the third (non-vanishing) column of M2 containing a leading entry (the first two such columns of M2 are the j(1)-th and the j(2)-th). Suppose it is the j(3)-th column of M2 , and let i(3) be the largest integer i among 1, . . . , m such that M2 has a leading entry at the position (i, j(3)). Equivalently, find the  largest  i among {1, . . . , m} \ {i(1), i(2)} such that the entry at position i, j(3) is nonzero. Then i(3) = i(1), i(2). So i(1), i(2) and i(3) are different integers and j(1) < j(2) < j(3). Also, the matrix M2 has leading entries at the positions (i(1), j(1)), (i(2), j(2)) and (i(3), j(3)). When there are nonzero entries in positions above the latter leading entry, that is in positions (i, j(3)) with i < j(3), we turn them into zeros by applying appropriate row operations. This comes down to multiplying M2 from the left with a monic m × m matrix L3 , resulting in the matrix M3 = L3 M2 = L3 L2 L1 M . The first j(3) − 1 columns of M3 are the same as the first j(3) − 1 columns of M2 . For the case when the rank r of the given matrix M is larger than 3, the subsequent steps are carried out accordingly. Proceeding in this way, ↑ one arrives in r steps at the canonical echelon form MrEch = LM , where L = Lr · · · L2 L1 M . This canonical form has leading entries at the positions       i(1), j(1) , i(2), j(2) , . . . , i(r), j(r)

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with 1 ≤ j(1) < j(2) < · · · < j(r) ≤ n and i(1), i(2), . . . , i(r) different integers among 1, 2, . . . , m. 2.4. An example We illustrate the material presented above (and the algorithm in particular) with a concrete example. Example 1. Take m = 6, n = 9, and consider M ∈ C6×9 given by ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ M =⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0

0

4

−2

8

−2

0

−9

0

2

−1 −6

2

3

0

3

0

0

0

2

−4

8

0

0

0

0

0

0

2

−4

0

6

0

0

0

3

−2

1

0

3

0

0

0

2

0

0

0

6

⎤

⎥ 1 ⎥ ⎥ ⎥ 3 ⎥ ⎥. ⎥ −1 ⎥ ⎥ 2 ⎥ ⎦ 1

The rank of this matrix is 4 and, correspondingly, we need four steps of the type indicated above. (In practice three genuine ones because one step – actually the third – turns out to be ‘empty’.) Carrying them out, writing u1 , . . . , u6 for the standard unit (column) vectors in C6 , yields, j(1) = 2,

i(1) = 6,



L1 = I6 + 3u2 u

6 − u3 u6 − 2u5 u6 ,

j(2) = 4,

i(2) = 2,

L2 = I6 − 2u1 u

2,

j(3) = 6,

i(3) = 5,

L3 = I 6

j(4) = 7,

i(4) = 4,



L4 = I6 − 2u1 u

4 − u2 u4 + 2u3 u4 ,

(empty step),

so that the product L = L4 L3 L2 L1 can be written as









I6 − 2u1 u

2 − 2u1 u4 − 6u1 u6 − u2 u4 + 3u2 u6 + 2u3 u4 − u3 u6 − 2u5 u6

leading to

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ L=⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1

−2

0

−2

0

0

1

0

−1

0

0

0

1

2

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

−6

⎤

⎥ 3 ⎥ ⎥ ⎥ −1 ⎥ ⎥, ⎥ 0 ⎥ ⎥ −2 ⎥ ⎦ 1

↑ and the canonical upward echelon form MrEch = LM of M can be computed as

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H. Bart, T. Ehrhardt and B. Silbermann ⎡

↑ MrEch

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0

0

0

0

0

0

0

2

−1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3

0

0

0

0

0

0

0

⎤

⎥ 5 ⎥ ⎥ ⎥ 0 0 0 0 ⎥ ⎥. ⎥ 0 2 −4 −1 ⎥ ⎥ −1 −2 1 0 ⎥ ⎦ 0

0

7

2

0

0

1

Here the leading entries are highlighted for clarity. Multiplying on the left with a suitable diagonal matrix, one obtains the canonical monic upward echelon form ⎡ ⎤ 0 0 0 0 0 0 0 0 0 ⎢ 7 5 ⎥ 0 0 1 − 12 0 0 ⎢ 0 2 2 ⎥ ⎥ ⎢ ⎢ ⎥ 0 0 0 0 0 0 0 0 0 ⎥ ⎢ ↑ ⎥. MmrEch =⎢ ⎥ ⎢ 0 0 0 0 0 1 −2 − 12 ⎥ ⎢ 0 ⎢ ⎥ ⎢ 0 0 0 0 0 1 2 −1 0 ⎥ ⎦ ⎣ 0

1

0

0

2 3

0

↑ Note that MmrEch is not a row permutation echelon form MrEch which is given by ⎡ 0 1 0 0 0 0 ⎢ 1 ⎢ 0 0 0 1 −2 0 ⎢ ⎢ 0 0 0 0 1 ⎢ 0 ⎢ ⎢ 0 0 0 0 0 ⎢ 0 ⎢ ⎢ 0 0 0 0 0 ⎢ 0 ⎣ 0 0 0 0 0 0

0

1 3

0

of the standard reduced row 0 0 0 1 0 0

0

1 3 5 2

⎤

⎥ ⎥ ⎥ ⎥ −1 0 ⎥ ⎥ ⎥ −2 − 12 ⎥ ⎥ ⎥ 0 0 ⎥ ⎦ 0 0 7 2

(compare the number of nonzero entries). → The canonical starboard echelon form McEch of M is given by ⎡

→ McEch

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−4 −12 −4

⎤

0

0

0

4

0

4

0

−9

0

2

0

0

0

0

0

3

0

0

0

0

−1

0

0

0

0

0

0

0

2

0

0

6

0

0

0

−1

0

0

⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥, ⎥ 0 ⎥ ⎥ 0 ⎥ ⎦

0

3

0

0

0

0

0

0

0

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85

where now the trailing entries are highlighted for emphasis. Observe that, ↑ indeed, the leading entry reduction of MrEch and the trailing entry reduction → of McEch coincide. ↑ Keeping only the leading entries in MrEch (or the trailing entries in → McEch ) and changing all other nonzero entries into zeros, we obtain the echelon frame MEchfr of M . Subsequently the canonical (monic echelon) frame Mfr of M comes then about by making all leading entries (or, if one prefers: all trailing) entries 1. 

3. Zero patterns and graph theoretical preparations We wish to consider canonical echelon-type forms (including echelon frames) of matrices satisfying certain zero pattern requirements. This section contains the necessary preparations. As before, m and n are positive integers. Also, the symbols M and N will be used for the sets {1, . . . , m} and {1, . . . , n}, respectively. 3.1. Zero pattern subspaces and algebras Let Z be a binary relation between the sets M and N , in other words, let Z ⊂ M × N . With Z we associate a subset Cm×n [Z] of Cm×n by stipulating n m×n that Cm×n [Z] is the collection of all matrices M = [mi,j ]m i=1 j=1 ∈ C m×n with mi,j = 0 whenever (i, j) ∈ / Z. Evidently, C [Z] is closed under scalar multiplication and addition. So, regardless of additional properties of Z, the set Cm×n [Z] is a linear subspace of the linear space Cm×n . In what follows, the binary relation Z will be identified with a directed bipartite graph between M to N . In line with this, the notation i →Z j is used as an alternative for (i, j) ∈ Z. In the same vein, i Z j signals that (i, j) ∈ / Z. If this happens to be convenient, we will write j ←Z i instead of i →Z j. Assume m = n. Then Z ⊂ N × N is a directed graph (often abbreviated to digraph) on the set of nodes N = {1, . . . , n}, in other words, with ground set N . We will call the digraph Z of upper triangular type when it is compatible with the standard linear order on N , i.e., i →Z j implies i ≤ j. Recall that Cn×n [Z] is a linear space, actually a linear subspace of n×n C . For being an algebra more is needed. Indeed, Cn×n [Z] is a subalgebra n×n of C if and only if the relation Z is transitive (see [8]; cf., also [11]). Clearly Cn×n [Z] contains the identity matrix In if and only if Z is reflexive. Thus Cn×n [Z] is a subalgebra of Cn×n containing the unit element in Cn×n if and only if Z is a preorder (i.e., both transitive and reflexive). Next we specialize to the situation where Z is a partial order. So besides being reflexive and transitive, Z is antisymmetric. It is common knowledge that this implies the existence of a linear order (also called total order) L on N which is compatible with Z, i.e., for which Z ⊂ L. (In the present situation with a finite ground set this is not difficult to see. When the ground set is infinite, things are more involved; in particular the axiom of choice

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H. Bart, T. Ehrhardt and B. Silbermann

is needed then. For details, see [14].) If L can be taken to be the standard linear order on N , so that Z is of upper triangular type, then Cn×n [Z] is a subalgebra of Cn×n upper . To put this in perspective, note that linear orders on N are always permutation similar to the standard linear order on N . 3.2. Echelon compatibility Returning to the general case where m and n are possibly different, suppose Z is any directed bipartite graph between M and N , and let M be a matrix in the zero pattern subspace Cm×n [Z]. A moment of reflection makes clear that one cannot expect echelon compatibility, meaning that the canonical echelon-type forms of M featuring in the previous section belong to Cm×n [Z]. Concrete counterexamples are easy to construct but (for the case m = n) can also be found in [4]. That paper, on the other hand, features positive results too where echelon compatibility does occur. These involve ultra-transitivity of the underlying digraph (actually a partial order in this context). It is a concept that comes in two types, one relevant in working with row operations (left equivalence; cf. Theorem 2.2) and one relevant for dealing with column operations (right equivalence; cf. Theorem 2.3). More about this in the next section where we will also introduce an encompassing generalization. 3.3. L-free directed bipartite graphs: definition and related notions Let Z be a directed bipartite graph between M = {1, . . . , m} and N = {1, . . . , n}. A quadruple (p, q, r, s) will be called an L for Z if p, q ∈ M , r, s ∈ N and p →Z r ←Z q →Z s,

p Z s.

p < q, r < s,

(3.1)

(Caveat: attention should be paid here to the directions of the arrows.) This terminology can be illustrated with the following picture in which m = n = 9 and Z is a partial order (in fact one of upper triangular type) given by the matrix diagram ⎛ ⎜ ⎜1 ⎜ ⎜2 ⎜ ⎜ ⎜3 ⎜ ⎜ ⎜4 Z=⎜ ⎜ ⎜5 ⎜ ⎜6 ⎜ ⎜ ⎜7 ⎜ ⎜ ⎝8 9

9

⎞

1

2

3

4

5

6

7

8

∗

0

0

0

0

0

0

0

0

∗

0

0

0



0

0

0

0

∗

0

0

0

0

0

0

0

0

∗

0

0

0

0

0

0

0

0

∗



0



0

0

0

0

0

∗

0

0

0

0

0

0

0

0

∗

0

0

0

0

0

0

0

0

∗

⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟. ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎠

0

0

0

0

0

0

0

0

∗

Here the quadruple (2, 5, 6, 8) is an L for Z (cf. the emphasized stars and zero).

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Going back to the general case, we say that the directed bipartite graph Z from M to N is L-free if Z does not feature any L. Thus Z is L-free if and only if ⎫ p, q ∈ M and r, s ∈ N ⎪ ⎪ ⎬ p ≤ q and r ≤ s ⇒ p →Z s. ⎪ ⎪ ⎭ p → Z r ←Z q →Z s In this, the inequalities p ≤ q and r ≤ s can be replaced by the strict ones p < q and r < s. The above definition is modeled after a notion that has appeared in the literature earlier, namely that of a digraph being N-free (cf. [9]). In our wider context of working with bipartite graphs, things amount to the following. The directed bipartite graph Z from M to N is N-free if it has no N’s, an N for Z being a quadruple (p, q, r, s) such that p, q ∈ M , r, s ∈ N and p →Z r ←Z q →Z s,

p = q, s = r,

p Z s.

(3.2)

In other words, Z is N-free if and only if

⎫ p, q ∈ M and r, s ∈ N ⎪ ⎪ ⎬ p = q and r = s ⇒ p →Z s. ⎪ ⎪ ⎭ p →Z r ←Z q →Z s

Here the conditions p = q and r = s are redundant and can be dropped. Proposition 3.1. Let Z be a directed bipartite graph from M to N . If Z is N-free, then Z is L-free as well. This is evident. The converse does not hold. Here is an example showing this and also illustrating the notion of an N as determined by (3.2). Example 2. Let the digraph Z, with ground matrix diagram ⎛ 1 2 3 4 ⎜ ⎜1 ∗ 0 ∗ ∗ ⎜ ⎜2 0 ∗ 0 ∗ ⎜ ⎜ Z=⎜ ⎜3 0 0 ∗ 0 ⎜ ⎜4 0 0 0 ∗ ⎜ ⎜ ⎝5 0 0 0 0 6

0

0

0

0

set {1, . . . , 6}, be given by the 5

6

⎞

∗

⎟ ∗⎟ ⎟ ∗⎟ ⎟ ⎟ 0⎟ ⎟. ⎟ 0⎟ ⎟ ⎟ 0⎠

0

∗

∗ ∗ 0 ∗

Then Z is a partial order. Inspection shows that Z is L-free. Note that Z is not N-free. Indeed, from the corresponding arrow diagram

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H. Bart, T. Ehrhardt and B. Silbermann 1

2

4 3

6

5

(in which the ubiquitous reflexivity loops are ignored) it is clear that the quadruple (2, 1, 4, 3) is an N for Z (dotted arrows).  The difference between (3.1) and (3.2) is that in (3.1) reference is made to the (standard) linear order on the ground set N whereas this is not the case in (3.2). Thus for N-free relations the specific form of the underlying sets is irrelevant. For a characterization of N-free partial orders, see Theorem 2 in [9]. Section 7 contains some observations about the possibility for characterizing the property of being L-free. We also make the connection with two other notions that have been considered (for the partial order situation) in earlier papers by the authors, namely [3] and [4]. A digraph Z is said to be in-ultra transitive if 9 p = q ⇒ p →Z q or q →Z p. p →Z r and q →Z r Also, Z is called out-ultra transitive if ⎫ v = w ⎬ u →Z v and u →Z w ⎭

⇒ v →Z w or w →Z v.

These definitions differ slightly from the ones given in [4]. In the situation considered there, in which Z is reflexive, they amount to the same. For partial orders, in-ultra transitivity and out-ultra transitivity can be characterized in terms of of the Hasse diagram (cf. [6]) and rooted trees. Details can be found in [3], Section 4. Proposition 3.2. Let Z be a transitive upper triangular-type directed graph on N . If Z is either in-ultra transitive or out-ultra transitive, then Z is L-free. So, for partial order of upper triangular type, the property of being L-free generalizes both in-ultra and out-ultra transitivity. Note that in-ultra transitivity and out-ultra transitivity are quite strong requirements. Indeed, L-free upper triangular partial orders need not be in-ultra or out-ultra transitive (see Example 2 above or the Examples 7 and 9 in Section 6). Neither is it true that upper triangular partial orders are automatically L-free. Here

L-free bipartite graphs and echelon-type canonical forms is an example of one which is not L-free: ⎛ 1 2 ⎜ ⎜1 ∗ 0 ⎜ ⎜ Z = ⎜2 0 ∗ ⎜ ⎜ ⎝3 0 0 4 0 0

3 ∗ ∗ ∗ 0

4

89

⎞

⎟ 0⎟ ⎟ ⎟ ∗ ⎟. ⎟ ⎟ 0⎠ ∗

Proof. Let Z be in-ultra transitive. Suppose p, q, r, s ∈ N and p →Z r ←Z q →Z s,

p < q,

r < s.

By in-ultra transitivity we get from p →Z r, q →Z r, p = q (in fact p < q) that either p →Z q or q →Z p. As Z is of upper triangular type, the first of these implies p ≤ q, and the second q ≤ p. But p < q, and it follows that p →Z q. We also have q →Z s. By assumption, Z transitive. Hence p →Z s, as desired. This covers the situation where Z is in-ultra transitive. For the case when Z is out-ultra transitive the argument is similar.  Notice also that condition of upper triangularity in Proposition 3.2 cannot be dropped. As simple example of a partial order which is both in-ultra transitive and out-ultra transitive, but not L-free (thus also not N-free) is given by ⎞ ⎛ 1 2 3 ⎟ ⎜ ⎜1 ∗ 0 0 ⎟ ⎟. Z=⎜ ⎟ ⎜ ⎝2 ∗ ∗ 0 ⎠ 3 ∗ ∗ ∗ In the same vein, we observe that the property of being N-free does not imply in-ultra or out-ultra transitivity (an example is easy to produce). Neither does one of these two (or even the combination of both) guarantee that the partial order in question is N-free, not even in the upper triangulartype case. Take for instance the standard linear order on N . 3.4. In-diagrams and out-diagrams Let Z be a directed bipartite graph from M to N . With Z we associate two digraphs, one with ground set M and one with ground set N . They are named the in-diagram of Z – written ZinD – and the out-diagram of Z – written ZoutD . The digraph ZinD (viewed as a relation on M ) consists of all reflexivity loops (r, r), r ∈ M and all pairs (k, l) ∈ M × M such that k < l and for which there exists j ∈ N with k →Z j and l →Z j. Similarly, ZoutD consists of all reflexivity loops (r, r), r ∈ N and all pairs (v, w) ∈ N × N such that v < w and for which there exists u ∈ M with u →Z v and u →Z w. Both ZinD and ZoutD are of upper triangular type (hence antisymmetric) and reflexive. Here is an example.

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Example 3. Take m = 7, n = 5, and let Z be the directed bipartite graph from M = {1, 2, 3, 4, 5, 6, 7} to N = {1, 2, 3, 4, 5} given by the matrix diagram ⎞ ⎛ 1 2 3 4 5 ⎟ ⎜ ⎜1 0 0 0 ∗ ∗⎟ ⎟ ⎜ ⎟ ⎜ ⎜2 0 ∗ 0 ∗ ∗⎟ ⎟ ⎜ ⎜3 0 0 0 0 0⎟ ⎟ ⎜ Z=⎜ ⎟. ⎜4 ∗ ∗ 0 ∗ ∗⎟ ⎟ ⎜ ⎟ ⎜ ⎜5 0 0 0 0 0⎟ ⎟ ⎜ ⎜6 ∗ 0 0 ∗ ∗⎟ ⎠ ⎝ 7 0 0 ∗ ∗ 0 Then Z is L-free. Also ⎛ 1 2 3 ⎜ ⎜1 ∗ ∗ 0 ⎜ ⎜ ⎜2 0 ∗ 0 ⎜ ⎜3 0 0 ∗ ⎜ ZinD = ⎜ ⎜4 0 0 0 ⎜ ⎜ ⎜5 0 0 0 ⎜ ⎜6 0 0 0 ⎝ 7 0 0 0

4

5

6

∗

0

∗

∗

0

∗

0

0

0

∗

0

∗

0

∗

0

0

0

∗

0

0

0

7

⎞

⎟ ∗⎟ ⎟ ⎟ ∗⎟ ⎟ 0⎟ ⎟ ⎟, ∗⎟ ⎟ ⎟ 0⎟ ⎟ ∗⎟ ⎠ ∗

⎛

ZoutD

1

2

3

4

5

⎞

⎜ ⎜1 ⎜ ⎜2 ⎜ =⎜ ⎜3 ⎜ ⎜ ⎝4

∗

∗

0

∗

0

∗

0

∗

0

0

∗

∗

0

0

0

∗

⎟ ∗⎟ ⎟ ∗⎟ ⎟ ⎟. 0⎟ ⎟ ⎟ ∗⎠

5

0

0

0

0

∗

The digraph ZinD is in-ultra transitive, ZoutD is out-ultra transitive. These observations corroborate our next proposition. Note that out that ZinD is transitive, but ZoutD is not. The latter appears from 3 →ZoutD 4 →ZoutD 5 but 3 ZoutD 5.  Proposition 3.3. Let Z be an L-free directed bipartite graph from M to N . Then ZinD and ZoutD are in-ultra transitive and out-ultra transitive, respectively. Proof. We give the argument only for ZinD . Suppose p →ZinD r, q →ZinD r, and p < q. We need to prove that p →ZinD q. Here is the argument. From the definition of ZinD we have that p, q < r and that there exist s, t ∈ N such that p →Z s, r →Z s and q →Z t, r →Z t. We now distinguish two cases. The first is that s ≤ t. Then, Z being L-free, it follows from p →Z s ←Z r →Z t,

p < r, s ≤ t

that p →Z t. Combining this with q →Z t and p < q, we get p →ZinD q. Next assume that s ≥ t. Then q →Z t ←Z r →Z s,

q < r, t ≤ s

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gives q →Z s. Together with p →Z s and p < q, we obtain again the desired conclusion, namely p →ZinD q.  Since the property of being N-free implies that of being L-free, the conclusion of Proposition 3.3 also holds when Z is N-free. In that situation we can prove more. A digraph Z is called ultra transitive if it is both in-ultra transitive and out-ultra transitive. Proposition 9.4 in [4] provides a characterization of such digraphs in terms of disjoint unions of linear orders. Proposition 3.4. Let Z be an N-free directed bipartite graph from M to N . Then ZinD and ZoutD are upper triangular-type ultra transitive partial orders. Proof. Again we concentrate on ZinD . By definition, ZinD is of upper triangular type (hence antisymmetric) and reflexive. So to prove that ZinD is a partial order, we only need to establish transitivity. Suppose k →ZinD l →ZinD j. We need to show that k →ZinD j. This is trivially true when l = k or l = j. So we assume that l = k, j. As ZinD is of upper triangular type, it follows that k < l < j. But then there exist u, v ∈ N such that k →Z u, l →Z u,

l →Z v, j →Z v.

In particular k →Z u ←Z l →Z v. As Z is N-free, it follows that k →Z v. Combining this with j →Z v and k < j, we get k →ZinD j. From Proposition 3.3 we already have that the digraph ZinD is in-ultra transitive. Its out-ultra transitivity is established as follows. Assume u →ZinD v, u →ZinD w and v < w. Then u < v, w and there exist s, t in N such that u →Z s, v →Z s, u →Z t, w →Z t. Thus v →Z s ←Z u →Z t, and hence v →Z t because Z is N-free. But then we have v →Z t and w →Z t. In combination with v < w this gives v →ZinD w, as desired.  For later reference (cf. the all but last paragraph prior to the proof of Theorem 4.1) we expand a bit on the previous proposition and its proof. Let Z be a directed bipartite graph from M to N . Then the extension trick described in Subsection 2.2 and the material presented above suggest to introduce the upper triangular-type digraph " # ZinD Z Z = , 0 ZoutD having ground set {1, . . . , m, m + 1, . . . , m + n} with s = m + n. Clearly Z is of upper triangular type and reflexive. If Z happens to be N-free, then Z is an ultra-transitive partial order. The proof of this goes along the lines of that of Proposition 3.4. The condition that Z is N-free is essential here. Examples showing this are easy to produce. Later on we will need that Cm×m [ZinD ] and Cn×n [ZoutD ] are unital n×n subalgebras of Cm×m upper and Cupper , respectively. For this we need that the digraphs ZinD and ZoutD are transitive and reflexive (see Subsection 3.1). Given the definition of ZinD and ZoutD , reflexivity is not a problem, but in

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general transitivity is. To be specific, in case Z is N-free, ZinD and ZoutD are transitive (cf. Proposition 3.4) but when Z is only L-free, this conclusion cannot be drawn (see Example 3). Thus it is necessary to consider transitive closures. Let Z be a digraph on N . The transitive closure of Z – written ZT – is the smallest transitive digraph on N containing Z. It is equal to the intersection of all transitive digraphs on N that contain Z. The collection of such digraphs is nonempty. Indeed, it contains N × N . When Z is of upper triangular type, then so is ZT . Indeed, in that case, the standard linear order L on N is a transitive digraph containing Z, hence ZT ⊂ L. For use in the proof of the lemma below, we point out that k →ZT m if and only if there exists a positive integer s and a chain l0 →Z l1 →Z · · · →Z ls such that l0 = k and ls = m. Lemma 3.5. Let Z be a directed graph on N . Suppose Z is in-ultra transitive, respectively out-ultra transitive. Then ZT is in-ultra transitive, respectively out-ultra transitive. Proof. We focus on the in-ultra transitive case and give the argument by reductio ad absurdum. So we assume that Z is in-ultra transitive but ZT is not. The latter means that there exist p, q and r in the ground set N such that p →ZT r, q →ZT r, p = q, p ZT q, q ZT p. Now take into account the description of ZT preceding the formulation of the lemma. This yields that there exist positive integers s, t, and chains p0 →Z p1 →Z · · · →Z ps ,

q0 → Z q 1 → Z · · · → Z q t ,

such that p0 = p, q0 = q and ps = qt = r. This makes clear that we can introduce the (nonempty) set C of all pairs of chains p0 →Z p1 →Z · · · →Z ps ,

q0 → Z q 1 → Z · · · → Z q t ,

such that p0 = p, q0 = q, and ps = qt (the latter not necessarily being equal to r). Therein we allow s and t to be non-negative integers. In the case s = 0 the first chain reduces to p = p0 = ps = qt , while in the case t = 0 the second chain reduces to q = q0 = qt = ps . Among all pairs of chains in C choose one for which s + t is minimal. To obtain a contradiction we argue as follows. First, let us consider some trivial cases. If s = 0, then q = q0 →Z q1 →Z · · · →Z qt = p, whence q →ZT p. Likewise, if t = 0, then p →ZT q. Both contradict the assumption made at the beginning. For completeness note that t = s = 0 is also impossible because it would imply p = q. Therefore, from now on we can assume that the chosen pair of chains has lengths s ≥ 1 and t ≥ 1. Then we can consider the elements ps−1 and

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93

qt−1 . Now distinguish two cases. If ps−1 = qt−1 , then p0 →Z p1 →Z · · · →Z ps−1 ,

q0 →Z q1 →Z · · · →Z qt−1

with p0 = p, q0 = q, ps−1 = qt−1 . This is a pair of chains in C of a total length s − 1 + t − 1 < t + s. We obtain a contradiction to the minimality condition of the original chain. Note that up to now the in-ultra transitivity on Z was not used. This will be different when we consider the other possibility, namely ps−1 = qt−1 . Indeed, the in-ultra transitivity on Z then guarantees that ps−1 →Z qt−1 or qt−1 →Z ps−1 . If the former of these holds, then we have chains p0 →Z p1 →Z · · · →Z ps−1 →Z qt−1 ,

q0 →Z q1 →Z · · · →Z qt−1

of lengths s and t − 1. If the latter holds, then the chains p0 →Z p1 →Z · · · →Z ps−1 ,

q0 →Z q1 →Z · · · →Z qt−1 →Z ps−1

have lengths s − 1 and t. In both case these pairs of chains belong to C and have total length s + t − 1. Again we get a contradiction with the minimality condition.  For a directed bipartite graph Z from M to N , we let ZTinD and ZToutD denote the transitive closures of ZinD and ZoutD , respectively. If Z is N-free, then ZTinD coincides with ZinD , and ZToutD coincides with ZoutD . This is immediate from Proposition 3.4. Theorem 3.6. Let Z be an L-free directed bipartite graph from M to N . Then ZTinD and ZToutD are upper triangular-type partial orders. Also ZTinD and ZToutD are in-ultra transitive and out-ultra transitive, respectively. The first part of the theorem implies that Cm×m [ZTinD ] and n×n C [ZToutD ] are subalgebras of Cm×m upper and Cupper , respectively, having the corresponding identity matrices as unit elements. n×n

Proof. The way ZinD and ZoutD were defined brings with it that these digraphs are reflexive and of upper triangular type. Taking the transitive closures ZTinD and ZToutD now yields digraphs that are reflexive and of upper triangular type too. Of course these transitive closures are transitive. This proves the first part of the theorem. The second part is obtained by combining Proposition 3.3 and Lemma 3.5.  3.5. L-closures Let Z be a directed bipartite graph Z from M to N . By Z Λ we denote the intersection of all L-free directed bipartite graphs from M to N which contain Z. Such bipartite graphs do exist. In fact M × N is one of them. Obviously, Z Λ is an L-free directed bipartite graph from M to N and Z ⊂ Z Λ . So Z Λ is the smallest L-free directed bipartite graph from M to N containing Z. It is called the L-closure of Z. We also introduce the L-reduction ZΛ of Z. Modeled after the familiar Hasse diagram (defined for partial orders), ZΛ is stipulated to be the complement in Z of the set of all pairs (p, s) ∈ M × N (or, what here leads to

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the same end result, (p, s) ∈ Z) for which there exists (q, r) ∈ M × N such that p →Z r ←Z q →Z s and p < q, r < s. In case m = n, so M = N , and Z is an upper triangular-type partial order, ZΛ is contained in the Hasse diagram of Z with the reflexivity loops added to it. Proposition 3.7. Let Z be a directed bipartite graph from M to N . Then (ZΛ )Λ ⊃ Z, with equality (ZΛ )Λ = Z holding if and only if Z is L-free. An immediate consequence is that two L-free directed bipartite graphs Z from M to N are identical if and only if their L-reductions coincide. Λ Λ Proof. Suppose Z is not : contained in (ZΛ ) , Λin; other words, Z \ (ZΛ ) is nonempty. Then D = s − p | (p, s) ∈ Z \ (ZΛ ) is nonempty too. Hence it has a smallest element, d say. Take (p, s) ∈ Z \ (ZΛ )Λ , such that s − p = d. Now ZΛ ⊂ (ZΛ )Λ , so (p, s) cannot be in ZΛ . This means that there exists a pair (q, r) ∈ M × N such that p →Z r ←Z q →Z s and p < q, r < s. Clearly all three integers r − p, r − q and s − q are smaller than s − p = d, therefore they are not in D. It follows that the pairs (p, r), (q, r) and (q, s) are not in Z \ (ZΛ )Λ . As they do belong to Z, we may conclude that (p, r), (q, r) and (q, s) are in (ZΛ )Λ . The fact that (ZΛ )Λ is L-free now implies that (p, s) ∈ (ZΛ )Λ too, contrary to (p, s) being a member of Z \ (ZΛ )Λ . With this the first statement in the proposition has been proved. It is now easy to establish the second too. The ‘if part’ of it is obvious. Indeed, if Z = (ZΛ )Λ , then Z is L-free along with (ZΛ )Λ . So let us consider the case where it is given that Z is L-free. Then (ZΛ )Λ ⊂ Z because ZΛ ⊂ Z. As the inclusion Z ⊂ (ZΛ )Λ was already obtained, we may conclude that Z = (ZΛ )Λ . 

4. L-free directed graphs: canonical forms As before M = {1, . . . , m} and N = {1, . . . , n}, where m and n are given positive integers. Theorem 4.1. Suppose Z is an L-free directed bipartite graph from M to N . Then for each matrix M in the linear space Cm×n [Z], its canonical forms ↑ ↑ → → MrEch , McEch , MmrEch , MmcEch , MEchfr , Mfr

are all in Cm×n [Z]. Also, there are left, respectively right, equivalence matrices corresponding to these canonical forms that belong to Cm×m [ZTinD ], respectively Cn×n [ZToutD ]. From Theorem 3.6 we know that ZTinD and ZToutD are upper triangulartype partial orders on M and N , respectively. Thus Cm×m [ZTinD ] is a subaln×n [ZToutD ] is a subalgebras of Cn×n gebra of Cm×m upper , and C upper (see Subsection 3.1). Both are unital, the unit element in Cm×m [ZTinD ] being Im , and the unit element in Cn×n [ZToutD ] being In . We also bring back to mind that ZTinD is in-ultra transitive and ZToutD is out-ultra transitive.

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Before we enter into the proof of the theorem, we make two remarks serving as caveats. As to the first one, suppose for the moment that m = n and that Z is an in-ultra transitive upper triangular-type partial order (hence L-free by Proposition 3.2). Then we know from [4], Theorem 7.1 that the left equivalence ma↑ can be chosen in such a way that it belongs to trix corresponding to MrEch ↑ n×n C [Z]. As the latter is an algebra, the conclusion MrEch = LM ∈ Cn×n [Z] is then immediate. It is not in the more general situation considered here where Z is L-free, the reason being that, as can be seen from Example 4 below, the left equivalence matrix L may lie (even necessarily) outside Cn×n [Z]. For the second remark we consider the possibility of directly relating the above theorem to the results of [4], Section 7 by taking advantage of the extension trick employed in Subsection 2.2 and further expanded in Subsection 3.4 after the proof of Proposition 3.4. So put s = m + n and define the . ∈ Cs×s and the reflexive digraph Z with ground set {1, . . . , s} by matrix M   " # 0 M ZTinD Z .  M= , Z= . 0 0 0 ZToutD . belongs to Cs×s [Z].  Now, if Z were Then Z is of upper triangular type and M a partial order, one could try to apply the results of [4]. This works under the rather restrictive condition that Z is N-free, in fact even when ZTinD and ZToutD are replaced by ZinD and ZoutD , respectively (cf. Proposition 3.4 and the remark directly after its proof). However, the approach fails when Z is just L-free. Indeed, in that case, Z need not even be transitive. Examples showing this are easy to construct. The upshot of these two remarks is that in order to prove Theorem 4.1 we must go into a deeper analysis of the way the echelon-type canonical forms of M considered here are obtained from M . Proof. Take M ∈ Cm×n [Z]. As is explained in Subsection 2.3, the (uniquely ↑ determined) canonical upward echelon form MrEch of M can be obtained from M by an algorithm involving elementary row operations and leading in an ↑ unambiguous way to a monic matrix LM ∈ Cn×n upper such that LM M = MrEch . ↑ The aim is to prove that LM belongs to Cm×m [ZTinD ] and MrEch to Cm×n [Z]. For this we need to describe LM more explicitly. The basis for this lies in the material contained in [4], Sections 4 and 7; cf. Subsections 2.3 and 2.4. (If desired, things can be reestablished via an induction argument.) Let u1 , . . . , um stand for the standard (column) unit vectors in Cm . ↑ Suppose the leading entries of the matrix MrEch are at the positions 

   i(1), j(1) , . . . , i(r), j(r) ,

where r is the rank of M (necessarily), i(s) = i(t),

s = t, s, t = 1, . . . , r,

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and j(k − 1) < j(k),

k = 2, . . . , r.

Then LM has the form " LM =

Im −



# αp,r up u

i(r)

"



· · · Im −

p=1,...,i(r)−1 p→Z j(r)

# αp,1 up u

i(1)

,

p=1,...,i(1)−1 p→Z j(1)

with αp,r+1−s ∈ C. Further, with   LM (s) = Im −

αp,s up u

i(s)

 ,

s = 1, . . . , r,

p=1,...,i(s)−1 p→Z j(s)

and LM (0) = Im (for convenience) the following holds: LM = LM (r) · · · LM (0), LM (s) · · · LM (0)M has leading entries at (i(1), j(1)), . . . , (i(s + 1), j(s + 1)). In the latter statement s is allowed to take the values 0, . . . , r − 1. We claim that   LM (s) · · · LM (0) M ∈ Cm×n [Z], s = 0, . . . , r,

(4.1)

so, in particular, LM M = (LM (r) · · · LM (0))M ∈ C [Z]. As LM M = ↑ ↑ MrEch , this gives MrEch ∈ Cm×n [Z], the first of the desired conclusions. The argument is by (finite) induction with s as induction parameter. For s = 0, the assertion (4.1) is evidently true. Indeed for this value of s it reduces to M ∈ Cn×n [Z] which is given. To carry out the induction step, we assume that (4.1) holds for some s among the integers 0, . . . , r − 1. We then have to show that the matrix (LM (s + 1)LM (s) · · · LM (0))M belongs to Cn×n [Z]. Put N = (LM (s) · · · LM (0))M . Then N ∈ Cm×n [Z] and N has leading entries at (i(1), j(1)), . . . , (i(s + 1), j(s + 1)). In particular, it has one at (i(s + 1), j(s + 1)). Also   LM (s + 1)LM (s) · · · LM (0) M = LM (s + 1)N m×n

 =

Im −



αp,s+1 up u

i(s+1)

 N

p=1,...,i(s+1)−1 p→Z j(s+1)

=N−



αp,s+1 up u

i(s+1) N.

p=1,...,i(s+1)−1 p→Z j(s+1)

It now suffices to make clear that, for the indicated values of p, the matrix m×n up u

[Z]. i(s+1) N belongs to C

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Assume up u

i(s+1) N has a nonzero entry at the position (h, t), i.e.,

u

h up ui(s+1) N ut = 0.

Then h = p, so h < i(s + 1) and h →Z j(s + 1). Also u

i(s+1) N ut = 0, i.e., the entry of N at the position (i(s + 1), t) is nonzero. As N ∈ Cm×n [Z], it follows that i(s + 1) →Z t. Recalling that N has a leading entry at the position (i(s + 1), j(s + 1)), we may also conclude that i(s + 1) →Z j(s + 1) and j(s+1) ≤ t. The upshot of all of this is h →Z j(s+1) ←Z i(s+1) →Z t, h < i(s + 1) and j(s + 1) ≤ t. By hypothesis, Z is L-free, and it follows that h →Z t, as desired. Let us now explain that LM belongs to Cm×m [ZTinD ]. As the latter is an algebra, it is sufficient to show that LM (s) ∈ Cm×m [ZTinD ],

s = 1, . . . , r.

Take s ∈ {1, . . . , r}, let g, k ∈ {1, . . . , n}, and suppose LM (s) has a nonzero entry at the position (g, k). Then g does not exceed k because the matrix LM (s) is upper triangular. We need to show that g →ZTin k. For g = k, this is immediate from the reflexivity of ZTin . So we may assume that g < k. The entry of LM (s) at the position (g, k) is equal to u

g LM (s)uk , so we have   

u

I − α u u m p,s p g i(s) uk = 0. p=1,...,i(s)−1 p→Z j(s)

Using the Kronecker delta notation, this expression can be rewritten as  δg,k − αp,s δg,p δi(s),k = 0, p=1,...,i(s)−1 p→Z j(s)

which, in view of g = k, leads to 

αp,s δg,p δi(s),k = 0.

p=1,...,i(s)−1 p→Z j(s)

From this it is immediate that k = i(s). It also follows that g belongs to the set over which the above sum is taken. Hence g < i(s) = k and g →Z j(s). ↑ We also have k = i(s) →Z j(s) because MrEch has a leading entry at the ↑ position (i(s), j(s)) and, as we have seen MrEch ∈ Cm×n [Z]. It is now clear that g →ZinD k. But then, a fortiori, g →ZTinD k as desired. It has now been proved that there exists a monic L ∈ Cm×m [ZTinD ] such ↑ that LM = MrEch ∈ Cm×n [Z]. As a counterpart one has that there exists n×n → a monic R ∈ C [ZToutD ] such that M R = McEch belongs to Cm×n [Z]. At the expense of sacrificing the monicity requirement on L and R in favor ↑ of plain invertibility, these statements remain true when one replaces MrEch ↑ → → and McEch by their monic versions MmrEch and MmcEch , respectively. The arguments involve left or right multiplication with a diagonal matrix (see the reasoning given in the proof of [4], Corollary 4.4).

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Next let us turn to the echelon frames. Let L be a monic matrix in ↑ Cm×m [ZTinD ] such that LM = MrEch ∈ Cm×n [Z]. Also, let R be a monic ma↑ ↑ n×n m×n trix in C [ZToutD ] for which MrEch R = (MrEch )→ [Z]. cEch ∈ C ↑ → Employing the identity MEchfr = (MrEch )cEch from Theorem 2.8, we see that MEchfr ∈ Cm×n [Z]. The canonical frame Mfr can (again) be handled via left and right multiplication with diagonal matrices (cf. the previous paragraph).  In connection with Theorem 4.1, certain questions immediately come to mind. Let us discuss them for the case where m = n and Z is an L-free digraph on N . The first question is: can the equivalence matrices be chosen in Cn×n [Z]; the second: is it true that (using a self-evident notation) Cn×n [ZTinD ] × Cn×n [Z] ⊂ Cn×n [Z]? For both questions, the answer is negative. Here is a simple example showing this. Example 4. Consider the situation where n = 4 and ⎛ ⎞ ⎡ ∗ 0 ∗ ∗ 0 0 ⎜ ⎟ ⎢ ⎜0 ∗ ∗ ∗⎟ ⎢0 0 ⎟ Z=⎜ M =⎢ ⎜0 0 ∗ 0⎟, ⎢0 0 ⎝ ⎠ ⎣ 0 0 0 ∗ 0 0

1 1 0 0

1

⎤

⎥ 0⎥ ⎥. 0⎥ ⎦ 0

Then Z is an upper triangular-type L-free digraph (in fact a partial order) on {1, 2, 3, 4} and M belongs to C4×4 [Z]. Introduce ⎤ ⎡ 1 −1 0 0 ⎥ ⎢ ⎢0 1 0 0⎥ ⎥. ⎢ L=⎢ ⎥ ⎣0 0 1 0⎦ 0 0 0 1 Then L is a monic ⎡ 0 ⎢ ⎢0 LM = ⎢ ⎢0 ⎣ 0

(hence invertible) matrix and ⎤ 0 0 1 ⎥ 0 1 0⎥ ⎥ = M ↑ (= M ↑ rEch mrEch = MEchfr = Mfr ). 0 0 0⎥ ⎦ 0 0 0

From the fact that ZTinD (= ZinD ) is given by ⎛ ∗ ∗ ∗ ⎜ ⎜0 ∗ ∗ ZTinD = ⎜ ⎜0 0 ∗ ⎝ 0

0

0

∗

⎞

⎟ ∗⎟ ⎟, 0⎟ ⎠ ∗

we see that L ∈ C [ZTin ], in line with Theorem 4.1. However, L does not  ∈ C4×4 [Z], invertible or not, belong to C4×4 [Z]. In fact, there is no matrix L ↑  with LM = MrEch . 4×4

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 ∈ C4×4 [Z] is such a matrix. Writing To see this, assume that L ⎡ ⎤ l1,1 0 l1,3 l1,4 ⎢ ⎥ ⎢ 0 l2,2 l2,3 l2,4 ⎥ ⎢ ⎥ =⎢ L ⎥, ⎢ 0 0 l3,3 0 ⎥ ⎣ ⎦ 0 we have

⎡

0

⎢ ⎢0  =⎢ LM ⎢ ⎢0 ⎣ 0

0

l1,1

0

l2,2

0

0

0

0

0

0

l4,4

⎤

⎡ 0 ⎥ ⎢ 0 ⎥ ⎢0 ⎥ ↑ ⎥ = MrEch = ⎢ ⎢0 ⎥ 0 ⎦ ⎣ 0 0

l1,1

0

0

0

1

0

0

0

0

1

⎤

⎥ 0⎥ ⎥. 0⎥ ⎦ 0

But then 0 = l1,1 = 1 which is impossible. The other issue we need to address is: when A ∈ C4×4 [ZTinD ] and M ∈ C4×4 [Z], does it follow that AM ∈ Cn×n [Z]? Again this is not the case. To illustrate this, consider ⎡ ⎤ ⎡ ⎤ 1 1 0 0 0 0 0 0 ⎢ ⎥ ⎢ ⎥ ⎢0 1 0 0⎥ ⎢0 1 0 0⎥ ⎥, ⎢ ⎥ A=⎢ M = ⎢0 0 1 0⎥ ⎢0 0 0 0⎥. ⎣ ⎦ ⎣ ⎦ 0 0 0 1 0 0 0 0 Then A ∈ C4×4 [ZTinD ] and M Z is L-free, ⎡ 0 ⎢ ⎢0 ⎢ ⎢ AM = ⎢ 0 ⎢ ⎢0 ⎣ 0

∈ C4×4 [Z]. However, in spite of the fact that 1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

⎤

⎥ 0⎥ ⎥ ⎥ 0⎥ ∈ / C4×4 [Z]. ⎥ 0⎥ ⎦ 0

Note that A has the additional property of being monic.



Given a directed bipartite graph Z from M to N , one can apply Theorem 4.1 to the L-closure Z Λ of Z (see Subsection 3.5). Here is what comes out of this. Corollary 4.2. Suppose Z is a directed bipartite graph from M to N . Then for each matrix M in the linear space Cn×n [Z], its canonical forms ↑ ↑ → → , McEch , MmrEch , MmcEch , MEchfr , Mfr MrEch

are all in Cm×n [Z Λ ]. Also, there are left, respectively right, equivalence matrices corresponding to these canonical forms that belong to Cm×m [(Z Λ )TinD ], respectively Cn×n [(Z Λ )ToutD ].

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In line with what we said after Theorem 4.1, we mention that (Z Λ )TinD and (Z Λ )ToutD are upper triangular-type partial orders, in-ultra transitive and out-ultra transitive, respectively. As a consequence, Cm×n [(Z Λ )TinD ] and Cn×n [(Z Λ )ToutD ] are subalgebras of Cn×n upper having the identity matrices as unit elements.

5. The upper triangular-type case In Proposition 3.2 we specialized to the case where m = n and the given directed bipartite graph Z is a digraph of upper triangular type. Here we proceed in this direction, also with the purpose to clarify the relation with the paper [4] exclusively devoted to studying the upper triangular-type case. As before, n stands for positive integer and N for the set {1, . . . , n}. 5.1. Observation concerning reflexivity Let Z be a digraph of upper triangular type on N . Here is an observation on how the property of being L-free combines with reflexivity. Proposition 5.1. Let Z be an L-free upper triangular-type directed graph on N . Then Z is a partial order if and only if Z is reflexive. Proof. As partial orders are reflexive by definition, the ‘only if part’ of the proposition is trivial. So we move to the ‘if part’. Suppose Z is reflexive. The digraph Z, being of upper triangular type, is antisymmetric. So it remains to prove that Z is transitive. Assume that k →Z l →Z j. As Z is of upper triangular type, we have k ≤ l ≤ j. Also l →Z l by the assumed reflexivity of Z. But then, Z being L-free, we get k →Z j, as desired.  5.2. Relationship of the in/out-diagrams to in/out-ultra closures From [4], Section 7 we borrow the notions in-ultra closure and out-ultra closure. The in-ultra closure Zin of Z is the smallest in-ultra transitive upper triangular-type partial order on N containing Z. Similarly, the outultra closure Zout of Z is the smallest out-ultra transitive upper triangulartype partial order on N containing Z. The inclusions Z ⊂ Zin and Z ⊂ Zout are generally strict, also when Z is L-free. Theorem 5.2. Let Z be a reflexive upper triangular-type directed graph with ground set N . Then Z ⊂ ZinD ⊂ ZTinD ⊂ Zin ,

Z ⊂ ZoutD ⊂ ZToutD ⊂ Zout .

Also, when Z is L-free, then ZTinD = Zin and ZToutD = Zout . In the second part of the theorem, the digraph Z is required to be L-free, reflexive and of upper triangular type, hence it is a is a partial order (see Proposition 5.1).

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101

Proof. We only consider the statements concerning ZinD , ZTinD and Zin . For ZoutD , ZToutD and Zout , the reasoning is similar. Let k →Z l. If k = l, then k < l because Z is of upper triangular type. Since Z is reflexive, we also have l →Z l. Thus k →Z l,

l →Z l,

k < l,

and it follows that k →ZinD l. If k = l, then certainly k →ZinD l. Indeed, ZinD contains all reflexivity loops. Thus Z ⊂ ZinD . As, clearly, ZinD ⊂ ZTinD , we will have established the desired inclusions once it is shown that ZTinD ⊂ Zin . As an intermediate step, we show that ZinD ⊂ Zin . Suppose k →ZinD l. The digraph Zin is a partial order, hence reflexive. So, in order to prove that k →Zin l, we may assume that k = l. Now ZinD is of upper triangular type. So k < l. Take j such that k →Z j and l →Z j. This is possible on account of the definition of ZinD . A fortiori we have k →Zin j and l →Zin j. Now Zin is in-ultra transitive. Hence k →Zin l. The digraph Zin , being a partial order, is transitive. As we have seen, it contains ZinD . But then it is immediate that ZTinD ⊂ Zin . We finish the proof by considering the case when Z is L-free. From Theorem 3.6 we then know that ZTinD is an in-ultra transitive upper triangular-type partial order. As it contains Z, it follows that Zin ⊂ ZTinD . The reverse inclusion was established above. Thus Zin = ZTinD , as desired.  Corollary 5.3. Let Z be an upper triangular-type partial order on N . If Z is in-ultra transitive, then ZTinD = ZinD = Z. Also, if Z is out-ultra transitive, then ZToutD = ZoutD = Z. Proof. If Z is in-ultra transitive, then Zin = Z, if Z is out-ultra transitive, then Zout = Z. Apply now the first part of Theorem 5.2.  5.3. L-closures of upper triangular-type partial orders If Z is a digraph with ground set N , then Z is of upper triangular type if and only Z is contained in the standard linear order on N . From this it is obvious that Z is of upper triangular type if and only if so is its L-closure Z Λ . Proposition 5.4. Let Z or, equivalently, Z Λ be an upper triangular-type directed graph with ground set N . If Z is transitive, then so is Z Λ (hence Cn×n [Z Λ ] is a subalgebra of Cn×n upper , possibly without unit element). Taking the upper triangular-type digraph ⎛ ⎞ 1 2 3 ⎜ ⎟ ⎜1 ∗ ∗ 0⎟ ⎜ ⎟ Z=⎜ ⎟ ⎝2 0 0 ∗⎠ 3 0 0 ∗

(5.1)

as an example, one sees that the transitivity requirement in the proposition is essential. Indeed Z is not transitive and, as Z and Z Λ coincide, Z is not transitive either.

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Proof. Assume that Z Λ is not transitive. Then there exists u, v, w ∈ N such that u →Z Λ v →Z Λ w, but u Z Λ w. Since Z Λ is upper triangular for any such u, v, w we have that u < v < w. Therefore the set ; : T = w − u | u →Z Λ v →Z Λ w, u Z Λ w, u, v, w ∈ N is a nonempty set of positive integers. Let d be its smallest element and choose u, v, w ∈ N such that d = w − u and u →Z Λ v →Z Λ w but u Z Λ w. Note that u Z v or v Z w. Indeed, otherwise we have u →Z v →Z w which implies u →Z w due to the transitivity of Z. Since Z ⊆ Z Λ we obtain u →Z Λ w, contradicting the choice of u, v, w. Our first claim is that p Z Λ v for all u < p ≤ v. Otherwise, there exists p such that u < p ≤ v and p →Z Λ v. We may illustrate the situation in the following contracted matrix diagram, based on the relevant rows and columns of Z Λ , ⎞ ⎛ v w ⎟ ⎜ ⎜u ∗ 0 ⎟ ⎟ ⎜ ⎜p ∗ ? ⎟. ⎠ ⎝ v ∗ Now one can ask whether p →Z Λ w or p Z Λ w. If the former is the case, so p →Z Λ w, then u →Z Λ v ←Z Λ p →Z Λ w,

u < p,

v < w,

which gives u →Z Λ w since Z Λ is L-free by definition. This is a contradiction. On the other hand, if p Z Λ w, then p →Z Λ v →Z Λ w while w − p < w − u and we get a contradiction to the minimality condition with the choice of (u, v, w). Indeed, the triple (p, v, w) would yield a number smaller than d. Therefore we have proved that p Z Λ v for all p satisfying u < p ≤ v. In a completely analogous manner it is possible to prove a second claim, namely that v Z Λ q for all v ≤ q < w. Recall that we have u Z v or v Z w. In case u Z v one can get a contradiction to the first claim, in case v Z w one can get a contradiction to the second claim. The arguments are similar to each other and without loss of generality we consider the first case only. Therefore, let us assume that u Z v. Put Y = Z Λ \ {(u, v)}. Then Y ⊇ Z. We are now going to show that Y, which is a proper subset of Z Λ , is L-free. But this is impossible due to the definition of L-closure. Here is the argument. Suppose Y is not L-free and notice that Z Λ = Y ∪ {(u, v)} is L-free. The only way this can occur is that the L in Y is given by some quadruple (u, u , v  , v) with certain u , v  ∈ N , u < u , v  < v. The situation can be

L-free bipartite graphs and echelon-type canonical forms illustrated by the contracted respectively ⎛ v ⎜u ∗ ⎝  u ∗

103

matrix diagrams corresponding to Y and Z Λ , v

⎞

0 ⎟ ⎠, ∗

⎞

⎛

v

v

⎜u ⎝ u

∗ ∗

∗ ⎟ ⎠. ∗

But from there we see that u →Z Λ v with u < u , while u ≤ v due to the upper triangularity of Z Λ . But this contradicts the first claim if we take p = u .  The first part of our next result is a counterpart to Proposition 5.4. Proposition 5.5. Let Z or, equivalently, Z Λ be an upper triangular-type directed graph with ground set N . If Z is reflexive, then Z Λ is an upper triangular-type partial order (hence Cn×n [Z Λ ] is a subalgebra of Cn×n upper having In as unit element). Also (Z Λ )TinD = (Z Λ )in = Zin ,

(Z Λ )ToutD = (Z Λ )out = Zout .

As can be seen from (5.1), the reflexivity condition in the proposition cannot be dropped. Proof. The digraph Z Λ is reflexive because it contains Z which is reflexive by hypothesis. Also Z Λ is L-free and of upper triangular type. But then Z Λ is a partial order by Proposition 5.1. Now apply Theorem 5.2 to Z Λ . This gives (Z Λ )TinD = (Z Λ )in and (Z Λ )ToutD = (Z Λ )out . It remains to prove that (Z Λ )in = Zin and (Z Λ )out = Zout . The partial order (Z Λ )in is of upper triangular type and contains Z Λ . A fortiori it contains Z. Also (Z Λ )in is in-ultra transitive. Hence Zin ⊂ (Z Λ )in . According to Proposition 3.2, the partial order Zin is L-free. It contains Z. Therefore Z Λ ⊂ Zin . But then (Z Λ )in ⊂ Zin as well. We conclude that (Z Λ )in = Zin , as desired. The identity (Z Λ )out = Zout can be obtained in the same way.  The upper triangularity condition in Propositions 5.4 and 5.5 is essential. This appears from the following simple example of a partial order Z and its L-closure, ⎛ ⎞ ⎛ ⎞ 1 2 3 1 2 3 ⎜ ⎟ ⎜ ⎟ ⎜1 ∗ 0 0 ⎟ ⎜1 ∗ ∗ ∗ ⎟ Λ ⎟, ⎜ ⎟, Z=⎜ Z = ⎜ ⎟ ⎜ ⎟ ⎝2 ∗ ∗ 0 ⎠ ⎝2 ∗ ∗ ∗ ⎠ 3

∗

0

∗

3

∗

0

∗

in which Z Λ is not transitive (3 →Z Λ 1 →Z Λ 2, but 3 Z Λ 2). Furthermore, the second statement in Proposition 5.5 cannot be extended to include (Z Λ )inD or (Z Λ )outD . In other words, the in-diagram or out-diagram of Z Λ need not be transitive. This becomes clear by taking Z

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as indicted below ⎛ ⎜ ⎜1 ⎜ ⎜ Z = ZΛ = ⎜ 2 ⎜ ⎜ ⎝3 4

1

2

3

∗

0

0

0

∗

∗

0

0

∗

0

0

0

4

⎞

⎟ ∗⎟ ⎟ ⎟ ∗ ⎟, ⎟ ⎟ 0⎠ ∗

⎛

ZinD = (Z Λ )inD

⎜ ⎜1 ⎜ ⎜ = ⎜2 ⎜ ⎜ ⎝3 4

1

2

3

∗

∗

0

0

∗

∗

0

0

∗

0

0

0

4

⎞

⎟ ∗⎟ ⎟ ⎟ ∗ ⎟. ⎟ ⎟ 0⎠ ∗

Note that Z is an (L-free) upper triangular-type partial order, but the (coinciding) digraphs ZinD and (Z Λ )inD are not transitive (1 →Z Λ 2 →Z Λ 3, but 1 Z Λ 3). In line with Proposition 5.5, their transitive closures coincide with the in-ultra closure of Z = Z Λ . 5.4. Canonical forms Recall from Proposition 5.1 that an L-free upper triangular-type directed graph is a partial order if and only if it is reflexive. Theorem 5.6. Let Z be an L-free upper triangular-type partial order with ground set {1, . . . , n}. Then for each matrix M in the linear space Cn×n [Z], its canonical forms ↑ ↑ → → MrEch , McEch , MmrEch , MmcEch , MEchfr , Mfr

all belong to Cn×n [Z]. Also, there are left, respectively right, equivalence matrices corresponding to these canonical forms that belong to Cn×n [Zin ], respectively Cn×n [Zout ]. As Zin and Zout are partial orders of upper triangular type, Cn×n [Zin ] and Cn×n [Zout ] are subalgebras of Cn×n upper containing the n×n identity matrix In as unit element. Proof. Combine Theorems 4.1 and 5.2.



From the above theorem one can quickly recover the results presented in [4], Section 7. Here is an indication of how this goes. Let Z be an upper triangular-type partial order on N , and suppose Z is in-ultra transitive. Then Z is L-free by Proposition 3.2. So we can apply Theorem 5.6. This ↑ gives, among other things, that MrEch is in (the algebra) Cn×n [Z] and that ↑ MrEch = LM for some monic matrix L ∈ Cn×n [Zin ]. However, in the present situation Zin = Z. So L ∈ Cn×n [Z]. With this we have obtained Theorem 7.1 in [4]. For the other results in [4], Section 7 the reasoning is similar. Corollary 5.7. Suppose Z is a reflexive upper triangular-type directed graph with ground set {1, . . . , n}. Then for each matrix M in the linear space Cn×n [Z], its canonical forms ↑ ↑ → → MrEch , McEch , MmrEch , MmcEch , MEchfr , Mfr

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105

all belong to Cn×n [Z Λ ]. Also, there are left, respectively right, equivalence matrices corresponding to these canonical forms that belong to Cn×n [Zin ], respectively Cn×n [Zout ]. For the proof, employ Theorem 5.6 and Proposition 5.5. As Zin and Zout are upper triangular-type partial orders, Cn×n [Zin ] and Cn×n [Zout ] are subalgebras of Cn×n upper containing the n×n identity matrix In as unit element. Corollary 5.7 covers several of the results of [4], Section 7, for instance Corollary 7.3 and 7.7. In case the digraph Z happens to be an in-ultra respectively an out-ultra, transitive partial order, then Zin = Z, Cn×n [Zin ] = Cn×n [Z], respectively Zout = Z, Cn×n [Zout ] = Cn×n [Z]; cf. the remark made after the proof of Theorem 5.6.

6. Application: counting equivalence classes We begin with an introductory remark. Let M, N ∈ Cm×n where, as before, m and n are positive integers. We call N equivalent to M – written N  M – if there exist invertible matrices L ∈ Cm×m and R ∈ Cn×n such that N = LM R. Clearly  is an equivalence relation in Cm×n . For M ∈ Cm×n , the -equivalence class in Cm×n containing M is denoted by [M ] . It is common knowledge that N  M if and only if N and M have the same rank, in standard notation rank N = rank M . The -equivalence classes in Cm×n are given by {M ∈ Cm×n | rank M = r}, r = 0, . . . , min{m, n}. Hence {[M ] | M ∈ Cm×n } = {rank M | M ∈ Cm×n },

(6.1)

were  is short-hand for cardinality. So the number of -equivalence classes in Cm×n is min{m, n} + 1, in other words {[M ] | M ∈ Cm×n } = min{m, n} + 1.

(6.2)

After these introductory remarks, we now focus on a different (and more restricted) type of equivalence. Let M, N ∈ Cm×n . We call N upper equivalent to M – written N upp M – if there exist invertible matrices L ∈ Cm×m upper and R ∈ Cn×n upper such that N = LM R. Clearly upp is an equivalence relation in Cm×n . For M ∈ Cm×n , the upp -equivalence class – also called the upper equivalence class – in Cm×n containing M is denoted by [M ]upp . If N upp M , then N  M as well. Hence [M ]upp is contained in [M ] , and the number of upp -equivalence classes in Cm×n is larger than or equal to number of  -equivalence classes in Cm×n , i.e., {[M ]upp | M ∈ Cm×n } ≥ {[M ] | M ∈ Cm×n }. In case m ≥ 2 or n ≥ 2, the inequality is strict. When m = n = 1 there is equality, both sides being equal to 2. Recall that when M ∈ Cm×n we write Mfr for the canonical (monic echelon) frame of M (cf. the last paragraph of Subsection 2.1). This canonical frame is an m × n matrix such that all its rows and all its columns have at most one nonzero entry, which is then equal to one.

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Let M, N ∈ Cm×n . From Theorem 2.7 we see that N upp M if and only if the canonical frames of N and M coincide. In other words, [N ]upp = [M ]upp if and only if Nfr = Mfr . If M and N happen to be monic echelon frames themselves, then [N ]upp = [M ]upp if and only if N = M . Indeed, in this situation Nfr = N and Mfr = M . Thus we have the following analogue of (6.1) {[M ]upp | M ∈ Cm×n } = {F | F monic echelon frame in Cm×n }. In words: counting the number of upper equivalence classes in Cm×n comes down to determining the number of monic echelon frames in Cm×n . It is easy to see that the number of such frames in Cm×n having rank r is equal to    m n r! . r r It follows that 

min{m,n}

{[M ]upp | M ∈ Cm×m } =

r=0

r!

   m n , r r

which is the counterpart of (6.2). We remark that this sum can be expressed as (−1)m U (−m, n+1−m, −1) = (−1)n U (−n, m+1−n, −1) where U (a, b, z) is the Tricomi confluent hypergeometric function. Given a (nonempty) subset C of Cm×n , we may ask for the number of upper equivalence classes corresponding to the matrices in C, i.e., for the number {[M ]upp | M ∈ C}. In this generality, we can only say that {[M ]upp | M ∈ Cn×n } = {Mfr | M ∈ C} ≥ {F | F monic echelon frame in C}, and that, as can be seen from trivial examples, it is possible that the latter inequality is strict. The latter cannot be the case, in other words there is equality, if and only if for each matrix in C, its canonical frame belongs to C as well. Taking into account Theorem 4.1, we now arrive at the following result. Theorem 6.1. If Z is an L-free directed bipartite graph from M to N , then {[M ]upp | M ∈ Cm×n [Z]} = {F | F monic echelon frame in Cm×n [Z]}.

(6.3)

Thus in the situation of the theorem, counting the number of upper equivalence classes associated with the matrices in Cm×n [Z] comes down to determining the number of monic echelon frames in Cm×n [Z]. We close this section with some examples. Example 5. Let L be the standard linear order on {1, . . . , n}. Then L is L-free (but not N-free) and Cn×n [L] = Cn×n upper . So (6.3) stands here for the number of upper equivalence classes in the algebra Cn×n upper . The number in question is equal to the number of monic echelon frames in Cn×n upper . Theorem 9.2 in [4]

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says that it is equal to what is called the (n + 1)-th Bell number Bn+1 . Using a well-known expression for the latter, we obtain

{[M ]upp | M ∈ Cn×n upper } =

n+1  k=0

  k  (−1)k−s k n+1 . s m! s s=0

A wealth of other expressions for the Bell numbers (and the related Stirling numbers of the second kind) is available in the literature (cf. [7], [12], [13]). In [4], the result indicated above, involves a Pascal-type triangle and the familiar recurrence relation holding for the Stirling numbers of the second kind. These numbers are intimately related to counting partitions of finite sets. This suggests that there should be also a relation of the latter with counting the number of monic echelon frames. And indeed there is. It is possible to prove that there is a one-to-one correspondence between the collection of rank r monic echelon frames in Cn×n upper and the collection of all possible partitions of the set {0, 1, . . . , n} into n + 1 − r subsets. The description of this correspondence is not completely straightforward. We refrain from giving it here but intend to present it somewhere else ([5]). The result in question can be used to given an alternative (though not simpler) proof of [4], Theorem 9.2.  Example 6. Let n, m ≥ 2, and let S(m, n) stand for the directed bipartite graph from M = {1, . . . , m} to N = {1, . . . , n} given by 

 {(k, l) | k = 1, m, l = 1, . . . , n} ∪ {(s, t) | s = 1, . . . , m, t = 1, n} \{(m, 1)}.

By way of illustration, we give the matrix diagram for m = 5 and n = 7 ⎛ ⎜ ⎜1 ⎜ ⎜ ⎜2 S(5, 7) = ⎜ ⎜ ⎜3 ⎜ ⎜4 ⎝ 5

1

2

3

4

5

6

∗

∗

∗

∗

∗

∗

∗

0

0

0

0

0

∗

0

0

0

0

0

∗

0

0

0

0

0

0

∗

∗

∗

∗

∗

7

⎞

⎟ ∗ ⎟ ⎟ ⎟ ∗ ⎟ ⎟. ⎟ ∗ ⎟ ⎟ ∗ ⎟ ⎠ ∗

Then the bipartite graph S(m, n) is L-free (but not N-free). Note the emphasized zero in the left lower corner. Returning to the general case, write φr (m, n) for the number of monic echelon frames in Cm×n (S(m, n)) having rank r. A little reflection reveals

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that

⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎪ 2m + 2n − 5, ⎪ ⎪ ⎪ ⎪ ⎨ 4(m − 2)(n − 2) + m2 + n2 − 2m − 2n + 1, φr (m, n) = ⎪ (m − 2)(n − 2)(2m + 2n − 9), ⎪ ⎪ ⎪ ⎪ ⎪ (m − 2)(n − 2)(mn − 3m − 3n + 9) ⎪ ⎪ ⎪ ⎩ 0,

r = 0, r = 1, r = 2, r = 3, r = 4, r ≥ 5.

Hence {[M ]upp | M ∈ Cm×n [S(m, n)]} = {F | F monic echelon frame in Cm×n [S(m, n)]} = φ0 (m, n) + φ1 (m, n) + φ2 (m, n) + φ3 (m, n) + φ4 (m, n) = (m − 2)(n − 2)(mn − m − n + 4) + (m2 + n2 − 3) = m2 n2 − 3m2 n − 3mn2 + 3m2 + 3n2 + 12mn − 12m − 12n + 13. For completeness, note that the (transitive closure) of the in-diagram of S(m, n) is the standard linear order L(M ) on M , i.e., S(m, n)inD = S(m, n)TinD = L(M ). Similarly S(m, n)outD = S(m, n)ToutD = L(N ) where L(N ) is the standard linear order on N . As an afterthought, we mention that adding the pair (m, 1) to S(m, n), the corresponding graph is not L-free any more. Its L-closure ‘explodes’ to M × N.  Example 7. Let D(n) be the upper triangular-type partial order on N given by D(n) = {(1, k) | k ∈ N } ∪ {(k, n) | k ∈ N } ∪ {(k, k) | k ∈ N }. This digraph has been considered too in [2]. By way both the arrow and matrix diagram for n = 6: ⎛ 1 1 ⎜ ⎜1 ∗ ⎜ ⎜2 0 ⎜ ⎜ ⎜3 0 2 3 4 5 ⎜ ⎜ ⎜4 0 ⎜ ⎜ ⎝5 0 6

6

0

of illustration, we give 6

⎞

2

3

4

5

∗

∗

∗

∗

∗

0

0

0

0

∗

0

0

0

0

∗

0

0

0

0

∗

⎟ ∗⎟ ⎟ ∗⎟ ⎟ ⎟ ∗⎟ ⎟. ⎟ ∗⎟ ⎟ ⎟ ∗⎠

0

0

0

0

∗

In the arrow diagram, again the reflexivity loops are ignored. Inspection shows that the partial order D(n) is L-free (but not N-free for values of n ≥ 2). For n one of the integers 1, 2 and 3, it is just the standard linear order on N .

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We now turn to determining the number {[M ]upp | M ∈ Cn×n [D(n)]} = {F | F monic echelon frame in Cn×n [D(n)]}.

(6.4)

For n = 1, the outcome is evident, namely 2. For n = 2, inspection shows that it is 5. Actually, for n ≥ 2, the number featuring in (6.4) is equal to 2n−4 (n2 + 5n + 6).

(6.5)

In the argument given below we may assume that n ≥ 3. Indeed, substituting n = 2 in 2n−4 (n2 + 5n + 6) yields 5. Decompose the set D(n) into the following four disjoint sets, : ; H(n) = (1, k) | k = 2, . . . n − 1 , : ; V (n) = (k, n) | k = 2, . . . n − 1 , : ; D(n) = (k, k) | k = 2, . . . n − 1 , T (n) = {(1, 1), (1, n) (n, 1)}. Now each monic echelon frame F in Cn×n [D(n)] belongs to precisely one of the following classes: (i) F has zero entries at all positions in H(n) ∪ V (n). Then there are 5 possibilities for placing zeros or ones at the positions in T (n), and independently, there are 2n−2 possibilities for putting zeros or ones at the n − 2 positions in D(n). Thus there are precisely 5 · 2n−2 such frames. (ii) F has a one at one position in H(n) and zeros at all positions in V (n). Then there are 2 possibilities for placing zeros or ones at the positions in T (n), and independently, there are 2n−3 possibilities for putting zeros or ones at the n − 2 positions in D(n). (Note that one of the diagonal entries must always be zero due to the one in H(n).) Moreover, the one in H(n) can occur at precisely n − 2 places. Hence there are precisely 2(n − 2)2n−3 such frames. (iii) F has zeros at all positions in H(n) and a one at one position in V (n). In the same way we get that the number of such frames is precisely 2(n − 2)2n−3 . (iv) F has a one at position (1, k) in V (n) and a one at position (k, n) in H(n), for some k = 2, . . . , n − 1. The positions in T (n) must all be zero, and the diagonal positions are freely available except for the single position (k, k). Hence there are (n − 2)2n−3 frames of this kind. (v) F has a one at position (1, k) in V (n) and a one at position (l, n) in H(n), for k, l = 2, . . . n−1 with k = l. In this case, the diagonal positions (k, k) and (l, l) must be zero. Thus there are precisely (n−2)(n−3)2n−4 frames of this kind.

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H. Bart, T. Ehrhardt and B. Silbermann Adding up all the quantities obtained in the cases (i)–(v), we obtain

5 · 2n−2 + 2(n − 2)2n−3 + 2(n − 2)2n−3 + (n − 2)2n−3 + (n − 2)(n − 3)2n−4 which equals 2n−4 (n2 + 5n + 6). Analogous to what we observed in Example 6, we have D(n)inD = D(n)TinD = L(N ) and, likewise, D(n)outD = D(n)ToutD = L(N ) where L(N ) is the standard linear order on N .  In Examples 5 and 7, the digraphs under consideration are transitive. Hence the corresponding linear subspaces of Cn×n are actually subalgebras of Cn×n . This gives rise to a remark. Let Z be an upper triangular-type reflexive transitive digraph on {1, . . . , n}. Then the corresponding algebra Cn×n [Z] is a subalgebra of Cn×n upper containing In as unit element. Let M, N ∈ Cn×n [Z]. We say that N is Z-equivalent to M – written N Z M – if there exist invertible matrices L, R ∈ Cn×n [Z] such that N = LM R. Clearly Z is an equivalence relation in Cn×n [Z]. If N Z M , then N upp M as well. For M ∈ Cn×n [Z], the Z -equivalence class in Cn×n [Z] containing M is denoted by [M ]Z . It is obviously contained in [M ]upp . Hence the number of Z -equivalence classes in Cn×n [Z] is larger than or equal to number of upper equivalence classes in n×n Cn×n [Z], i.e., upper associated with the matrices from C {[M ]Z | N ∈ Cn×n [Z]} ≥ {[M ]upp | N ∈ Cn×n [Z]}.

(6.6)

The inequality is strict if and only if there exist K, L ∈ Cn×n [Z] such that K upp L but not K Z L. Anticipating on what we shall see in Examples 8 and 9 below, we mention that such a situation can indeed occur. It cannot, however, when Z is ultra transitive. Let us elaborate on this. Recall that the upper triangular-type partial order Z is ultra transitive when it is both in-ultra transitive and out-ultra transitive. Suppose this is the case, and take M ∈ Cn×n [Z]. Corollary 7.11 in [4] then says that Mfr belongs to Cn×n [Z] and Mfr is Z-equivalent to M . Now, if K, L ∈ Cn×n [Z] and K upp L, then Kfr = Lfr , hence K Z Kfr = Lfr Z L, so K Z L. It follows that [M ]Z = [M ]upp ∩ Cn×n [Z],

M ∈ Cn×n [Z],

and we also get {[M ]Z | M ∈ Cn×n [Z]} = {[M ]upp | M ∈ Cn×n [Z]} = {Mfr | M ∈ Cn×n [Z]} = {F | F monic echelon frame in Cn×n [Z]}. As to the latter identity, recall that Ffr = F for all monic echelon frames F in n×n [Z]. The upshot of all of this Cn×n upper , so a fortiori for all such frames in C is that, when Z is ultra transitive, counting the number of Z -equivalence classes in Cn×n [Z] comes down to determining the (finite) number of monic echelon frames in Cn×n [Z]. Proposition 9.3 in [4] states that the outcome of

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the counting process in question is always a product of Bell numbers. For a concrete example, see [4], Section 9 (pages 110 and 111 in particular), the Bell numbers being there 2 and 5. The standard linear order L featuring in Example 5 is ultra transitive too. We close this section with two examples, both showing that the inequality (6.6) can be strict when Z fails to be ultra transitive. In the first example (Example 8), the left hand side of (6.6) is finite, in the second (Example 9) it is infinite. Example 8. Let Z be the upper triangular-type partial order on {1, 2, 3, 4} given by ⎞ ⎛ 1 2 3 4 ⎟ ⎜ ⎜1 ∗ ∗ ∗ ∗ ⎟ ⎟ ⎜ ⎟ ⎜ Z = ⎜2 0 ∗ 0 ∗ ⎟, ⎟ ⎜ ⎜3 0 0 ∗ ∗ ⎟ ⎠ ⎝ 4 0 0 0 ∗ i.e., Z = D(4) from Example 7. As was already observed there the partial order Z is L-free. It is not ultra transitive, however. In fact, Z is neither inultra transitive (2 →Z 4, 3 →Z 4, 2 Z 3) nor out-ultra transitive (1 →Z 2, 1 →Z 3, 2 Z 3). Put ⎤ ⎡ ⎤ ⎡ 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎢0 0 0 0 ⎥ ⎢0 0 0 1 ⎥ ⎥. ⎢ ⎥ ⎢ N =⎢ M =⎢ ⎥ ⎥, ⎣0 0 0 1 ⎦ ⎣0 0 0 1 ⎦ 0 0 0 0 0 0 0 0 Then M, N ∈ C4×4 [Z]. As

⎡

1

⎢ ⎢0 N =⎢ ⎢0 ⎣ 0

0

0

1

−1

0

1

0

0

0

⎤

⎥ 0⎥ ⎥ M, 0⎥ ⎦ 1

we have N upp M and, in fact, N = Mfr . Note that the first term in the above product is not in C4×4 [Z] while, nevertheless, N = Mfr ∈ C4×4 [Z], the latter corroborating Theorem 4.1. One easily verifies that [N ]upp is the set of all 4 × 4 matrices of the type ⎤ ⎡ 0 0 0 u ⎥ ⎢ ⎢0 0 0 v ⎥ ⎥ ⎢ u, v, w ∈ C, w = 0, ⎢0 0 0 w ⎥, ⎦ ⎣ 0 0 0 0

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which, by the way, is contained in C4×4 [Z]. Also, [N ]Z consists of all 4 × 4 matrices ⎤ ⎡ 0 0 0 x ⎥ ⎢ ⎢0 0 0 0 ⎥ ⎥ ⎢ x, y ∈ C, y = 0. ⎢0 0 0 y ⎥, ⎦ ⎣ 0 0 0 0 In particular M ∈ / [L]Z or, stated otherwise, N = Mfr is not Z -equivalent to M . The upshot of all of this is that ; : ; :  [M ]Z | M ∈ C4×4 [Z] >  [M ]upp | M ∈ C4×4 [Z] ,

(6.7)

affirming what was already mentioned after the inequality (6.6). As Z is the partial order D(4) from Example 7, it can be derived from (6.5) that the right hand side of (6.7) is 42. The left hand side of (6.7) is finite (too) and does not exceed 512 (but see below). This follows from the fact – the proof of which is left to the reader – that a matrix in Cn×n [Z] is always Z-equivalent to a matrix in Cn×n [Z] featuring only ones as nonzero entries.  Elaborating on the last observation in the above example, we mention that for D(n) as in Example 7, the number of D(n) -equivalence classes in the algebra Cn×n [D(n)] is finite, in fact equal to 2 when n = 1 and at most 23(n−1) = 8n−1 when n ≥ 2. Indeed, generalizing what was said in Example 8, it can be proved that a matrix in Cn×n [D(n)] is always D(n)-equivalent to a matrix in Cn×n [D(n)] featuring only ones as nonzero entries. A more refined analysis (the details of which are omitted) gives that the number of D(n) -equivalence classes equals dn = 2 · 2n−2 + 2 · 3n−2 + 5n−2 ,

n ≥ 2.

Note that d2 = 5, d3 = 15, d4 = 51, and d5 = 195. Example 9. Let Z be the upper triangular-type given by the arrow and matrix diagrams: ⎛ 1 1 2 ⎜ ⎜1 ∗ ⎜ ⎜ ⎜2 0 ⎜ ⎜3 0 ⎝ 3 4 4 0

partial order on {1, 2, 3, 4} 2

3

0

∗

∗

∗

0

∗

0

0

4

⎞

⎟ ∗⎟ ⎟ ⎟ ∗ ⎟. ⎟ 0⎟ ⎠ ∗

Once more, the reflexivity loops are not shown. Clearly the partial order Z is L-free. It is not ultra transitive, however. In fact, Z is neither in-ultra transitive (1 →Z 3, 2 →Z 3, 1 Z 2) nor out-ultra transitive (1 →Z 3, 1 →Z 4, 3 Z 4). A simple (but tedious) count shows that the number of monic echelon frames in C4×4 [Z] is 34. Thus {[M ]upp | M ∈ C4×4 [Z]}, that is the number

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of left/right equivalence classes in C4×4 upper associated with the matrices in 4×4 4×4 the subalgebra C [Z] of Cupper , is equal to 34. However, the number of Z -equivalence classes is infinite. Here is the reasoning. For η ∈ C, put ⎡ ⎤ 0 0 1 1 ⎢ ⎥ ⎢0 0 1 η⎥ ⎢ ⎥. Z(η) = ⎢ ⎥ ⎣0 0 0 0⎦ 0 0 0 0 Then Z(η) ∈ C4×4 [Z]. Obviously {[N ]Z | N ∈ C4×4 [Z]} ≥ {[Z(η)]Z | η ∈ C}. So it suffices to show that the cardinality in the right hand side of this inequality is infinite. In fact, as we shall see, it is the same as the cardinality of C, hence uncountable. Our task has been fulfilled, once we have established that Z(η1 ) Z Z(η2 ) implies η1 = η2 . Assume there exist invertible matrices L and R in C4×4 [Z] such that Z(η2 ) = LZ(η1 )R. Write ⎡ ⎡ ⎤ ⎤ l1,1 r1,1 0 l1,3 l1,4 0 r1,3 r1,4 ⎢ ⎢ ⎥ ⎥ ⎢ 0 l2,2 l2,3 l2,4 ⎥ ⎢ 0 r2,2 r2,3 r2,4 ⎥ ⎢ ⎢ ⎥ ⎥. , R=⎢ L=⎢ 0 l3,3 0 ⎥ 0 r3,3 0 ⎥ ⎣ 0 ⎣ 0 ⎦ ⎦ 0 0 0 l4,4 0 0 0 r4,4 A straightforward (2 × 2 block) calculation gives      1 1 l1,1 0 1 1 r3,3 = 1 η2 0 l2,2 1 η1 0  =

l1,1 r3,3

l1,1 r4,4

l2,2 r3,3

η2 l2,2 r4,4

0



r4,4

 ,

which amounts to the identities l1,1 r3,3 = l1,1 r4,4 = l2,2 r3,3 = 1,

η2 l2,2 r4,4 = η1 .

But then η1 = η2 l2,2 (l1,1 r3,3 )r4,4 = η2 (l2,2 r3,3 )(l1,1 r4,4 ) = η2 , as desired.



7. Concluding remarks and open problems This section contains some remarks on the material presented above. Also, possibilities for further research are identified. As before M = {1, . . . , m} and N = {1, . . . , n}, where m and n are given positive integers.

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7.1. L-free directed bipartite graphs: characterization In Subsection 3.3, it was mentioned that in-ultra transitivity and out-ultra transitivity can be characterized in terms of rooted trees. Indeed, it has been proved in [3] that a partial order is in-ultra transitive, respectively out-ultra transitive, if and only if its Hasse diagram is an out-tree, respectively an in-tree. From Proposition 3.2 we know that for reflexive digraphs, the property of being L-free generalizes both in-ultra transitivity and outtransitivity. Thus the question arises whether being L-free also allows for an adequate characterization in terms of the Hasse diagram. This question gets further profile by the fact that Theorem 2 in [9] contains a characterization of N-free partial orders involving the Hasse diagram too. For a discussion on the relationship between super transitivity and the property of being N-free we again refer to Subsection 3.3, see Proposition 3.1 in particular. Proposition 3.7 gives a necessary and condition for being L-free in terms of the L-reduction). This, however, is not a substantial characterization. Moreover, we have reason to believe that a characterization of the property of being L-free in terms of the Hasse diagram might be out of reach. It is possible, though, to obtain a quite different type of characterization via the notion of echelon compatibility hinted at in the abstract. A directed bipartite graph Z on from M to N is called echelon compatible if for each M ∈ Cm×n all ↑ ↑ → → the canonical forms MrEch , McEch , MmrEch , MmcEch , MEchfr and Mfr belong to m×n C [Z]. Theorem 7.1. Suppose Z is a directed bipartite graph from M to N . Then Z is L-free if and only if Z is echelon compatible. In fact, the following statements are equivalent: (1) Z is L-free. ↑ (2) For each M ∈ Cm×n [Z], the canonical upward echelon form MrEch of m×n M belongs to C [Z]. (3) For each M ∈ Cm×n [Z], the canonical monic upward echelon form ↑ MmrEch of M belongs to Cm×n [Z]. → (4) For each M ∈ Cm×n [Z], the canonical starboard echelon form McEch of m×n M belongs to C [Z]. (5) For each M ∈ Cm×n [Z], the canonical monic starboard echelon form → McEch of M belongs to Cm×n [Z]. (6) For each M ∈ Cm×n [Z], the canonical echelon frame MEchfr of M belongs to Cm×n [Z]. (7) For each M ∈ Cm×n [Z], the canonical (monic echelon) frame Mfr of M belongs to Cm×n [Z]. Proof. Assume Z is not L-free. This means that there are p, q ∈ M and r, s ∈ N such that p →Z r ←Z q →Z r and p < q, r < s, but p Z s. Now introduce the m × n matrix M by stipulating that all its entries are zero except for those on the positions (p, r), (q, r) and (q, s) where the values are −1, 1 and 1, respectively. Then M ∈ Cm×n [Z], but none of the matrices

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↑ ↑ → → MrEch , McEch , MmrEch , MmcEch , MEchfr and Mfr is. Indeed, they all have a nonzero entry (with value 1) at the position (p, s) while p Z s.

The upshot of this is that each of (2)–(7) implies (1). From Theorem 4.1 we know that, conversely, (1) implies each of (2)–(7). But then the statements (1)–(7) are all equivalent. 

7.2. Issues for further research As before, let m and n be positive integers. A very common type of equivalence relation in Cm×n comes about by considering left multiplication by invertible matrices in Cn×n , right multiplication by invertible matrices in Cm×m , or by multiplication on both the left and the right by invertible matrices in Cm×m and Cn×n , respectively. In the present paper and in [4] (there for the case m = n), we focussed on what happens when one takes into account only invertible, or even monic, upper triangular matrices. In other m×m words, when only invertible, or even monic, elements from the algebras Cupper n×n and of Cupper are admitted. We also encountered situations where the equivalence matrices happened to lie in a zero pattern algebra of upper triangular matrices. In view of all of this, we now raise the possibility of defining canonical forms for matrices in Cm×n when using only invertible matrices from a given zero pattern algebra (or algebras when we allow both multiplication from the left and the right). To make the issue more concrete, let Z be directed bipartite graph from M = {1, . . . , m} to N = {1, . . . , n} so that Cn×n [Z] is a linear subspace of Cm×n . Following up on a definition given in Section 6, we now introduce left Z-equivalence, right Z-equivalence, and left/right Z-equivalence, denoted by lZ , Zr and lZr . For M, N ∈ Cm×n this means that N lZ M , N Zr M , N lZr M , if respectively, there exists an invertible L ∈ Cm×m [Z], an invertible R ∈ Cn×n [Z] and invertible matrices L ∈ Cm×m [Z], R ∈ Cn×n [Z] such that N = LM, N = M R and N = LM R. In this way, indeed, equivalence relations are obtained. The question is now: is it possible to identify ‘canonical’ representatives for the corresponding equivalence classes? Or, given a subset S of Cm×n (for instance a linear subspace generated by a directed bipartite graph), can one identify such representatives for the classes corresponding to the matrices in S? In case the given directed bipartite digraph Z actually is an upper triangular-type partial order, a refinement can be made by considering monic (upper triangular invertible) equivalence matrices. We will not pursue these issues here but bring them up as directions for further research. Still we do want to point out another possible line of investigation which might have some bearing on what was said above or perhaps is of interest in its own right.

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Returning to the beginning of Subsection 3.3, we bring back to mind that a directed bipartite graph Z from M to N is L-free if and only if ⎫ p, q ∈ M and r, s ∈ N ⎪ ⎪ ⎬ p ≤ q and r ≤ s ⇒ p →Z s. ⎪ ⎪ ⎭ p → Z r ←Z q →Z s Denoting the underlying linear orders in M and N by L(M ) and L(N ), respectively, we can rewrite this as ⎫ p, q ∈ M and r, s ∈ N ⎪ ⎪ ⎬ p →L(M ) q and r →L(N ) s ⇒ p →Z s. (7.1) ⎪ ⎪ ⎭ p →Z r ←Z q →Z s By allowing L(M ) and L(N ) to be arbitrary digraphs with ground sets M and N , respectively, it is possible to introduce the property of being L-free relative to two underlying digraphs, just by stipulating that (7.1) must be fulfilled. For the case when m = n and Z, L(M ) and L(N ) are all partial orders, in line with the discussion in the previous subsection, one may ask whether there is an interesting characterization of this new notion in terms of the Hasse diagrams of Z, L(M ) and L(N ). This question derives its weight from the fact that partial orders are uniquely determined by their Hasse diagrams. Characterizations of a different type might also be of interest. As an afterthought we mention that the directed bipartite graph Z from M to N is N-free if and only if it is L-free relative to the complete directed graphs on M and N .

References [1] H. Bart, T. Ehrhardt, B. Silbermann, Logarithmic residues in Banach algebras, Integral Equations and Operator Theory 19 (1994), 135–152. [2] H. Bart, T. Ehrhardt, B. Silbermann, Sums of idempotents and logarithmic residues in zero pattern matrix algebras, Linear Algebra Appl. 498 (2016), 262– 316. [3] H. Bart, T. Ehrhardt, B. Silbermann, Rank decomposition in zero pattern matrix algebras, Czechoslovak Mathematical Journal 66 (2016), 987–1005. [4] H. Bart, T. Ehrhardt, B. Silbermann, Echelon type canonical forms in upper triangular matrix algebras, Operator Theory: Advances and Applications 259, 79–124 (2017). [5] H. Bart, T. Ehrhardt, B. Silbermann, Stirling numbers of the second kind and equivalence classes of upper triangular matrices, in preparation. [6] G. Birkhoff, Lattice Theory. Third edition, American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967.

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[7] C.A. Charalambides, Enumerative combinatorics, CRC Press Series on Discrete Mathematics and its Applications, Chapman and Hall/CRC, Boca Raton, FL, 2002. [8] R.L. Davis, Algebras defined by patterns of zeros, J. Combinatorial Theory 9 (1970), 257–260. [9] M. Habib, R. Jegou, N -free posets as generalizations of series-parallel posets, Discrete Appl. Math. 12 no. 3 (1985), 279–291. [10] R.A. Horn, C.R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1994. [11] T.J. Laffey, A structure theorem for some matrix algebras, Linear Algebra Appl. 162–164 (1992), 205–215. [12] J. Riordan, Combinatorial identities, John Wiley and Sons, Inc., New York– London–Sydney, 1968. [13] R.P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge University Press, Cambridge, 1997. [14] E. Szpilrajn (later called E. Marczewski), Sur l’extension de l’ordre partiel, Fundamenta Mathematicae 16 (1930), 386–389 [French]. (Available at http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm16125.pdf.) Harm Bart Econometric Institute, Erasmus University Rotterdam P.O. Box 1738, 3000 DR Rotterdam The Netherlands e-mail: [email protected] Torsten Ehrhardt Mathematics Department, University of California Santa Cruz, CA-95064 U.S.A. e-mail: [email protected] Bernd Silbermann Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at Chemnitz 09107 Chemnitz Germany e-mail: [email protected]

Extreme individual eigenvalues for a class of large Hessenberg Toeplitz matrices J.M. Bogoya, S.M. Grudsky and I.S. Malysheva Abstract. In a previous work we studied the asymptotic behavior of individual inner eigenvalues of the n-by-n truncations of a certain family of infinite Hessenberg Toeplitz matrices as n goes to infinity. In the present work we deal with the extreme eigenvalues. The generating function of the Toeplitz matrices is supposed to be of the form 1 a(t) = (1 − t)α f (t) (t ∈ T), where 0 < α < 1 and f is a smooth t function in H ∞ . Mathematics Subject Classification (2010). Primary 47B35. Secondary 15A15, 15A18, 47N50, 65F15. Keywords. Toeplitz matrix, eigenvalue, Fourier integral, asymptotic expansion.

1. Introduction The n×n Toeplitz matrix generated by a complex-valued function a ∈ L1 (T), on the unit circle T, is the square matrix Tn (a) = (aj−k )n−1 j,k=0 , where ak is the kth Fourier coefficient of a, that is,   1 π 1 iθ −ikθ a(e )e dθ = a(t)t−(k+1) dt ak = 2π −π 2πi T

(k ∈ Z).

The function a is referred to as the symbol of the matrices Tn (a). For a real-valued symbol a, the matrices Tn (a) are all Hermitian, and in this case a number of results on the asymptotics of the eigenvalues of Tn (a) are known; see, for example, [5, 6, 11, 13, 14, 15, 16, 18, 19, 21, 22, 24, 25]. If a is a rational function, papers [10, 12, 17] describe the limiting behavior of the eigenvalues of Tn (a). If a is a non-smooth symbol, papers [1, 23] are The second author was supported by CONACYT grant 238630. The third author was supported by the Ministry of Education and Science of the Russian Federation, Southern Federal University (Project No. 1.5169.2017/8.9).

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_4

119

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devoted to the asymptotic eigenvalue distribution. If a ∈ L∞ (T) and R(a) does not separate the plane, the papers [20, 23] in particular show that the eigenvalues of Tn (a) approximate R(a). Many of the results of the papers cited above can also be found in the books [4, 7, 8]. The paper [9] concerns asymptotic formulas for individual eigenvalues of Toeplitz matrices whose symbols are complex-valued and have a so-called Fisher–Hartwig singularity. These are special symbols that are smooth on T minus a single point but not smooth on the entire circle T; see [7, 8]. We here consider genuinely complex-valued symbols, in which case less is known. As we noted in [2], Dai, Geary, and Kadanoff [9] considered symbols of the form  γ 1 (−1)β+3γ a(t) = 2 − t − (−t)β = (1 − t)2γ (t ∈ T), t tγ−β (n)

where 0 < γ < −β < 1. They conjectured that the eigenvalues λ = λj satisfy  1  1 (n) λj ∼ a n n (2γ−1) e− n 2πij (j = 0, . . . , n − 1), (1.1) and confirmed this conjecture numerically. Note that in (1.1) the argument of a can be outside of T. This is no problem, since a can be extended analytically to a neighborhood of T {1} not containing the singular point 1. Denote by H ∞ the usual Hardy space of (boundary values of) bounded analytic functions over the unit disk D. For a function a ∈ C(T), let windλ (a) be the winding number of a about the point λ ∈ C R(a) where R(a) stands for the range of a, and let D(a) be the set {λ ∈ C R(a) : winda (λ) = 0}. We take the multi-valued complex function z → z β (β ∈ R) with the branch specified by −π < arg z β  π. Let B(z0 , r) be the set {z ∈ C : |z − z0 | < r}. In the present paper we study the extreme (closest to zero) eigenvalues of Tn (a) for symbols of the form 1 a(t) = (1 − t)α f (t) (t ∈ T) (1.2) t satisfying the following properties: 1. The function f is in H ∞ with f (0) = 0 and for some ε > 0, f has an analytic continuation to the region Kε := B(1, ε) {x ∈ R : 1 < x < 1 + ε} ˆ ε := B(1, ε) {x ∈ R : 1 < x  1+ε}. Additionally, and is continuous in K fϕ (x) := f (1+xeiϕ ) belongs to the algebra C 2 [0, ε) for each −π < ϕ  π. 2. Let 0 < α < 1 be a constant and take −απ < arg(1 − z)α  απ when −π < arg(1 − z)  π. 3. R(a) is a Jordan curve in C and windλ (a) = −1 for each λ ∈ D(a). In this paper we show that the problem to find the extreme eigenvalues of Tn (a), as n goes to infinity, can be reduced to the solution of a certain equation in a fixed complex domain not depending on n. In this sense our results extend to the complex-valued case the well known results of Parter [16] and Widom [23] for the real-valued case. Moreover, we show that the

Extreme eigenvalues for large Hessenberg Toeplitz matrices

121

conjecture of Dai, Geary, and Kadanoff (1.1) [9] is not true for the extreme eigenvalues.

2. Main results Note that, in general, limϕ→0+ fϕ (x) = limϕ→0− fϕ (x), thus f cannot be continuously extended to the ball B(1, ε). Without loss of generality, we assume that f (1) = 1. Let Dn (a) be the determinant of Tn (a). Since a has a power singularity at t = 1, it cannot be analytically extended to any neighborhood containing T, and hence, according to [23], the spectrum of Tn (a) (i.e., the zeros of Dn (a−λ)) has canonical distribution. That is, the Hausdorff distance between the spectrum of Tn (a) and R(a) goes to zero as n goes to infinity (see Figure 4). Note that when β = γ − 1, α = 2γ, and f = (−1)4γ−1 , our symbol coincides with the one of [9]. Before we go further, we need to give a((1 + ε)T)

(1 + ε)T 0.3

0.3

0.2

0.2

0.1

0.1

arg(z − 1) = δ

0.0

a(T) arg z =

1 2

απ

arg z = απ γ−

0.0

arg(z − 1) = −δ - 0.1

-0.1

T

- 0.2

γ+

-0.2 arg z = −απ

- 0.3

1

arg z = − απ 2

-0.3

0.9

1.0

1.1

1.2

1.3

1.4

-0.4

-0.2

Figure 1. The behavior of the function a(t) = near the point t = 1.

0.0

1 (1 t

0.2 3

− t) 4

the required understanding to the symbol a near to the point t = 1. For 0  x  ε take a+ (x) := lim a(1 + xeiδ ), + δ→0

γ+ := a+ ([0, ε]),

a− (x) := lim− a(1 + xeiδ ), δ→0

γ− := a− ([0, ε]).

Figure 1 shows the situation. Note that γ+ and γ− are very close to the lines arg z = ∓απ, respectively, but they are not the same. In [2] we proved that if λ ∈ D(a) is bounded away from 0, then a can be extended bijectively to a certain neighborhood of T {1} not containing the point 1, but if λ is arbitrarily close to 0 the situation is much more complicated. The map z → z α transforms the real negative semi-axis into the lines arg z = ±απ generating bijectivity limitations to a, see Figure 1. Moreover, Lemma 2.1 of [2] tell us that a maps C D into D(a). Let ρ < sup{|a(z)| : z ∈ Kε } be a positive constant and consider the regions S0 := B(0, ρ) D(a) and

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S := D(a) ∩ B(0, ρ), which we split as follows (see Figure 2 right): S1 is the subset of S enclosed by the curves ρT, R(a), and γ− , including γ− only; S2 is the subset of S enclosed by the curves ρT, R(a), and γ+ , including γ+ only; and S3 is the open subset of S enclosed by the curves ρT, γ− , and γ+ . We thus have S = S1 ∪ S2 ∪ S3 . It is easy to see that, for every sufficiently large n, we have no eigenvalues of Tn (a) in S0 . Since wind(a−λ) = 0 for each λ ∈ S0 , the operator T (a−λ) must be invertible and the finite section method is applicable (see [3, pg. 22]), which means that for every sufficiently large n the matrix Tn (a − λ) is invertible and hence, λ is not an eigenvalue of Tn (a). The regions S1 , S2 , and S3 will be our working sets for λ. We have the following conjecture: For every sufficiently large n, Tn (a) has no eigenvalues in S3 . We will prove this conjecture for the cases with | arg λ| >

π 2

1 2

< α < 1 and 0 < α 

0.4

0.2

R(a)

S1

0.2

U2 0.0

S3

0.0

U1

- 0.4

0.9

S0 S2

- 0.2

- 0.4

T 0.8

1 2

(see Theorem 4.6). In order to simplify our calculations,

0.4

- 0.2

(2.1)

1.0

1.1

1.2

1.3

1.4

- 0.4

- 0.2

0.0

0.2

0.4

Figure 2. The bijectivity regions of the symbol a near to the point t = 1. throughout the paper we use the parameter 1

Λ := (n + 1)λ α

(2.2)

divided in two cases: m  |Λ|  M for certain constants 0 < m < M < ∞ (depending only on the symbol a) and |Λ| → 0, and the case |Λ| → ∞ including the situation where λ is bounded away from zero, which was studied in [2]. Throughout the paper, let ψ be the argument of λ, δ a small positive constant (see Figure 1), and consider the sets R1 := {λ ∈ S : α(π − δ)  ψ < απ}; R2 := {λ ∈ S :

− απ < ψ  −α(π − δ)}.

The following are our main results.

Extreme eigenvalues for large Hessenberg Toeplitz matrices

123

Theorem 2.1. Let a be the symbol (1.2) satisfying the properties 1– 3. A point λ ∈ (S1 R1 ) ∪ (S2 R2 ) is an eigenvalue of Tn (a) if and only if there exist numbers m, M (depending only on the symbol a) satisfying 0 < m  |Λ|  M , and ∞ 2πi iψ( 1 −1) Λ e α e = e−|Λ|v b(v, ψ) dv + Δ1 (λ, n), α 0 where e−iαπ eiαπ b(v, ψ) := α − , v − ei(ψ−απ) v α − ei(ψ+απ) ψ = arg λ, and Δ1 (μ, n) is  a function defined for μ ∈ (S1 R1 ) ∪ (S2 R2 )  1 and satisfying Δ1 (μ, n) = O α as n → ∞ uniformly in μ. n

The previous theorem is important when doing numerical experiments, but using a variable change and doing some rotations, we can re-write Theorem 2.1 in terms of Λ alone with the disadvantage of having complex integration paths. Corollary 2.2. point λ ∈ (S1 exists positive m  |Λ|  M ,

Let a be the symbol (1.2) satisfying the properties 1– 3. A R1 ) ∪ (S2 R2 ) is an eigenvalue of Tn (a) if and only if there numbers m, M (depending only on the symbol a) satisfying and  2πi Λ e = e−Λu β(u) du + Δ2 (λ, n), α C

where

1 1 − , uα eiαπ − 1 uα e−iαπ − 1 3 the integration path C is the straight line from 0 to ∞e−i 4 π if λ ∈ S1 or the 3 straight line from 0 to ∞ei 4 π if λ ∈ S2 , and Δ2 (μ, n) is a function   which is 1 defined for μ ∈ (S1 R1 )∪(S2 R2 ) and satisfies Δ2 (μ, n) = O α as n → ∞ n uniformly in μ. β(u) :=

To get the eigenvalues of Tn (a) from the previous corollary we proceed as follows. Consider the function  2πi Λ F (Λ) := e − e−Λu β(u) du, (2.3) α C where C and β are as in Corollary 2.2. Consider the complex sets 1 Sˆ := {Λ = (n + 1)λ α : λ ∈ S and m  |Λ|  M } ( = 1, 2, 3).

For each sufficiently large n, the function F is analytic in Sˆ1 ∪ Sˆ2 . We can think of Δ2 as a function of Λ with parameter n which, for each sufficiently (n) (n) large n, will be analytic in Sˆ1 ∪ Sˆ2 also. Let λ1 , . . . , λn be the eigenvalues (n) of Tn (a), then according to Corollary 2.2, if λj ∈ (S1 R1 ) ∪ (S2 R2 ) 1 (n) := (n + 1)(λ(n) ) α ∈ Sˆ1 ∪ Sˆ2 will be a zero of then the corresponding Θ F (·) − Δ2 (·, n).

j

j

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J.M. Bogoya, S.M. Grudsky and I.S. Malysheva

Theorem 2.3. Under the same assumptions as in Theorem 2.1, consider the function F in (2.3) and suppose that Λj (1  j  k) are its roots located in Sˆ1 ∪ Sˆ2 with F  (Λj ) = 0 for each j. We then have α  Λj (n) λj = (1 + Δ3 (Λj , n)), n+1   1 where Δ3 (Λ, n) = O α as n → ∞ uniformly in Λ. n

The previous theorem gives us a simple method to get the extreme (n) eigenvalues of Tn (a). To approximate λj , for every sufficiently large n, we only need to calculate numerically (see Table 1) the extreme zeros Λj of F once. Analogous results were obtained by S. Parter [15, 16] and H. Widom [23] for real-valued symbols.

3. An example The symbol studied by Dai, Geary, and Kadanoff [9] was γ  1 a(t) = 2 − t − (−t)β = (−1)3γ+β tβ−γ (1 − t)2γ , t where 0 < γ < −β < 1. In the case β = γ − 1, this function a becomes our symbol with α = 2γ, times the constant (−1)4γ−1 which produces a rotation. The conjecture of [9] is that 1

1

zλj,n ∼ n n (2γ+1) e− n 2πij . In [2] we showed that this conjecture is true when λ is bounded away from zero, but now, Theorem 2.1 and Corollary 2.2 show that the conjecture is false when λ → 0. 40

20

0

-20

-40

-9 1 α

-8

-7

Figure 3. Left: the norm of F ((n + 1)(·) ) for n = 512. We see 3 zeros corresponding to 3 consecutive extreme eigenvalues. Right: the 16 zeros of F closest to zero.

-6

Extreme eigenvalues for large Hessenberg Toeplitz matrices

125

In order to approximate the extreme eigenvalues of Tn (a), we worked with the function F in Theorem 2.3. See Figure 3. Consider the symbol 3 1 a(t) = (1 − t) 4 (t ∈ T). t 3

This symbol satisfies the properties 1–3 with α = and f (t) = 1. According 4 to [23] the eigenvalues of Tn (a) must approximate (in the Hausdorff metric) R(a) as n increases. See Figure 4 (right). In this case the Fourier coefficients 3  can be calculated exactly as ak = (−1)k+1 /k4 . −5.4682120370856014060824201941002 ± 5.7983682817148888207896459067784 i −6.5314428236842426830338089371926 ± 12.367528740074554797742518382959 i −7.2146902524700029142376134506139 ± 18.766726622277519575303569592433 i −7.7391832574277648348440150030617 ± 25.107047817964583436614118399184 i −8.1801720679740042575992012537452 ± 31.419065936016327475853819485556 i −8.5727223117580707859817744737871 ± 37.714934295174649424694165724166 i −8.9360890295369466170427530738561 ± 44.000518944333248611110872372448 i −9.2820006335018468357176990494608 ± 50.279021560318150412713405426181 i

Table 1. The 16 zeros of F closest to zero with 32 decimal places (see Figure 3 right). 0.10

(512)

1.0

λ3

(512)

λ2 0.05

(512) λ1

0.5

0.00

0.0

(512)

λ512

-0.05

-0.5 (512)

λ511

(512)

λ510

-0.10 -0.04

-0.02

0.00

-1.0

0.02

0.04

-1.5

1

-1.0

-0.5

0.0

3

Figure 4. Range of a(t) = (1 − t) 4 (black curve) and t spectrum of Tn (a) (blue dots) for n = 512 (left) and n = 64 (right). (n)

(n)

Let λ1 , . . . , λn be the eigenvalues of Tn (a) numbered counterclockwise starting from the closest one to zero with positive imaginary part. See Figure 4 (left). Note that when f is real-valued, then the eigenvalues of Tn (a) as well as the zeros of F come in conjugated pairs. Let Λ1 , . . . , Λn be the zeros of F , and take α  Λj ˆ (n) := (j = 1, . . . , n) λ j n+1

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J.M. Bogoya, S.M. Grudsky and I.S. Malysheva

*

0.10

* 0.05

arg z =

3 4

*

π

arg z =

*

3 8

π

0.00 3

arg z = − π

*

-0.05 arg z = − 3 π

8

4

* *

-0.10 -0.04

*

0.00

-0.02

1 t

0.02

0.04

3

Figure 5. Range of a(t) = (1−t) 4 (black curve), a few ex(512)

treme exact and approximated eigenvalues λj ˆ (512) (orange stars), respectively. and λ

(blue dots)

j

be the approximated eigenvalues obtained from the zeros of F . Finally let (n) (n) εj and εˆj be our individual and relative individual errors, respectively, i.e., (n) ˆ (n) | |λ − λ j (n) (n) ˆ (n) | and εˆ(n) := j εj := |λj − λ . j j (n) |λj | See Figure 5 and Tables 2 and 3. The data was obtained with Wolfram Mathematica. n

128

256

512

1024

2048

4096

(n) ε1 (n) ε2 (n) ε3 (n) ε4 (n) ε5 (n) ε6 (n) ε7 (n) ε8

4.39·10−3

1.28·10−3

3.58·10−4

9.16·10−5

1.82·10−5

1.79·10−6

1.20·10−2

3.51·10−3

9.81·10−4

2.51·10−4

4.99·10−5

4.92·10−6

2.28·10−2

6.66·10−3

1.86·10−3

4.77·10−4

9.48·10−5

9.36·10−6

3.64·10−2

1.07·10−2

2.99·10−3

7.65·10−4

1.52·10−4

1.50·10−5

5.27·10−2

1.55·10−2

4.33·10−3

1.11·10−3

2.20·10−4

2.18·10−5

7.16·10−2

2.10·10−2

5.89·10−3

1.51·10−3

3.00·10−4

2.96·10−5

9.29·10−2

2.73·10−2

7.65·10−3

1.96·10−3

3.89·10−4

3.84·10−5

1.16·10−1

3.43·10−2

9.61·10−3

2.46·10−3

4.89·10−4

4.83·10−5

(n)

Table 2. The error εj for the 8 eigenvalues of Tn (a) closest to zero and with positive imaginary part. Here 3 1 a(t) = (1−t) 4 . t

Extreme eigenvalues for large Hessenberg Toeplitz matrices n

128

256

512

1024

2048

4096

(n) εˆ1 (n) εˆ2 (n) εˆ3 (n) εˆ4 (n) εˆ5 (n) εˆ6 (n) εˆ7 (n) εˆ8

3.63·10−2

1.71·10−2

8.18·10−3

3.50·10−3

1.17·10−3

1.94·10−4

6.54·10−2

3.16·10−2

1.47·10−2

6.31·10−3

2.10·10−3

3.48·10−4

9.48·10−2

4.58·10−2

2.13·10−2

9.13·10−3

3.04·10−3

5.05·10−4

1.24·10−1

6.00·10−2

2.80·10−2

1.20·10−2

3.99·10−3

6.62·10−4

1.54·10−1

7.43·10−2

3.46·10−2

1.48·10−2

4.94·10−3

8.19·10−4

1.84·10−1

8.87·10−2

4.13·10−2

1.77·10−2

5.89·10−3

9.77·10−4

2.14·10−1

1.03·10−1

4.80·10−2

2.05·10−2

6.84·10−3

1.13·10−3

2.44·10−1

1.17·10−1

5.47·10−2

2.34·10−2

7.80·10−3

1.29·10−3

127

(n)

Table 3. Relative individual error εˆj for the 8 eigenvalues of Tn (a) closest to zero and with positive imaginary part. We 3 1 worked here with a(t) = (1 − t) 4 . t

The rest of the paper is devoted to the proofs of the results of Section 2.

4. Proof of the main results We start this section with a technical result that enables us to invert a in a certain neighborhood of 0. Lemma 4.1. Let ρ be a small positive constant and a be the symbol in (1.2) satisfying the properties 1–3. Then (i) there exist U1 , U2 subsets of Kε D such that a(U1 ) ⊆ S1 and a(U2 ) ⊆ S2 , and a restricted to U1 ∪ U2 is a bijective map; moreover, for some small positive δ and each λ ∈ S1 R1 , S2 R2 , there exists a unique zλ in U1 , U2 , respectively, such that a(zλ ) = λ; (ii) for some small positive μ we have, π 2

− μ < arg(1 − z)  π π 2

−π  arg(1 − z) < − + μ

for every

z ∈ U1 ,

for every

z ∈ U2 ;

that is, the sets U1 , U2 are located as in Figure 2; (iii) zλ is a simple zero of a − λ. Proof. (i) Let U1 := a−1 (S1 ) and U2 := a−1 (S2 ), see Figure 2. By property 1, f has an analytic continuation to Kε , thus a has a continuous (but not ˆ ε. analytic) extension to K Let’s show the uniqueness of zλ . Suppose that there exist zλ and z˜λ in U1 ∪ U2 satisfying a(zλ ) = a(˜ zλ ) = λ, thus  a(zλ ) − a(˜ zλ ) = 0 = a (z) dz, (4.1) γλ

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J.M. Bogoya, S.M. Grudsky and I.S. Malysheva

where γλ is some closed polygonal curve in U1 ∪ U2 from z˜λ to zλ . Since f is an arbitrarily smooth function with f (1) = 1, as λ → 0, we have   α 1−z (1 − z)f  (z) a (z) = − (1 − z)α−1 f (z) 1 + − z αz αf (z) = − α(1 − z)α−1 (1 + O(|1 − z|))

(z → 1).

Putting together (4.1) and (4.2) we get    a(zλ ) − a(˜ zλ ) = −α (1 − z)α−1 dz + O |1 − z|α | dz| γλ γ  λ  = (1 − zλ )α − (1 − z˜λ )α + O |1 − z|α | dz| .

(4.2)

(4.3)

γλ

In order to reach a contradiction, we work separately with the terms in the right of (4.3). We begin by showing that there exists a positive constant c satisfying |(1 − zλ )α − (1 − z˜λ )α |  c|zλ − z˜λ |. (4.4) Suppose first that λ ∈ S1 . Then zλ , z˜λ ∈ U1 . Let Iλ be the closed line segment π from z˜λ to zλ . We thus have − − μ  arg(z − 1)  0 for some small positive 2  π μ and every z ∈ Iλ , which implies −(1−α)π  arg(1−z)α−1  (1−α) μ− 2 for every z ∈ Iλ . Then : ; c inf |Im(1 − z)α−1 | = inf |1 − z|α−1 | sin(arg(1 − z)α−1 )|  > 0 z∈Iλ z∈Iλ α for some positive c. Using the parametrization r(t) = tzλ + (1 − t)˜ zλ for 0  t  1, we get     α α α−1  |(1 − zλ ) − (1 − z˜λ ) | = α  (1 − z) dz  Iλ  1    = α|zλ − z˜λ | (1 − tzλ − (1 − t)˜ zλ )α−1 dt 0  1    α−1   α|zλ − z˜λ | Im(1 − tzλ − (1 − t)˜ zλ ) dt 0

1

 α|zλ − z˜λ |

inf |Im(1 − z)α−1 | dt

0 z∈Iλ

 c|zλ − z˜λ |, which proves (4.4). The case λ ∈ S2 follows readily. On the other hand, noticing that      α    kλα |1 − z| dz | dz| = O(kλα |zλ − z˜λ |),   γλ

γλ

where kλ := sup{|1 − z| : z ∈ γλ } satisfies kλ → 0 as λ → 0, we obtain   |1 − z|α dz = o(|zλ − z˜λ |) (λ → 0). (4.5) O γλ

Extreme eigenvalues for large Hessenberg Toeplitz matrices

129

Combining the relations (4.3), (4.4), and (4.5) we obtain c zλ )|  (c − o(1))|zλ − z˜λ | > |zλ − z˜λ |, |a(zλ ) − a(˜ 2 which contradicts (4.1). Note that because of the power ramification at the real positive semi-axis, a cannot be analytically extended to Kε . We have proven that for some small positive δ and every λ ∈ S1 R1 , S2 R2 there exists zλ ∈ U1 , U2 , respectively, satisfying a(zλ ) = λ. (ii) Recall that ψ = arg λ. We know that the point zλ is located outside of the unit disk D, zλ → 1 as λ → 0, and that 1  λzλ α zλ = 1 − , f (zλ ) which, by the smoothness of the continuation of f , gives us 1 2 1 ψ zλ = 1 − λ α + O(|λ| α ) and arg(1 − zλ ) = + O(|λ| α ), (4.6) α 1 2

as λ → 0. If λ ∈ S1 , for a small positive μ, we must have απ −αμ  ψ  απ. 1 π 2

In this case, the second relation in (4.6) tells us

− μ < arg(1 − zλ )  π. 1 2

A similar procedure applies when λ ∈ S2 , we have −απ  ψ  − απ + αμ 1

and thus −π  arg(1 − zλ ) < − π + μ. Then the sets U1 and U2 are located 2 as in Figure 2. (iii) Note that zλ is a simple zero of a − λ if and only if a (zλ ) = 0. From (4.2) we get −α (1 + O(|1 − zλ |)) (λ → 0), a (zλ ) = (1 − zλ )1−α which combined with zλ → 1 as λ → 0, gives us lim |a (zλ )| = ∞. λ→0

Figure 6. Contour ϑ



130

J.M. Bogoya, S.M. Grudsky and I.S. Malysheva

The previous proof also shows that if λ ∈ S0 ∪ S3 , then there is no point zλ with a(zλ ) = λ. From Lemma 2.1 in [2] we know that   −(n+2) t−(n+1) t 1 1 (−1)n Dn (a − λ) = dt = dt (4.7) 2πi T (1 − t)α f (t) − λt 2πi T a(t) − λ where λ ∈ D(a). To deal with the Fourier integral in (4.7), we consider the contour shown in Figure 6. That is, ϑ1 := {1 + xeiϕ : 0  x  ε}, ϑ2 := {1 + xe−iϕ : 0  x  ε}, ϑ3 := {xeiε + (1 − x)(1 + εeiϕ ) : 0  x  1} ∪ {eiθ : ε  θ  2π − ε} ∪ {x(1 + εe−iϕ ) + (1 − x)e−iε : 0  x  1}, ϑ := ϑ1 ∪ ϑ2 ∪ ϑ3 .

0.4

0.2

R(a)

arg z = α(π −δ) Gδ

arg z = α(π +δ) Bδ

0.0

Gδ Bδ

- 0.2

Gδ

arg z = −α(π +δ) arg z = −α(π −δ)

- 0.4 - 0.6

- 0.4

- 0.2

0.0

0.2

Figure 7. The regions in S used to determine the value of ϕ. If λ belongs to Gδ , Bδ we take ϕ = 0, ϕ = 2δ, respectively. Give ϑ the positive orientation and choose ϕ in the following way (see Figure 7): 1. let Gδ ⊂ S be the set of all λ ∈ S1 ∪ S2 (equivalently zλ ∈ U1 ∪ U2 ) with | arg(zλ − 1)| > δ (equivalently |ψ ± απ| > αδ) and all the λ ∈ S3 with |ψ ± απ| > αδ (green regions in Figure 7); if λ ∈ Gδ take ϕ = 0; 2. let Bδ ⊂ S be the set (S1 ∪ S2 ∪ S3 ) Gδ (blue regions in Figure 7); if λ ∈ Bδ take ϕ = 2δ. Let g(z) :=

z −(n+2) . a(z) − λ

According to Lemma 4.1 for λ ∈ (S1 R1 ) ∪ (S2 R2 ), the

function g has a simple pole at zλ . By (4.7) and the Cauchy residue theorem, for every λ ∈ (S1 R1 ) ∪ (S2 R2 ), we obtain, (−1)n Dn (a − λ) = − res(g, zλ ) + I1 + I2 + I3 ,

(4.8)

Extreme eigenvalues for large Hessenberg Toeplitz matrices  1 g(z) dz 2πi ϑj If λ ∈ R1 ∪ R2 ∪ S3 we will simply get

131

where

Ij :=

(j = 1, 2, 3).

(−1)n Dn (a − λ) = I1 + I2 + I3 .

(4.9)

We know that λ ∈ C is an eigenvalue of Tn (a) if and only if Dn (a − λ) = 0, thus we are interested in the zeros of the right-hand sides of (4.8) and (4.9). The following lemmas evaluate, one by one, the terms in the right-hand sides of (4.8) and (4.9). Lemma 4.2. Suppose that λ ∈ (S1 R1 ) ∪ (S2 R2 ). (i) If there exist positive constants m, M (depending only on the symbol a) satisfying m  |Λ|  M , then  2     |Λ| 1 1 |Λ| res(g, zλ ) = − λ α −1 eΛ 1 + O +O α n n as n → ∞ uniformly in λ. res(g, zλ ) 1 (ii) lim =− . 1 −1 α |Λ|→0 λα Proof. (i) Since λ ∈ (S1 R1 ) ∪ (S2 R2 ), Lemma 4.1 guarantees the existence of zλ . A direct calculation reveals that −(n+2)

res(g, zλ ) =

−(n+1)

zλ zλ (zλ − 1)f (zλ ) = .  a (zλ ) λ((α − 1)zλ f (zλ ) + f (zλ ) + zλ (zλ − 1)f  (zλ ))

Using the equation (4.6) and the smoothness of the continuation of f in Kε , we get 1 1 f (zλ ) = 1 + O(|λ| α ) and f  (zλ ) = f  (1) + O(|λ| α ), which combined with log(1 − z) = −z + O(|z|2 ) (z → 0) gives us 1

1

res(g, zλ ) =

2

1

−λ α −1 e−(n+1) log(1−λ α +O(|λ| α )) (1 + O(|λ| α )) 1

1

1

1

(α − 1)(1 + O(|λ| α )) + 1 + O(|λ| α ) − λ α f  (1)(1 + O(|λ| α ))2 1 2 1 1 α 1 2 1 1 − λ α −1 eΛ (1 + O(|λ| α ) + O(n|λ| α )). α

= − λ α −1 eΛ (1 + O(|λ| α ))(1 + O(n|λ| α )) =

Finally, recalling Λ from (2.2) we obtain the first part of the lemma. The limit in (ii) can be calculated directly.  Let ϑˆ1 := log ϑ1 . Thus ϑˆ1 is a path from 0 to log(1 + εeiϕ ) = εˆeiϕˆ with εˆ and ϕˆ satisfying εˆ = ε + O(ε2 )

and ϕˆ = ϕ + O(ε). Analogously, let ϑˆ2 := log ϑ2 . Thus ϑˆ2 is a path from log(1 + εe−iϕˆ ) = εˆe−iϕˆ to 0. For −π < β  π let ∞eiβ be lims→∞ seiβ . The following lemma is the heart of the calculation. It gives us asymptotic expansions for I1 and I2 with the disadvantage of handling complex integration paths.

132

J.M. Bogoya, S.M. Grudsky and I.S. Malysheva

Lemma 4.3. Suppose that λ ∈ S1 ∪ S2 ∪ S3 . (i) If there exist positive constants m and M (depending only on the symbol a) satisfying m  |Λ|  M , then   ∞eiϕˆ   |Λ|1−α e−|Λ|v 1 I1 = , dv + O 1−α −iαπ α iψ 2πi(n + 1) e v −e n 0   ∞e−iϕˆ   |Λ|1−α e−|Λ|v 1 I2 = − . dv + O iαπ v α − eiψ 2πi(n + 1)1−α e n 0 (ii) If |Λ| → 0, then I1 ∼

eiαπ Γ(1 − α) 2πi(n + 1)1−α

and

I2 ∼

eiαπ Γ(1 − α) . 2πi(n + 1)1−α

Where all the asymptotic relations work with n → ∞ uniformly in λ. Proof. All the order terms in this proof work with n → ∞ and λ → 0. Consider first the integral I1 and make the variable change v = eu . Then  e−(n+1)u 2πi I1 = du. (4.10) ˆ1 a(eu ) − λ ϑ We can write a(eu ) = e−u (1 − eu )α f (eu ) = (−u)α fˆ(u),  α f (eu ) u u2 where fˆ(u) = 1+ + + ··· which, by property 1, belongs to u e 2 6 C 2 (ϑˆ1 ). Note that fˆ(0) = f (1) = 1 and that (−u)α equals e−iαπ uα when u ∈ ϑˆ1 and eiαπ uα when u ∈ ϑˆ2 . Using the function k(u, λ) :=

1 (−u)α fˆ(u) − λ

−

1 (−u)α − λ

we split I1 as 2πi I1 = I1,1 + I1,2 , where

 I1,1 :=

ˆ1 ϑ

e−(n+1)u du (−u)α − λ

 and

I1,2 :=

(4.11) k(u, λ)e−(n+1)u du.

ˆ1 ϑ

As we will see, in norm, the integral I1,2 is much smaller than I1,1 . Thus we need to estimate I1,2 and we will do it by finding a uniform bound for |k|. To this end, note that fˆ(u) = 1 + O(u) (u → 0) and consider another variable 1 change: u = |λ| α v, thus (−u)α = |λ|(−v)α and 1

k(|λ| α v, λ) =

1

O(|λ| α −1 |v|α+1 ) 1

((−v)α − eiψ )((−v)α − eiψ + O(|λ| α |v|α+1 ))

.

The path ϑˆ1 is close to the line segment given by {xeiϕˆ : 0  x  εˆ}. Thus for u ∈ ϑˆ1 we have arg(−u)α = arg(−v)α ∼ α(ϕˆ − π) and we are ready to show that the denominator of |k| is bounded away from 0. Suppose that

Extreme eigenvalues for large Hessenberg Toeplitz matrices

133

λ ∈ Gδ (see Figure 7). Then ϕˆ = 0, (−v)α lies arbitrarily close to the ray with argument −απ, and eiψ lies on T with |ψ − απ| > αδ, giving us |(−v)α − eiψ | >

αδ . 2

(4.12)

If λ ∈ Bδ (see Figure 7), then ϕˆ = 2δ, (−v)α lies arbitrarily close to the ray with argument α(2δ − π), and eiψ lies on T with |ψ − απ|  αδ, giving us (4.12) again. For the second term in the denominator of |k|, note that 1 |(−v)α − eiψ + O(|λ| α |v|α+1 )| attains its minimum value when |v| ∼ 1 and 1 1 thus the order term will be bounded by |λ| α < ρ α , which can be taken arbitrarily small. Then we get 1

|(−v)α − eiψ + O(|λ| α |v|α+1 )| >

αδ . 4

1

(4.13) 1

Using (4.12) and (4.13) we get the bound |k(|λ| α v, λ)|  c2 |λ| α −1 |v|α+1 (or equivalently |k(u, λ)|  c2 |λ|−2 |u|α+1 ) where c2 is a positive constant not depending on λ or v. Thus,  |I1,2 |  |k(u, λ)e−(n+1)u | du ˆ1 ϑ  c2  |u|α+1 |e−(n+1)u | du |λ|2 ϑˆ1  ∞eiϕˆ c2 |w|α+1 |e−w | dw  (n + 1)α+2 |λ|2 0 ∞ c2 = wα+1 e−w dw (n + 1)α+2 |λ|2 0 c2 Γ(α + 2) = , (n + 1)α+2 |λ|2 where in the third line we shifted to the variable w = (n + 1)u, in the forth line we changed the path of integration by noticing that the corresponding integral over the line segment joining ∞ with ∞eiϕˆ is zero, and Γ is the well-known Gamma function. The previous calculation gives us     1 1 I1,2 = O α+2 2 = O 2−α 2α (4.14) n |λ| n |Λ| uniformly in λ. Now we work with I1,1 . Write I1,1 = I1,1,1 − I1,1,2

(4.15)

where I1,1,1 :=

 ∞eiϕˆ 0

e−(n+1)u du (−u)α − λ

and

I1,1,2 :=

 ∞eiϕˆ εˆeiϕˆ

e−(n+1)u du. (−u)α − λ

134

J.M. Bogoya, S.M. Grudsky and I.S. Malysheva

For I1,1,2 consider the variable change w = ue−iϕˆ . Thus    ∞ iϕˆ −(n+1)eiϕˆ w   e e  dw |I1,1,2 | =  i ϕ ˆ α εˆ (−we ) − λ ∞ 1  α e−(n+1)w cos ϕˆ dw |ˆ ε − |λ|| εˆ =

e−(n+1)ˆε cos ϕˆ . (n + 1)|ˆ εα − |λ|| cos ϕˆ

(4.16)

Since ϕˆ is a small non-negative constant and |λ| < ρ, which can be chosen  − 12 εˆn  e satisfying ρ < εˆα , equation (4.16) shows that I1,1,2 = O uniformly in 1

n

λ. Taking again the variable change u = |λ| α v, and putting together (4.11), (4.14), (4.15), and (4.16) we obtain   ∞eiϕˆ   1 1 e−|Λ|v |λ| α −1 dv + O (4.17) I1 = 2πi e−iαπ v α − eiψ n|Λ|α+1 0 uniformly in λ. A similar result for I2 can be obtained readily by changing every ϕˆ by −ϕ, ˆ getting   ∞e−iϕˆ   1 1 e−|Λ|v |λ| α −1 I2 = − dv + O (4.18) 2πi eiαπ v α − eiψ n|Λ|α+1 0 uniformly in λ. For proving (i) suppose that m  |Λ|  M . Then the result is immediate from (4.17) and (4.18). For proving (ii) take |Λ| → 0 and assume first that λ ∈ Gδ (see Figure 7), thus ϕ = ϕˆ = 0. From equation (4.10), with fˆ(u) = fˆ(0) + fˆ (0)O(u) = 1 + O(u)

(u → 0)

1 α

and the variable change u = |λ| v, we get  e−(n+1)u 2πi I1 = du ˆ1 e−iαπ uα fˆ(u) − λ ϑ  1 e−|Λ|v −1 α dv = |λ| 1 −iαπ v α fˆ(|λ| α v) − eiψ ϑ1 e  1 e−|Λ|v = |λ| α −1 dv 1 −iαπ v α − eiψ + O(|λ| α |v|α+1 ) ϑ1 e 1

= |λ| α −1 (J1,1 + J1,2 ) where ϑ1 is a continuous path in C starting at 0 and ending at εˆ|λ|  e−|Λ|v J1,1 := dv, 1 −iαπ v α − eiψ + O(|λ| α |v|α+1 ) ϑ1,1 e  e−|Λ|v dv; J1,2 := 1 −iαπ v α − eiψ + O(|λ| α |v|α+1 ) ϑ1,2 e

(4.19) 1 −α

, and

(4.20)

Extreme eigenvalues for large Hessenberg Toeplitz matrices

135 1

here ϑ1,1 and ϑ1,2 are the portions of ϑ1 from 0 to 1 and from 1 to εˆ|λ|− α , respectively. We proceed to find order bounds for J1,1 and J1,2 . The former will be easy but the latter will requiere a lot more work. 1 1 Consider the integral J1,1 . The term O(|λ| α |v|α+1 ) = O(|λ| α ) is arbitrarily small and the denominator in the integrand of J1,1 in (4.20) has a zero at some point close to v = ei(απ+ψ) . For λ ∈ Gδ we have |απ + ψ| > αδ, thus 1

1

|e−iαπ v α − eiψ + O(|λ| α )|  |e−iαπ v α − eiψ | − O(|λ| α ) 1

> αδ − O(|λ| α ) 1 2

> αδ. We thus have |J1,1 | 

2 αδ

 ϑ1,1

2 . αδ

e−|Λ|v dv 

(4.21)

To find an order bound for J1,2 we will go through three steps: In the first one, we split it as J1,2,1 + J1,2,2 , in the second step we bound J1,2,1 , and in 1

1

the third step we study J1,2,2 for the cases 0 < α < , α = , and 2 2 separately. Finally we will put all together.

1 2

π 2

and hence

cos2 (απ) cos2 ψ

< 1 and cos ψ < 0,

which shows that Re(e−iψ κ(v, ψ)) < 0, making the integrand in (4.29) strictly 1 π negative, which yields the theorem in this case. If 0 < α  and |ψ| > , 2 2 then a similar analysis applies and we get the theorem in this case also.  Proof of Theorem 2.1. Let m, M be constants (depending only on the symbol a) satisfying m  |Λ|  M . In this proof all the order terms work with n → ∞ uniformly in λ. Suppose that λ ∈ (S1 R1 ) ∪ (S2 R2 ). Using Lemmas 4.2 part (i), 4.4, and 4.5 in (4.8) we get that λ is an eigenvalue of Tn (a) if and only if       1 1 |λ| α −1 ∞ −|Λ|v 1 1 −1 Λ 1 α +O = λ e 1+O e b(v, ψ) dv, α n n 2πi 0 where b(v, ψ) := Noticing that

vα

e−iαπ eiαπ − α . i(ψ−απ) −e v − ei(ψ+απ)

 1−α Λ      1 1 −1 Λ |Λ| 1 |e | 1 =O = O 2−α , λα e O α n n2−α n    1−α    1 n 1 O =O =O α , 1 n|Λ|1−α n n|λ| α −1

140

J.M. Bogoya, S.M. Grudsky and I.S. Malysheva

we get the theorem in this case. Finally, suppose that |Λ| → 0. Using the part (ii) of the Lemmas 4.2 and 4.3 in (4.8) we get that λ is an eigenvalue of Tn (a) if and only if   res(g, zλ ) 1 0 = lim − + 1 −1 (I1 + I2 + I3 ) 1 |Λ|→0 λ α −1 λα   res(g, zλ ) 2eiαπ Γ(α − 1) 1 = − lim + lim (1 + o(1)) + lim O 1 1 −1 iψ( −1) 1−α n→∞ n |Λ|→0 |Λ|→0 |Λ| λα e α = ∞, 

thus, we don’t get eigenvalues in this case. iψ α

Proof of Corollary 2.2. Considering the variable change v = ue in Theorem 2.1, as n → ∞ uniformly in λ, we obtain    2πi Λ 1 −Λu e = e β(u) du + O α , α n D where 1 1 β(u) := α iαπ − α −iαπ u e −1 u e −1 ψ and the integration path D is the straight line from 0 to ∞e−i α . Assume that απ λ ∈ S1 . Then there exists a small constant μ satisfying − αμ < ψ < απ ψ

2

π

and hence −π < − < − + μ. In order to make the integration path α 2 independent of λ, we make a path rotation by integrating over the triangle T ψ 3 with vertices 0, ∞e−i α , and ∞e−i 4 π . Since the singularities of β are u = e±iπ , the integrand e−Λu β(u) is analytic on T and, moreover, the corresponding ψ 3 integral over the segment joining ∞e−i α and ∞e−i 4 π is clearly 0. We thus have    2πi Λ 1 −Λu e β(u) du + O α , e = α n D1 3

where D1 is the straight line from 0 to ∞e−i 4 π . Finally, If λ ∈ S2 , a similar calculation produces    2πi Λ 1 −Λu e = e β(u) du + O α , α n D2 3

where D2 is the straight line from 0 to ∞ei 4 π .



Proof of Theorem 2.3. Suppose that Λs for 1  s  k with k ! n, are the zeros of F located in Sˆ1 ∪ Sˆ2 and satisfying F  (Λs ) = 0 for each s (i.e. each Λs is simple). We can pick a neighborhood Us of each Λs with continuous and smooth boundary ∂Us satisfying |F (·)| > |Δ2 (·, n)| over ∂Us . In this ˆ s in Us . case the Rouch´e theorem says that F (·) − Δ2 (·, n) must have a zero Λ ˆ s corresponds to an eigenvalue λ(n) By Corollary 2.2, we know that each Λ j of Tn (a). If necessary, re-enumerate Λs in order to get s = j. To prove the theorem, note that ˆ j ) − F (Λj ) = Δ2 (Λ ˆ j , n) = F  (Λj )(Λ ˆ j − Λj ) + O(|Λ ˆ j − Λj |2 ), F (Λ

Extreme eigenvalues for large Hessenberg Toeplitz matrices which produces



O

1 nα



141

  ˆ j − Λj ) F  (Λj ) + O(|Λ ˆ j − Λj |) . = (Λ

By hypothesis we have F  (Λj ) = 0, thus ˆ j − Λj = (n + 1)(λ(n) ) α1 − Λj = O Λ j



 1 . nα

(n)

Finally, solving for λj (n) λj

=

we get  # " 1 α Λj + O α n

n+1

as n → ∞ uniformly in Λ.

 =

Λj n+1

α 



1 1+O α n





References [1] E.L. Basor and K.E. Morrison, The Fisher–Hartwig conjecture and Toeplitz eigenvalues, Linear Algebra Appl. 202 (1994), 129–142. [2] J.M. Bogoya, A. B¨ ottcher, and S.M. Grudsky, Asymptotics of individual eigenvalues of a class of large Hessenberg Toeplitz matrices, in: Recent progress in operator theory and its applications, volume 220 of Oper. Theory Adv. Appl., pp. 77–95, Birkh¨ auser/Springer Basel AG, Basel, 2012. [3] A. B¨ ottcher and S.M. Grudsky, Toeplitz matrices, asymptotic linear algebra, and functional analysis, Birkh¨ auser Verlag, Basel, 2000. [4] A. B¨ ottcher and S.M. Grudsky, Spectral properties of banded Toeplitz matrices, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. [5] A. B¨ ottcher, S.M. Grudsky, and E.A. Maximenko, Inside the eigenvalues of certain Hermitian Toeplitz band matrices, J. Comput. Appl. Math. 233 no. 9 (2010), 2245–2264. [6] A. B¨ ottcher, S.M. Grudsky, E.A. Maximenko, and J. Unterberger, The first order asymptotics of the extreme eigenvectors of certain Hermitian Toeplitz matrices, Integral Equations Operator Theory 63 no. 2 (2009), 165–180. [7] A. B¨ ottcher and B. Silbermann, Introduction to large truncated Toeplitz matrices, Universitext, Springer-Verlag, New York, 1999. [8] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz operators, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2nd ed., 2006. Prepared jointly with Alexei Karlovich. [9] H. Dai, Z. Geary, and L.P. Kadanoff, Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices, J. Stat. Mech. Theory Exp. 2009 (2009), May 2009, P05012. [10] K.M. Day, Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions, Trans. Amer. Math.Soc. 209 (1975), 175– 183.

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[11] U. Grenander and G. Szeg˝ o, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. [12] I.I. Hirschman, Jr., The spectra of certain Toeplitz matrices, Illinois J. Math. 11 (1967), 145–159. [13] M. Kac, W.L. Murdock, and G. Szeg˝ o, On the eigenvalues of certain Hermitian forms, J. Rational Mech. Anal. 2 (1953), 767–800. [14] A.Yu. Novosel’tsev and I.B. Simonenko, Dependence of the asymptotics of extreme eigenvalues of truncated Toeplitz matrices on the rate of attaining an extremum by a symbol, St. Petersburg Math. J. 16 no. 4 (2005), 713–718. [15] S.V. Parter, Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations, Trans. Amer. Math. Soc. 99 (1961), 153–192. [16] S.V. Parter, On the extreme eigenvalues of Toeplitz matrices, Trans. Amer. Math. Soc. 100 (1961), 263–276. [17] P. Schmidt and F. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial, Math. Scand. 8 (1960), 15–38. [18] S. Serra, On the extreme spectral properties of Toeplitz matrices generated by L1 functions with several minima/maxima, BIT 36 no. 1 (1996), 135–142. [19] S. Serra, On the extreme eigenvalues of Hermitian (block ) Toeplitz matrices, Linear Algebra Appl. 270 (1998), 109–129. [20] P. Tilli, Some results on complex Toeplitz eigenvalues, J. Math. Anal. Appl. 239 no. 2 (1999), 390–401. [21] E.E. Tyrtyshnikov and N.L. Zamarashkin, Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships, Linear Algebra Appl. 270 (1998), 15–27. [22] H. Widom, On the eigenvalues of certain Hermitian operators, Trans. Amer. Math. Soc. 88 (1958), 491–522. [23] H. Widom, Eigenvalue distribution of nonselfadjoint Toeplitz matrices and the asymptotics of Toeplitz determinants in the case of nonvanishing index, in: Topics in operator theory: Ernst D. Hellinger memorial volume, volume 48 of Oper. Theory Adv. Appl., pp. 387–421, Birkh¨ auser, Basel, 1990. [24] N.L. Zamarashkin and E.E. Tyrtyshnikov, Distribution of the eigenvalues and singular numbers of Toeplitz matrices under weakened requirements on the generating function, Sb. Math. 188 no. 8 (1997), 1191–1201. [25] P. Zizler, R.A. Zuidwijk, K.F. Taylor, and S. Arimoto, A finer aspect of eigenvalue distribution of selfadjoint band Toeplitz matrices, SIAM J. Matrix Anal. Appl. 24 no. 1 (2002), 59–67.

J.M. Bogoya Pontificia Universidad Javeriana Departamento de Matem´ aticas, Bogot´ a Colombia e-mail: [email protected]

Extreme eigenvalues for large Hessenberg Toeplitz matrices S.M. Grudsky CINVESTAV del I.P.N., Departamento de Matem´ aticas Apartado Postal 14-740, 07000 Ciudad de M´exico M´exico e-mail: [email protected] I.S. Malysheva Southern Federal University, Mathematics department Post code 344006, Rostov-on-Don Russia e-mail: [email protected]

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How to solve an equation with a Toeplitz operator? Albrecht B¨ottcher and Elias Wegert Abstract. An equation with a Hardy space Toeplitz operator can be solved by Wiener–Hopf factorization. However, Wiener–Hopf factorization does not work for Bergman space Toeplitz operators. The only way we see to tackle equations with a Toeplitz operator on the Bergman space is to have recourse to approximation methods. The paper is intended as a review of and an illustrated tour through several such methods and thus through some beautiful topics in the intersection of operator theory, complex analysis, differential geometry, and numerical mathematics. MSC 2010. Primary 47B35. Secondary 30E10, 30H20, 65R20 Keywords. Toeplitz operator, Hardy space, Wiener–Hopf factorization, Bergman space, Bergman kernel, complex approximation

For Rien Kaashoek on his 80th birthday

1. A personal note The name of Rien Kaashoek has been known to us since our times as graduate students, but it was only after the fall of the German wall that we made our personal acquaintance with him. The first of us met Rien at the 1989 Oberwolfach meeting “Wiener–Hopf-Probleme, Toeplitz-Operatoren und Anwendungen”, which was organized by Israel Gohberg, Erhard Meister, and Rien Kaashoek, while the first meeting of the second of us with Rien was in 1990 at the workshop on H ∞ control and optimization (the C.I.M.E. Session “Recent Developments in H ∞ Control Theory”), which took place in Como, Italy. For both of us, it was great pleasure to realize that Rien is not only an outstanding and extremely productive mathematician but also an extraordinarily likeable person. Rien is one of the great wizards of factorization. The two volumes [10] have accompanied us since the early 1990s, and his magical achievements, especially those obtained with the state space method, have always been © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_5

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around us. The more recent works [1, 11] summarize much of what Rien, in collaboration with many colleagues, has done over the years. The present article is an expository paper. When treating Toeplitz operators on the Hardy space, we confine ourselves to the scalar case and thus to a problem that is levels down the matrix-valued or operator-valued situations studied by Rien and co-authors. However, we embark on numerical aspects of Wiener–Hopf factorization, which are important in control theory and thus provide us with a nice link to one of Rien’s favorite topics. The results on Bergman space Toeplitz operators cited here were actually established more than two decades ago. But we think they are a nice contribution to this volume, because, first, in the Bergman space nothing like Wiener– Hopf factorization works and, secondly, we think a refreshed presentation of these results could be useful. During the last 27 years, we have met and corresponded with Rien on many occasions. It has always been pleasant to hear Rien’s opinion on the matters, both mathematical and non-mathematical ones, and to feel that he is a man who appreciates our opinions and takes care of us younger colleagues. As said, he is a truly unrivaledly likeable person. We wish Rien many more years with good health, pleasant life, and passion for mathematics.

2. Operators on the Hardy space The Hardy space H 2 (T) of the unit circle is the closed subspace of L2 (T) consisting of the functions f for which all coefficients with negative indices in the Fourier series ∞  f (t) = fn tn , t = eiϕ , n=−∞

are zero. The Toeplitz operator generated by a function a in L∞ (T) is defined by T (a) : H 2 (T) → H 2 (T), T (a)f = P (af ), where P : L2 (T) → H 2 (T) is the orthogonal projection. How can we solve an equation T (a)f = g ? 2.1. Wiener–Hopf factorization One method to solve the equation is Wiener–Hopf factorization. Represent a in the form a = a− a+ with a− ∈ GH ∞ (T), a+ ∈ GH ∞ (T), where H ∞ (T) stands for (the non-tangential limits of) the bounded analytic functions in the unit disk D and GA denotes the invertible elements of a unital Banach algebra A. Such a factorization exists if, for instance, a is sufficiently smooth, 0∈ / a(T), and the winding number of a(T) about the origin is zero. Then T (a) −1 is invertible, the inverse being [T (a)]−1 = T (a−1 + )T (a− ), and consequently, −1 the solution f of the equation T (a)f = g is given by f = a−1 + P (a− g); see, for example, [5, Section 1.5] or [9, Section I.7.3].

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If a is rational, a Wiener–Hopf factorization can be constructed explicitly by calculating the zeros and poles of a and sorting them according to whether they are inside or  outside T. For simplicity, suppose a is a trigonom metric polynomial, a(t) = k=−n ak tk . Passing into the complex plane we may write a(z) = z −n b(z) = z −n (b0 + b1 + · · · + bN z N ),

N = n + m,

and T (a) is invertible if and only if b(z) has exactly n zeros in |z| < 1 and m zeros in |z| > 1. Our problem is to find the coefficients of two polynomials v(z) = v0 + v1 z + · · · + vn z n ,

u(z) = u0 + u1 z + · · · + um z m

such that b(z) = v(z)u(z) and v(z) has its zeros in |z| < 1, u(z) has its zeros in |z| > 1. Then a(z) = a− (z)a+ (z) with a− (z) = z −n v(z), a+ (z) = u(z) is a Wiener– Hopf factorization. Figure 1 shows the phase plots (a short explanation of this visualization technique is given in the appendix) of two polynomials b(z). These polynomials were introduced in [2]. Their zeros (which are the points where all colors meet) are very close to T, and hence they are serving as benchmark test polynomials for factorization algorithms.

  n−1  m Figure 1. Polynomials b(z) = 2z n+ k=0 z k 2+ k=1 z k for n = 60, m = 20 (left) and n = 200, m = 20 (right). The problem is that if n and m are large, then determining the zeros λ1 ,. . ., λn in D and μ1 , . . . , μm in C \ D of b(z) is accompanied by rounding errors, and these are amplified when building v(z) = (z − λ1 ) · · · (z − λn ),

u(z) = bN (z − μ1 ) · · · (z − μm ).

Frank Uhlig [17] showed that things are already critical even for cubic polynomials: the polynomial z 3 − 122.63 z 2 + 20.8 z + 224220

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has the roots 100, 60, −37.37, and perturbing the roots by at most 1% to 100, 60.6, −37.0 we arrive at the polynomial z 3 − 123.6 z 2 + 117.8 z + 224220, which has a perturbation of about 500% in the coefficient of z. Various methods for the numerical Wiener–Hopf or spectral factorization (which is Wiener–Hopf factorization in the case where a(T) is a subset of the positive real line) are known [2, 8, 12, 13, 15]. These methods give very good results for n ≤ 1 000, m ≤ 1 000. We believe the algorithm of [4] is at present the most efficient method for Wiener–Hopf factorization. It works for n ≤ 10 000, m ≤ 10 000. Note that the adjective “real” in the title of [4] may be deleted. With the coefficients of v(z) and u(z) we have the coefficients of a± (z). To get a−1 ± we finally have to solve the following problem: suppose the power series 1 + c1 z + c2 z 2 + · · · converges in a neighborhood of the origin, find the coefficients of the power series 1 + d1 z + d2 z 2 + · · · such that (1 + c1 z + c2 z 2 + · · · )(1 + d1 z + d2 z 2 + · · · ) = 1. This is a triangular system for the dj , which gives d1 , . . . , dn with O(n2 ) flops. In paper [4], a matricial modification of the algorithm of Sieveking [16] and Kung [14] was developed. It reads d1 = −c1      d2 1 c2 c1 1 =− d3 d1 1 c3 c2 d1 ⎛ ⎞ ⎛ ⎞⎛ 1 c4 c3 d4 ⎜ d5 ⎟ ⎜ d1 1 ⎟ ⎜ c5 c4 ⎟⎜ ⎜ ⎟ ⎜ ⎝ d6 ⎠ = − ⎝ d2 d1 1 ⎠ ⎝ c6 c5 d7 d3 d2 d1 1 c7 c6 ...



c2 c3 c4 c5

⎞⎛ c1 1 ⎜ d1 c2 ⎟ ⎟⎜ c 3 ⎠ ⎝ d2 c4 d3

⎞ ⎟ ⎟ ⎠

and yields d1 , . . . , dn in a superfast way, that is, with O(n log2 n) flops.

2.2. Finite section method

√ k In the orthonormal basis {ek }∞ k=0 with ek (t) = t / 2π we have the matrix representation T (a) = (aj−k )∞ j,k=0 , that is, T (a) is given by what is usually called an infinite Toeplitz matrix. Let Tn (a) = (aj−k )nj,k=0 denote the principal (n + 1) × (n + 1) truncation of the infinite Toeplitz matrix and let Pn be the orthogonal projection of H 2 (T) onto the span of {e0 , e1 , . . . , en }. The finite section method for the solution of the equation T (a)f = g consists in replacing this equation by the equations  Pn T (a)Pn f (n) = Pn g. Equivalently, n (n) = j=0 xj ej such that the first n + 1 we look for a linear combination f Fourier coefficients of T (a)f (n) and g coincide. This amounts to solving the

Equations with a Toeplitz operator finite Toeplitz system ⎛ a0 a−1 ⎜ a1 a0 ⎜ ⎜ .. .. ⎝. . an

an−1

... ... .. .

a−n a−n+1 .. .

...

a0

⎞⎛ ⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

x0 x1 .. .

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎠ ⎝

xn

149

g0 g1 .. .

⎞ ⎟ ⎟ ⎟. ⎠

gn

For what follows, it will be crucial to consider compactly perturbed Toeplitz operators. Thus, let K be a compact operator on H 2 (T). We say that the finite section method converges for T (a) + K if T (a) + K is invertible on H 2 (T), if the operators (matrices) Pn (T (a) + K)Pn are invertible on Ran Pn for all sufficiently large n, and if [Pn (T (a) + K)Pn ]−1 Pn → [T (a) + K]−1 strongly. The following result by Gohberg and Feldman solves the problem for continuous functions a; proofs are in [5, Section 2.4] and [9, Sections II.3 and III.2.1]. Theorem 1 (Gohberg and Feldman). If a ∈ C(T), K is compact, and T (a)+K is invertible, then the finite section method for T (a) + K converges.

3. Operators on the Bergman space The Bergman space A2 (D) of the unit disk is the closed subspace of L2 (D, dA) consisting of the functions that are analytic in D. Here dA = (1/π)r dr dθ is the normalized area measure. In other words, A2 (D) is the Hilbert space of all analytic functions f on D for which  f 2 := |f (z)|2 dA(z) < ∞. D

The Toeplitz operator generated by a function a in L∞ (D) is the operator T (a) : A2 (D) → A2 (D),

T (a)f = P (af ),

where P : L (D) → A (D) is the orthogonal projection. How can we solve an equation T (a)f = g? Something like Wiener–Hopf factorization does not work because the orthogonal complement of A2 (D) in L2 (D) is not of the same size as A2 (D). We therefore try approximation methods. Let T : H → H be a Hilbert space operator. A Galerkin–Petrov method for solving T f = g approximately consists in choosing two systems of elements in H, 2

2

(0)

u0 (1) (1) u 0 , u1 (2) (2) (2) u 0 , u1 , u2 ............... basis functions

(0)

v0 (1) (1) v 0 , v1 (2) (2) (2) v 0 , v1 , v2 ............... test functions,

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and determining an approximate solution (n) (n)

(n) (n)

(n) f (n) = x0 u0 + x1 u1 + · · · + x(n) n un (n)

(n)

(n)

in the span of u0 , u1 , . . . , un (n) (T f (n) , vj )

(1)

such that (n)

= (g, vj ) for j = 0, 1, . . . , n.

(2)

This amounts to solving n + 1 linear equations with the n + 1 unknown coeffi(n) (n) (n) cients x0 , x1 , . . . , xn . The method is said to converge if T is invertible, if there is an n0 such that the equations (2) are uniquely solvable for all n ≥ n0 and all g ∈ H, and if the solutions f (n) converge in H to the solution f of T f = g. 3.1. Finite section method The Galerkin–Petrov method for T (a) on A2 (D) arising from specifying the basis and test functions to χ0 χ0 χ 0 , χ1 χ 0 , χ1 χ 0 , χ 1 , χ2 χ 0 , χ1 , χ2 ............... ............... basis functions test functions, where χn (z) = z n , is called the finite section method. It is equivalent to looking for an approximate solution f (n) in the form of a polynomial of degree n such that the first n + 1 Taylor coefficients (at zero) of T (a)f (n) and g coincide. 2 An √ orthonormal basis {ek }∞ k=0 of A (D) is formed by the monomials k ek (z) = k + 1 z . The matrix representation of T (a) in this basis has the j, k entry  = √ ajk = k + 1 j + 1 a(z)z k z j dA(z). (3) D

If a is the harmonic extension of a function a ∈ L∞ (T), this becomes √ √ 2 k+1 j+1 ajk = aj−k . |j − k| + j + k + 2

(4)

We denote the matrix with the entries (3) by T 1 (a) and think of it as an operator on 2 (Z+ ). Given a function b ∈ L∞ (T), let T 0 (b) stand for the operator on 2 (Z+ ) defined by the Toeplitz matrix (bj−k )∞ j,k=0 . The following result is due to Coburn [7, Lemma 3]. Theorem 2 (Coburn). If a ∈ C(D), then T 1 (a) = T 0 (a|T)+K with a compact operator K. The finite section method for the solution of the equation T (a)f = g in A2 (D) is nothing but passage to the finite linear system Tn1 (a)x(n) = Pn g, where Tn1 (a) is the principal (n + 1) × (n + 1) truncation of the matrix T 1 (a) (n) with the entries (3), x(n) is the column formed by the coefficients xj in (1),

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and Pn g is the column composed by the first n + 1 Taylor coefficients of g. Thus, combining Theorems 1 and 2, we arrive at the following result from [3]. Corollary 3. If a ∈ C(D) and T (a) is invertible on A2 (D), then the finite section method for T (a) on A2 (D) converges. 3.2. The Bergman kernel The orthogonal projection P : L2 (D) → A2 (D) is given by  (P f )(z) = (1 − zw)−2 f (w) dA(w), z ∈ D. D

It follows in particular that if f ∈ A2 (D), then  f (z) = (1 − zw)−2 f (w) dA(w), D

z ∈ D.

Thus, Kw (z) = k(z, w) = (1 − zw)−2 is the reproducing kernel for A2 (D): we have f (z) = (f, Kz ) for every f ∈ A2 (D). This kernel was introduced by Stefan Bergman in 1922 and is now called the Bergman kernel. See Figure 2.

Figure 2. Phase plot of the Bergman kernel Kw with w = 0.65 eiπ/4 (left) and the sum Kw + K−w (right). We see the poles of second order at ±1/w where all colors meet. (Note the reverse orientation of colors at zeros and poles.)

3.3. Polynomial collocation Another Galerkin–Petrov method results from choosing a system of points (0)

z0 (1) (1) z0 , z1 (2) (2) (2) z0 , z1 , z2 ...............

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in the unit disk D, and then defining the basis and test functions by χ0 χ 0 , χ1 χ 0 , χ1 , χ2 ............... basis functions where

(n) Kj

(0)

K0 (1) (1) K0 , K1 (2) (2) (2) K0 , K1 , K2 ............... test functions, (n)

= Kz(n) is the Bergman kernel of zj . We are now looking for a j n (n) polynomial f (n) (z) = j=0 xj z j such that   (n)  (n)  T (a)f (n) , Kj = g, Kj for j = 0, 1, . . . , n, which, by the reproducing property (f, Kz ) = f (z), is the same as   (n)   (n)  T (a)f (n) zj = g zj for j = 0, 1, . . . , n. Thus, the Galerkin–Petrov method at hand is in fact nothing else than a polynomial collocation method: the approximate solution f (n) is sought as a polynomial of degree n so that T (a)f (n) and g coincide at n + 1 prescribed (n) (n) collocation points z0 , . . . , zn . The following was established in [6]. Theorem 4. Suppose a ∈ C(D) and T (a) is invertible on A2 (D). Fix r ∈ (0, 1) (n) (n) and choose the collocation points z0 , . . . , zn as the roots of z n+1 − rn+1 . Then polynomial collocation for T (a) converges. The idea of the proof is as follows. Polynomial collocation may be interpreted as a projection method Ln T (a)f (n) = Ln g with f (n) ∈ Ran Pn . Here Ln is the projection of A2 (D) onto the polynomials of degree at most n (n) (n) that sends f to the polynomial Ln f determined by (Ln f )(zj ) = f (zj ) for j = 0, 1, . . . , n. We know that the finite section method Pn T (a)f (n) = Pn g converges, and the convergence of polynomial collocation will follow once we can prove that Ln − Pn  → 0. Proving this is analysis. 3.4. The Bergman metric The Bergman metric on D is given by its line element ds with 1 1 1 ds2 = k(z, z)|dz|2 = |dz|2 . π π (1 − |z|2 )2 It is also known as the hyperbolic or Poincar´e metric. The distance between z, w in D in this metric is    z−w  1 1 + (z, w) . dist(z, w) = √ log with (z, w) =  1 − (z, w) 1 − zw  2 π The geodesic circle Cb (r) is defined as Cb (r) = {z ∈ D : dist(z, b) = r}. The √ circle C0 (r) is an Euclidean circle with the radius tanh(r π). Here is another result of [6]. Let Γ = Cb (r) be a fixed geodesic circle in D and suppose that for each n the collocation points are the vertices of a hyperbolically regular (n + 1)-gon inscribed in Γ, that is, suppose the points

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153

are equidistantly distributed over Γ in the Bergman metric. If polynomial collocation converges for every invertible operator T (a) with a ∈ C(D), then necessarily b = 0, that is, Γ is an Euclidean circle centered at the origin. When passing from A2 (D) to A2 (G) over more general domains G, singling out “Euclidean circles centered at the origin” is a hopeless venture. Thus, let us replace polynomial collocation by another method. 3.5. Analytic element collocation We choose a system of points in D, (0)

z0 (1) (1) z0 , z1 (2) (2) (2) z0 , z1 , z2 ............... and then define the basis and test functions by (0)

(0)

K0 (1) (1) K 0 , K1 (2) (2) (2) K 0 , K1 , K2 ............... test functions,

K0 (1) (1) K0 , K1 (2) (2) (2) K0 , K1 , K2 ............... basis functions (n)

where Kj

denotes the Bergman kernel Kz(n) . This time the approximate

solution is sought as a linear combination

j

(n)

f (n) (z) = (n)

x0

(n)

(1 − z 0 z)2

(n)

+ ··· +

xn

(n)

(1 − z n z)2

(n)

and the coefficients x0 , . . . , xn are determined by the n + 1 equations   (n)  (n)  = g, Kj for j = 0, 1, . . . , n, T (a)f (n) , Kj which, by the reproducing property (f, Kz ) = f (z), is the same as   (n)   (n)  T (a)f (n) zj = g zj for j = 0, 1, . . . , n. We call the Galerkin–Petrov method at hand analytic element collocation: the approximate solution f (n) is taken from the linear span of the analytic (n) (n) functions K0 , . . . , Kn so that T (a)f (n) and g coincide at n + 1 prescribed collocation points. The following was proved in [6]. Theorem 5. Let Γ = Cb (r) be a fixed geodesic circle in D and suppose that for each n the collocation points are the vertices of a hyperbolically regular (n + 1)-gon inscribed in Γ. If a ∈ C(D) and T (a) is invertible on A2 (D), then analytic element collocation converges. Figure 3 shows phase plots of sums of Bergman kernels. In the image on the right of Figure 3, the kernels are Kj (z) = 1/(1 − rwj z)2 with r = 0.8,

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Figure 3. The sums of 4 and 50 Bergman kernels. wj = exp(2πij/n) (j = 0, . . . , n − 1) and n = 50. The red color indicates that the sum is almost a positive real constant. Indeed we have n−1 n−1 1 1  1 Kj (z) = . n j=0 n j=0 (1 − re−2πij/n z)2

This is a Riemannian integral sum with the limit  1  1 ∞  dx k = (k + 1)(rz) e−2πikx dx = 1, −2πix z)2 (1 − r e 0 0 k=0

for |rz| < 1. Thus, asymptotically the sum equals n, which is a number with argument zero and therefore gets red color in the phase plot.

Figure 4. The sum of Bergman kernels Kw with 50 randomly distributed points w with |w| = 0.8 (left), and a linear combination of 50 such kernels with random coefficients (right).

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The image on the left of Figure 4 is a phase plot of the sum of 50 Bergman kernels Kw with randomly distributed points w on a circle. Somewhat surprising is the appearance of an almost monochromatic (blue) circle through the poles. This property is lost when we consider a linear combination of such kernels with random real coefficients (uniformly distributed in [0, 1]), as is shown on the right. We conjecture that analytic element collocation does not converge if the collocation points are equidistantly distributed on a diameter (for polynomial collocation this was proved in [6]). Figure 5 shows sums of the corresponding   Bergman kernels. In this case we have, for z ∈ C \ (−∞, −1] ∪ [1, +∞) ,  n  1 1 dx 1 1 1 → = . (5) 2n + 1 j=−n (1 − kz/(n + 1))2 2 −1 (1 − xz)2 1 − z2 Our conjecture is based on the observation that the kernels (1/n)Kj (z) in the picture on the right of Figure 3 are a very good approximate partition of the unit, which is not at all the case for the normalized kernels giving Figure 5.

Figure 5. The sums of 15 and 150 Bergman kernels from a diameter. 3.6. Test computations To test the methods, we consider the equation T (a)f = g with the known solution f = Ku . Let χk (z) = z k for k ≥ 0. Clearly (T (χk )Ku )(z) = χk (z)Ku (z) =

zk . (1 − uz)2

On the other hand, (T (χk )Ku )(z) = (P (χk Ku ), Kz ) = (χk Ku , Kz ) = (Ku , χk Kz ) = (χk Kz , Ku ) = χk (u)Kz (u) =

uk . (1 − uz)2

Now take the harmonic function (the “Kaashoek symbol”) defined in D by a(z) = 19 z 2 + 37 z + 80 i + 10 z + 11 z 2 .

(6)

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In that case g = T (a)Ku is given by g(z) =

19 u2 + 37 u + 80 i + 10 z + 11 z 2 . (1 − uz)2

(7)

Since T (b) ≤ b∞ and 19 + 37 + 10 + 11 < 80, the operator T (a) is invertible. (n)

AEM. Put zj = zj = r e2πi(j−1)/n with 0 < r < 1 for j = 1, . . . , n. In the nth step of analytic element collocation we look for a linear combination of Bergman kernels n  (n) f = x k Kz k k=1

such that (T (a)f (n) )(zj ) = g(zj ) for j = 1, . . . , n. This is the linear system n 

ajk xk = g(zj ),

j = 1, . . . , n

k=1

with ajk = (T (a)Kzk )(zj ) =

19 z 2k + 37 z k + 80 i + 10 zj + 11 zj2 . (1 − z k zj )2

Figure 6 shows the phase plots of two approximate solutions f (n) for the righthand side (7) with u = 0.65 eiπ/4 and n = 50 collocation points uniformly distributed on circles with radii r = 0.4 and r = 0.8, respectively. The visual

Figure 6. Approximate solutions f (50) of AEM with g from (7) and 50 collocation points on circles with radii 0.4 and 0.8 coincidence of f (n) with the exact solution f = Ku inside the unit disk is apparent. In the image on the left, we see a number of zeros of f (n) in the exterior of the unit disk. The almost radial isochromatic lines (and the orthogonal contour lines of |f (n) |) indicate (polynomial) growth near the boundary of the square. The poles of the basis functions (Bergman kernels) cannot be seen in the left image since they are outside the region depicted. In the image on the right, these (double) poles are visible; they form a circular

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chain which separates two families of zeros. Poles and zeros together generate a gear-like structure. The corresponding error functions f (50) −f are displayed

Figure 7. Error functions f (50) − f for the solutions in Figure 6. In the brightest domain the error is less than 10−4 , the contour lines correspond to an increment by a factor of 10. in Figure 7. We used a special color scheme which allows one to estimate the magnitude of the error. The highlighted contour lines represent the levels |f (n) − f | = 10k with k = −4, −3, −2, −1, 0, 1, 2. According to the collocation

  Figure 8. Residual error functions T (a) f (50) − f corresponding to the solutions in Figure 6. condition, the residual error functions T (a)f (n) − g should have zeros at the collocation points. This is confirmed for r = 0.8 in the picture on the right of Figure 8, but for r = 0.4 (on the left) not even the number of zeros is correct. The reason is the severe ill-conditioning of the matrix (ajk ) for small values of r (even for moderate numbers n of collocation points), which causes relatively large errors in the solution of the linear system. The condition numbers of these matrices as functions of the radius r are shown in Figure 9

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Figure 9. Decadic logarithm of condition numbers of the matrices (ajk ) for n = 15, 30, 45, 60, 75 collocation points (left), and log10 of norm of error with 50 collocation points and |u| = 0.25, 0.40, 0.55, 0.70, 0.85 (right) as functions of r ∈ [0.1, 0.9]

(left) for several values of n. Looking at these graphs one may wonder why we at all get reasonable results for r < 0.8 (we used the Moore–Penrose routine of Matlab to solve the linear system, the “backslash operator” leads to different results). The diagram on the right of Figure 9 displays the decadic logarithm of the error for several values of |u| (which controls the position of the pole of f = Ku ) as functions of r. Though, theoretically, the AEM method converges for all r as n → ∞, good results can be expected only if one takes care for an interplay between n and r. If, for some fixed n, the radius r is too small, the Kzj are almost linearly dependent, which not only causes a huge condition number, but also bad approximation properties. On the other hand, if r is too close to 1, approximation is again bad, now because f (n) has poles near the boundary of D, which produces ‘ripples’ of the approximating function. Finding the optimal relation between n and r seems to be a challenging problem. √ FSM. In the orthonormal basis { n + 1χn }∞ n=0 , the matrix T (a) is given by (4) with j, k = 0, 1, . . .. The n × n truncation of this matrix is √  n 2 jk aj−k . B = (bjk )nj,k=1 = |j − k| + j + k j,k=1 Note that we changed j, k to running through 1, 2, . . .. In our case aj−k is 19, 37, 80i, 10, 11 for j − k = −2, −1, 0, 1, 2, respectively, and aj−k = 0 whenever |j − k| ≥ 3. The nth step of the finite section method requires the solution of the linear system n  bjk xk = gj , j = 1, . . . , n, k=1

Equations with a Toeplitz operator where the gj ’s are the coefficients in T (a)f = g = we get

∞

j=1 gj

159 √

j χj−1 . From (7)

g(z) = (19u2 + 37u + 80i + 10z + 11z 2 )(1 + 2uz + 3u2 z 2 + 4u3 z 3 + · · · ), which yields that g1 = 19u2 + 37u + 80 i,

1 g2 = √ (2 · 19 u3 + 2 · 37 u2 + 2 · 80 i u + 10), 2

and that gj equals  1  √ 19 j uj+1 + 37 j uj + 80i j uj−1 + 10 (j − 1) uj−2 + 11 (j − 2) uj−3 j for j ≥ 3. The approximate solution is given by √ √ f (n) (z) = x1 + x2 2 z + · · · + xn n z n−1 , while the exact solution is √ √ √ √ Ku (z) = 1 + 2 u 2 z + · · · + n un−1 n z n−1 + . . . . Figure 10 depicts phase plots of the approximate solution for the right-hand

Figure 10. Phase plots of approximate solutions of FSM with u = 0.65 eiπ/4 for n = 12 (left) and n = 50 (right, compare with Figure 2 and Figure 6) side g from (7) with u = 0.65 eiπ/4 for n = 12 and n = 50, respectively, the corresponding error functions are displayed in Figure 11. In the picture on the left, the error in the inner region is of magnitude 10−6 . In the image on the right it is about 10−15 in the central domain (the shimmering colors in the almost circular domain in the middle indicate that the values have random character, which is due to numerical cancelation effects). Figure 12 shows the behavior of the maximum norm of the error for |u| = 0.25, 0.40, 0.55, 0.70, 0.85 as function of the truncation parameter n for 3 ≤ n ≤ 100. Comparing this with Figure 9 (with the same number of basis functions), we see that the approximation properties FSM are much better than those of AEM. Compared with AEM, also the stability of FSM is a notch above. Remarkably, the

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Figure 11. Phase plots of error functions for FSM with |u| = 0.65 and n = 12 (left), respectively n = 50 (right, compare with Figure 7)

Figure 12. Logarithm of maximum norm of error for FSM with |u| = 0.25, 0.40, 0.55, 0.70, 0.85 as functions of n (left) and phase plot of solution with |u| = 0.95 for n = 100 (right) (numerically computed) condition number of the matrix (bjk ) is less than 2 for n ≤ 100. So, at least in our experiments, FSM outperforms AEM in every respect. This is demonstrated with the image on the right-hand side of Figure 12, which depicts an approximate solution for |u| = 0.95, computed with FSM and n = 100 basis functions – this would be impossible with AEM. PCM. We consider the equation T (a)f = g with f = Ku . An approximation f (n) to f is sought in the form f (n) (z) =

n 

xk z k−1 .

k=1

Thus, letting cjk = (T (a)χk−1 )(zj ),

j, k = 1, . . . , n,

Equations with a Toeplitz operator we arrive at the system n 

cjk xk = g(zj ),

161

j = 1, . . . , n.

k=1

Straightforward computation shows that if k ≥ 1 and m ≥ 0, then (T (χm )χk−1 )(z) = z m+k−1 and (T (χm )χk−1 )(z) = 2

k−m k−1−m k z

if k − 1 ≥ m,

0

if k − 1 < m. 2

Taking a(z) = 19 z + 37 z + 80i + 10z + 11z , we get the right-hand side g(z) =

19 u2 + 37 u + 80i + 10z + 11z 2 (1 − uz)2

and the matrix (cjk ) with cj1 = 80 i + 10zj + 11zj2 , 37 + 80 izj + 10zj2 + 11zj3 , cj2 = 2 19(k − 2) k−3 37(k − 1) k−2 cjk = zj + zj + 80izjk−1 + 10zjk + 11zjk+1 k k for 3 ≤ k ≤ n. Figure 13 shows the phase plots of two approximate solutions f (n) for the right-hand side (7) with u = 0.65 eiπ/4 and n = 50 collocation points uniformly distributed on circles with radii r = 0.4 and r = 0.8, respectively. While the approximate solution f (50) for r = 0.8 on the right-hand side shows

Figure 13. Approximate solutions of PCM for g from (7) and 50 collocation points on circles with radii r = 0.4 and r = 0.8 good visual accord with the Bergman kernel inside the unit disk, this is not the case for r = 0.4 (left). This is also apparent in the corresponding error

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Figure 14. Error functions corresponding to the solutions in Figure 13. functions f (n) −f depicted in Figure 14. Due to the collocation condition, the residual errors T (a) (f (n) − f ) (not depicted) should vanish at the collocation points. We find it somewhat surprising that for n = 50 and r = 0.8 (and other constellations of r and n) the zeros of f (n) − f are also very close to these points, as can be seen in the image on the right. If r is small, the

Figure 15. Decadic logarithm of condition numbers of the matrices (cjk ) for n = 15, 30, 45, 60, 75 collocation points (left), and log10 of norm of error f (n) − f for PCM with 50 collocation points and |u| = 0.25, 0.40, 0.55, 0.70, 0.85 (right) as functions of r ∈ [0.1, 0.9] conditions of the matrices (cjk ) are extremely large, which becomes apparent from the image on the left of Figure 15. The diagram on the right displays the decadic logarithm of the error for several values of |u|. In contrast to the AEM method, as a rule, the results are getting better the closer the collocation points are to the boundary of the domain. 3.7. General domains Finally, analytic element collocation is stable under conformal mappings, and this allows us to transfer Theorem 5 to more general domains. Let G be

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an open, connected, and simply connected subset of C. The Bergman space A2 (G) is the Hilbert space consisting of the analytic functions in L2 (G) with 2 area measure dA = dx dy. If {en }∞ n=0 is any orthonormal basis in A (G), then ∞  Kw (z) = kG (z, w) = en (z)en (w) (8) n=0

is independent of the choice of the basis and is the reproducing kernel of A2 (G): the orthogonal projection P : L2 (G) → A2 (G) acts by the rule  (P f )(z) = (f, Kz ) = kG (z, w)f (w)dA(w), z ∈ G. G

The Bergman metric on G is defined by ds2 = kG (z, z)|dz|2 . For a ∈ L∞ (G), the Toeplitz operator T (a) : A2 (G) → A2 (G) is the bounded linear operator given by T (a)f = P (af ). Analytic element collocation in A2 (G) is defined in (n) the natural fashion: we choose collocation points zj ∈ G (j = 0, . . . , n) and then take the reproducing kernels Kj = Kz(n) given by (8). Herewith another j

result of [6]. Theorem 6. Let Γ = Cb (r) be a fixed geodesic circle in G and suppose that for each n the collocation points are the vertices of a regular (n + 1)-gon in the Bergman metric inscribed in Γ. If a ∈ C(G) and T (a) is invertible, then analytic element collocation for T (a) on A2 (G) converges.

Appendix: Phase plots Most illustrations in this paper are so-called phase plots, which depict a complex function f : G → C as an image on its domain G by color-coding the values of f . We give a very brief description of this technique, readers interested in details and further results are invited to consult [18], [19], [20]. In a pure phase plot, each point z ∈ G is colored according to the phase f (z)/|f (z)| of f (z) using the standard hsv-color scheme, which can be seen in the leftmost picture of the upper row of Figure 16, showing a phase plot of f (z) = z in the square |Re z| < 2, |Im z| < 2. The image below is a phase plot of g(z) = (1 − z)5 /(z 2 + z + 1) in the same domain. Interpreting the upper and the lower squares as parts of the complex w-plane and the z-plane, respectively, the image in the lower row can be seen as a pull-back of the image in the top row via the function g. A number of properties of a meromorphic function can be read off easily from its phase plot. For instance, those points of G where all colors meet are either zeros or poles, both can be distinguished by the orientation of colors in their neighborhood, and finding their multiplicity is a simple counting exercise. Figures 1 and 4 are such pure phase plots, and the reader may wish to look at them again. Though phase plots neglect the modulus of a function, meromorphic functions can (in principle) be uniquely recovered from their phase plot up to a positive constant factor. Nevertheless it is convenient to incorporate additional information. The pairs of images in the middle and in the right

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↑ g

↑g

↑g

Figure 16. Generation of different types of phase plots by pulling back some standard colorings of the w-plane to the z-plane via w = g(z) = (1 − z)5 /(z 2 + z + 1) column are such enhanced phase plots, where we generated contour lines of the modulus by adding shades of gray. In the middle pair the levels of contour lines are scaled logarithmically, so that moving from one line to the next increases (or decreases) |f | by some factor. Figures 2 and 6 are of this type. In the two images in the right column of Figure 16 we have depicted only a finite number of contour lines at the levels 10−4 , 10−3 , 10−2 , 0.1, 1, 10, 100 (in the upper picture only a few are visible), which is helpful in estimating the modulus of the error function or the residue depicted in Figures 7 and 8. A fourth color scheme is used in Figure 5, where contour lines of the modulus and the phase of f are added, which generates a conformal tiling.

References [1] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of Matrix and Operator Functions: The State Space Method. Oper. Theory Adv. Appl. 178, Birkh¨ auser Verlag, Basel 2008. [2] D.A. Bini, G. Fiorentino, L. Gemignani, and B. Meini, Effective fast algorithms for polynomial spectral factorization. Numer. Algorithms 34 (2003), 217–227. [3] A. B¨ ottcher, Truncated Toeplitz operators on the polydisk, Monatshefte f. Math. 110 (1990), 23–32.

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[4] A. B¨ ottcher and M. Halwass, Wiener–Hopf and spectral factorization of real polynomials by Newton’s method, Linear Algebra Appl. 438 (2013), 4760–4805. [5] A. B¨ ottcher and B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Universitext, Springer-Verlag, New York 1999. [6] A. B¨ ottcher and H. Wolf, Galerkin–Petrov methods for Bergman space Toeplitz operators, SIAM J. Numer. Analysis 30 (1993), 846–863. [7] L.A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973), 433–439. [8] L. Ephremidze, N. Salia, and I. Spitkovsky, Some aspects of a novel matrix spectral factorization algorithm, Proc. A. Razmadze Math. Inst. 166 (2014), 49–60. [9] I. Gohberg and I.A. Feldman, Convolution Equations and Projection Methods for Their Solution, Amer. Math. Soc., Providence, RI, 1974 [Russian original: Nauka, Moscow 1971]. [10] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Vol. I: Birkh¨ auser Verlag, Basel 1990; Vol. II: Birkh¨ auser Verlag, Basel 1993. [11] I. Gohberg, M. A. Kaashoek, and I.M. Spitkovsky, An overview of matrix factorization theory and operator applications, in: Oper. Theory Adv. Appl. 141, pp. 1–102, 2003. [12] T.N.T. Goodman, C.A. Micchelli, G. Rodriguez, and S. Seatzu, Spectral factorization of Laurent polynomials, Adv. Comput. Math. 7 (1997), 429–454. [13] G. Janashia, E. Lagvilava, and L. Ephremidze, A new method of matrix spectral factorization, IEEE Trans. Inform. Theory 57 (2011), 2318–2326. [14] H.T. Kung, On computing reciprocals of power series, Numer. Math. 22 (1974), 341–348. [15] A.H. Sayed and T. Kailath, A survey of spectral factorization methods, Numer. Linear Algebra Appl. 8 (2001), 467–496. [16] M. Sieveking, An algorithm for division of power series, Computing 10 (1972), 153–156. [17] F. Uhlig, Are the coefficients of a polynomial well-conditioned functions of its roots? Numer. Math. 61 (1992), 383–393. [18] E. Wegert, Visual Complex Functions. An Introduction with Phase Portraits, Birkh¨ auser and Springer, Basel, 2012. [19] E. Wegert, Visual Exploration of Complex Functions, in: Mathematical Analysis, Probability and Applications. Plenary Lectures ISAAC 2015, Macau, China (T. Qian and L. G. Rodino Eds.), pp. 253–279, Springer, 2016. [20] E. Wegert and G. Semmler, Phase plots of complex functions: a journey in illustration, Notices of the Amer. Math. Soc. 58 (2011), 768–781.

Albrecht B¨ ottcher Technische Universit¨ at Chemnitz, Fakult¨ at f¨ ur Mathematik 09107 Chemnitz Germany e-mail: [email protected]

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Elias Wegert Technische Universit¨ at Bergakademie Freiberg, Fakult¨ at f¨ ur Mathematik und Informatik, Institut f¨ ur Angewandte Analysis 09596 Freiberg Germany e-mail: [email protected]

On the maximal ideal space of even quasicontinuous functions on the unit circle Torsten Ehrhardt and Zheng Zhou Dedicated to Marinus A. Kaashoek on the occasion of his eightieth birthday.

Abstract. Let P QC stand for the set of all piecewise quasicontinuous functions on the unit circle, i.e., the smallest closed subalgebra of L∞ (T) which contains the classes of all piecewise continuous functions P C and all quasicontinuous functions QC = (C + H ∞ ) ∩ (C + H ∞ ). We analyze the fibers of the maximal ideal spaces M (P QC) and M (QC) over  where QC  stands for the C ∗ -algebra of maximal ideals from M (QC),  is all even quasicontinuous functions. The maximal ideal space M (QC) described and partitioned into various subsets corresponding to different descriptions of the fibers. Mathematics Subject Classification (2010). Primary 46J10; Secondary 46J20, 47B35. Keywords. quasicontinuous function, piecewise quasicontinuous function, maximal ideal space.

1. Introduction Let C(T) stand for the class of all (complex valued) continuous functions on the unit circle T = { t ∈ C : |t| = 1 }, let L∞ (T) stand for the C ∗ -algebra of all Lebesgue measurable and essentially bounded functions defined on T, and let P C stand for the set of all piecewise continuous functions on T, i.e., all functions f : T → C such that the one-sided limits f (τ ± 0) = lim f (τ e±iε ) ε→+0

exist at each τ ∈ T. The class of quasicontinuous functions is defined by QC = (C + H ∞ ) ∩ (C + H ∞ ), where H ∞ stands for the Hardy space consisting of all functions f ∈ L∞ (T) > 2π 1 such that their Fourier coefficients fn = 2π f (eix )e−inx dx vanish for all 0 © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_6

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n < 0. The space H ∞ is the Hardy space of all functions f ∈ L∞ (T) such that fn = 0 for all n > 0. The Toeplitz and Hankel operators T (a) and H(a) with a ∈ L∞ (T) acting on 2 (Z+ ) are defined by the infinite matrices T (a) = (aj−k )∞ j,k=0 ,

H(a) = (aj+k+1 )∞ j,k=0 .

Quasicontinuous functions arise in connection with Hankel operators. Indeed, it is known that both H(a) and H(˜ a) are compact if and only if a ∈ QC (see, e.g., [1, Theorem 2.54]). Here, and in what follows, a ˜(t) := a(t−1 ), t ∈ T. Sarason [9], generalizing earlier work of Gohberg/Krupnik [6] and Douglas [2], established necessary and sufficient conditions for Toeplitz operators T (a) with a ∈ P QC to be Fredholm. This result is based on two ingredients. Firstly, due to Widom’s formula T (ab) = T (a)T (b) + H(a)H(˜b), Toeplitz operators T (a) with a ∈ QC commute with other Toeplitz operators T (b), b ∈ L∞ (T), modulo compact operators. Hence C ∗ -algebras generated by Toeplitz operators can be localized over QC. Secondly, in the case of the C ∗ -algebra generated by Toeplitz operators T (a) with a ∈ P QC, the local quotient algebras arising from the localization allow an explicit description, which is facilitated by the characterization of the fibers of the maximal ideal space M (P QC) over maximal ideals ξ ∈ M (QC). These underlying results were also developed by Sarason [8, 9], and we are going to recall them in what follows. Let A be a commutative C ∗ -algebra, and let B be a C ∗ -subalgebra such that both contain the same unit element. Then there is a natural continuous map between the maximal ideal spaces, π : M (A) → M (B),

α → α|B

defined via the restriction. For β ∈ M (B) introduce Mβ (A) = { α ∈ M (A) : α|B = β } = π −1 (β), which is called the fiber of M (A) over β. The fibers Mβ (A) are compact subsets of M (A), and M (A) is the disjoint union of all Mβ (A). Because A and B are C ∗ -algebras, π is surjective, and therefore each fiber Mβ (A) is non-empty (see, e.g., [1, Sect. 1.27]). Corresponding to the embeddings between the C ∗ -algebras C(T), QC, P C, and P QC, which are depicted in first diagram below, there are natural maps between the maximal ideal spaces shown in the second diagram: P QC

QC

M (P QC)

M (QC)

PC

C(T)

M (P C) ∼ = T × {+1, −1}

M (C(T)) ∼ =T

Therein the identification of y ∈ M (P C) with (τ, σ) ∈ T × {+1, −1} is made through y(f ) = f (τ ± 0) for σ = ±1, f ∈ P C.

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Let Mτ (QC) stand for the fiber of M (QC) over τ ∈ T, i.e., Mτ (QC) = { ξ ∈ M (QC) : ξ(f ) = f (τ ) for all f ∈ C(T) }, and define

? @ Mτ± (QC) = ξ ∈ M (QC) : ξ(f ) = 0 whenever lim sup |f (t)| = 0 and f ∈ QC . t→τ ±0

Both Mτ+ (QC) and Mτ− (QC) are closed subsets of Mτ (QC). Sarason introduced another subset Mτ0 (QC) (to be defined in (2.3) below) and established the following result (see [9], or [1, Prop. 3.34]). Proposition 1.1. Let τ ∈ T. Then Mτ0 (QC) = Mτ+ (QC)∩Mτ− (QC),

Mτ+ (QC)∪Mτ− (QC) = Mτ (QC). (1.1)

The previous definitions and observations are necessary to analyze the fibers of M (P QC) over ξ ∈ M (QC). In view of the second diagram above, for given z ∈ M (P QC) we can define the restrictions ξ = z|QC , z|C(T) ∼ = τ ∈ T, and y = z|P C ∼ = (τ, σ) ∈ T×{+1, −1}. Note that ξ ∈ Mτ (QC). Consequently, one has a natural map z ∈ M (P QC) → (ξ, σ) ∈ M (QC) × {+1, −1}.

(1.2)

This map is injective because P QC is generated by P C and QC. Therefore, M (P QC) can be identified with a subset of M (QC) × {+1, −1}. With this identification, the fibers Mξ (P QC) = { z ∈ M (P QC) : z|QC = ξ } are given as follows (see [8], or [1, Thm. 3.36]). Theorem 1.2. Let ξ ∈ Mτ (QC), τ ∈ T. Then (a) Mξ (P QC) = { (ξ, +1) } for ξ ∈ Mτ+ (QC) \ Mτ0 (QC); (b) Mξ (P QC) = { (ξ, −1) } for ξ ∈ Mτ− (QC) \ Mτ0 (QC); (c) Mξ (P QC) = { (ξ, +1), (ξ, −1) } for ξ ∈ Mτ0 (QC). In order to describe the content of this paper, let us consider what happens if one wants to develop a Fredholm theory for operators from the C ∗ -algebra generated by Toeplitz and Hankel operators with P QC-symbols [10]. In this situation, one cannot use localization over QC because the commutativity property fails. However, one can localize over A = { a ∈ QC : a = a QC ˜ }, the C ∗ -algebra of all even quasicontinuous functions. Indeed, due to the idenA commutes with tity H(ab) = T (a)H(b) + H(a)T (˜b), any T (a) with a ∈ QC ∞ any H(b), b ∈ L (T), modulo compact operators. When faced with the problem of identifying the local quotient algebras, it is necessary to understand A This is what this paper is about. the fibers of M (P QC) over η ∈ M (QC).  A and the C ∗ -algebra C(T) of all even continuous functions are When QC added to the picture, one arrives at the following diagrams:

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P QC

QC

 QC

M (P QC)

M (QC)

PC

C(T)

 C(T)

M (P C)

M (C) ∼ =T

Ψ

Ψ

 M (QC)

 ∼ M (C) = T+

As before, the diagram on the left shows the embeddings of the C ∗ -algebras, and the one on the right displays the corresponding (surjective) mappings between the maximal ideal spaces. Here T+ = { t ∈ T : Im(t) > 0 } and T+ = T+ ∪ {+1, −1}. The map Ψ is defined in such a way that the preimage of τ ∈ T+ equals the set {τ, τ }, which consists of either one or two points. Recall that Theorem 1.2 gives a description of the fibers of M (P QC) over ξ ∈ M (QC). Hence if we want to understand the fibers of M (P QC) over A it is sufficient to analyze the fibers of M (QC) over η ∈ M (QC). A η ∈ M (QC), Let A ξ → ξˆ := ξ|  , (1.3) Ψ : M (QC) → M (QC), QC A be the (surjective) map shown in the previous diagram. For η ∈ M (QC) define M η (QC) = { ξ ∈ M (QC) : ξˆ = η }, (1.4) A over the fiber of M (QC) over η. Let us also define the fibers of M (QC) τ ∈ T+ , A = { η ∈ M (QC) A : η(f ) = f (τ ) for all f ∈ C(T)  Mτ (QC) }. Notice that we have the disjoint unions   M (QC) = M η (QC) = Mτ (QC),  η∈M (QC)

A = M (QC)

(1.5)

(1.6)

τ ∈T



A Mτ (QC).

(1.7)

τ ∈T+

Furthermore, it is easy to see that Ψ maps Mτ (QC) ∪ Mτ¯ (QC)

(1.8)

A for each τ ∈ T+ . onto Mτ (QC) The main results of this paper concern the description of the fibers A into disjoint sets, analogous to M η (QC) and the decomposition of Mτ (QC) the decomposition of Mτ (QC) into the disjoint union of Mτ0 (QC),

Mτ+ (QC) \ Mτ0 (QC),

and

Mτ− (QC) \ Mτ0 (QC)

(1.9)

(see Proposition 1.1). This will be done in Section 3. In Section 2 we establish auxilliary results. In Section 4 we decribe the fibers M η (P QC) of M (P QC) A over η ∈ M (QC). A were alSome aspects of the relationship between M (QC) and M (QC) ready mentioned by Power [7]. They were used by Silbermann [10] to establish a Fredholm theory for operators from the C ∗ -algebra generated by Toeplitz

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and Hankel operators with P QC-symbols. Our motivation for presenting the results of this paper comes from the goal of establishing a Fredholm theory and a stability theory for the finite section method for operators taken from the C ∗ -algebra generated by the singular integral operator on T, the flip operator, and multiplication operators by (operator-valued) P QC-functions [5]. This generalizes previous work [3, 4] and requires the results established here.

2. Approximate identities and VMO A we need In order to examine the relationship between M (QC) and M (QC), to recall some results and definitions concerning QC and M (QC). For τ = eiθ ∈ T and λ ∈ Λ := [1, ∞) let us define the moving average,  θ+π/λ λ (mλ a)(τ ) = a(eix ) dx. (2.1) 2π θ−π/λ Since each pair (λ, τ ) ∈ Λ×T induces a bounded linear functional δλ,τ ∈ QC ∗ , δλ,τ : QC → C,

a → (mλ a)(τ ),

(2.2)

∗

the set Λ × T can be identified with a subset of QC . In fact, we have the following result, where we consider the dual space QC ∗ with the weak-∗ topology (see [1, Prop. 3.29]). Proposition 2.1. M (QC) = (clos QC ∗ (Λ × T)) \ (Λ × T). For τ ∈ T, let Mτ0 (QC) denote the points in M (QC) that lie in the weak-∗ closure of Λ × {τ } regarded as a subset of QC ∗ , Mτ0 (QC) = M (QC) ∩ clos QC ∗ (Λ × {τ }).

(2.3)

Mτ0 (QC)

Obviously, is a compact subset of the fiber Mτ (QC). We remark that here and in the above proposition one can use arbitrary approximate identities (in the sense of Section 3.14 in [1]) instead of the moving average (see [1, Lemma 3.31]). For a ∈ L1 (T) and τ = eiθ ∈ T, the integral gap γτ (a) of a at τ is defined by     θ  1 θ+δ  1   (2.4) γτ (a) := lim sup  a(eix ) dx − a(eix ) dx .   δ δ δ→+0 θ θ−δ It is well known [8] that QC = V M O ∩ L∞ (T), where V M O ⊂ L1 (T) refers to the class of all functions with vanishing mean oscillation on the unit circle T. We will not recall its definition here, but refer to [8, 9, 1]. In the following lemma (see [9] or [1, Lemma 3.33]), V M O(I) stands for the class of functions with vanishing mean oscillation on an open subarc I of T. Furthermore, we identify a function q ∈ QC with its Gelfand transform, a continuous function on M (QC).

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Lemma 2.2. (a) If q ∈ V M O, then γτ (q) = 0 for each τ ∈ T. (b) If q ∈ V M O(a, τ ) ∩ V M O(τ, b) and γτ (q) = 0, then q ∈ V M O(a, b). (c) If q ∈ QC such that q|Mτ0 (QC) = 0 and if p ∈ P C, then γτ (pq) = 0. Let χ+ (resp., χ− ) be the characteristic function of the upper (resp., lower) semi-circle. The next lemma is based on the preceding lemma. Lemma 2.3. Let q ∈ QC. (a) If q is an odd function, i.e., q(t) = −q(1/t), then q|M10 (QC) = 0 and 0 (QC) = 0. q|M−1 0 (QC) = 0, then pq ∈ QC whenever p ∈ P C ∩ C(T \ {±1}). (b) If q|M±1 0 (QC) = 0, then qχ+ , qχ− ∈ QC. (c) If q|M10 (QC) = 0 and q|M−1 Proof. For part (a), since q ∈ QC is an odd function, it follows from (2.1) that δλ,±1 (q) = (mλ q)(±1) = 0 for all λ ≥ 1. Therefore, by (2.2) and (2.3), q vanishes on Λ × {±1} ⊆ QC ∗ and hence on 0 its closure, in particular, also on M±1 (QC). 0 (QC) = 0. We are going to use the fact For part (b) assume that q|M±1 that QC = V M O ∩L∞ . It follows from the definition of V M O-functions that the product of a V M O-function with a uniformly continuous function is again V M O. Therefore, pq is V M O on the interval T\{±1}. By Lemma 2.2(c), the integral gap γ±1 (pq) is zero. Hence pq is V M O on all of T by Lemma 2.2(b). This implies pq ∈ QC. For case (c) decompose q = qc1 + qc−1 such that c±1 ∈ C(T) vanishes identically in a neighborhood of ±1. Then apply the result of (b).  We will also need the following lemma. A for each fixed λ ∈ [1, ∞) and Lemma 2.4. δλ,τ is not multiplicative over QC τ ∈ T. Proof. Let τ = eiθ and consider φ(eix ) = eikx +e−ikx with k ∈ N. Apparently, A Note that the moving average is generated by the function φ ∈ QC.  ∞ 1 K(x) = χ(−π,π) (x), δλ,τ (q) = (mλ q)(eiθ ) = λK(λx)q(ei(θ−x) ) dx. 2π −∞ Hence, by formula 3.14(3.5) in [1], or by direct computation, δλ,τ (φ2 ) − δλ,τ (φ)δλ,τ (φ)    2 ˆ 2k )e2kiθ + K(− ˆ k )ekiθ + K(− ˆ 2k )e−2kiθ + 2 − K( ˆ k )e−kiθ = K( λ λ λ λ "   2 # 2 sin(kπ/λ) sin(kπ/λ) sin(2kπ/λ) − +2−2 , = 2 cos(2kθ) 2kπ/λ kπ/λ kπ/λ

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ˆ is the Fourier transform of the above K. Note that sin x → 0 as where K x x → ∞. Hence, for each fixed λ, one can choose a sufficiently large k ∈ N, such that with the corresponding φ, δλ,τ (φ2 ) − δλ,τ (φ)δλ,τ (φ) > 1. 

Therefore δλ,τ is not multiplicative for each λ and τ .

A 3. Fibers of M (QC) over M (QC) Now we are going to describe the fibers M η (QC). To prepare for it, we make the following definition. Given ξ ∈ M (QC), we define its “conjugate” ξ  ∈ M (QC) by ξ  (q) := ξ(˜ q ), q ∈ QC. (3.1)  ˆ ˆ A Recalling also definition (1.3), it is clear that ξ = ξ ∈ M (QC). Furthermore, the following statements are obvious: (i) If ξ ∈ Mτ (QC), then ξ  ∈ Mτ¯ (QC). (ii) If ξ ∈ Mτ± (QC), then ξ  ∈ Mτ¯∓ (QC). (iii) If ξ ∈ Mτ0 (QC), then ξ  ∈ Mτ¯0 (QC). For the characterization of the fibers M η (QC) we have to distinguish whether A with τ ∈ {+1, −1} or with τ ∈ T+ . In this connection recall η ∈ Mτ (QC) formula (1.7). A τ ∈ {+1, −1} 3.1. Fibers over Mτ (QC), A the following result is For the description of M η (QC) with η ∈ M±1 (QC) crucial. + Proposition 3.1. If ξ1 , ξ2 ∈ M±1 (QC) and ξˆ1 = ξˆ2 , then ξ1 = ξ2 .

Proof. Each q ∈ QC admits a unique decomposition q + q˜ q − q˜ q= + =: qe + qo , 2 2 where qe is even and qo is odd. By Lemma 2.3(ac), we have qo χ− ∈ QC, and ξ1 (q) = ξ1 (qe ) + ξ1 (qo ) = ξ1 (qe ) + ξ1 (qo − 2qo χ− ) = η(qe ) + η(qo − 2qo χ− ) = ξ2 (qe ) + ξ2 (qo − 2qo χ− ) = ξ2 (qe ) + ξ2 (qo ) = ξ2 (q). A and that lim qo (t)χ− (t) = 0, Note that qo − 2qo χ− = qo (χ+ − χ− ) ∈ QC t→1+0

whence ξi (qo χ− ) = 0. It follows that ξ1 = ξ2 . A Then either Theorem 3.2. Let η ∈ M±1 (QC). 0 (a) M η (QC) = {ξ} with ξ = ξ  ∈ M±1 (QC), or (b) M η (QC) = {ξ, ξ  } with + − (QC) \ M±1 (QC) ξ ∈ M±1

and

− + ξ  ∈ M±1 (QC) \ M±1 (QC).



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A implies Proof. From the statement (1.8) it follows that ξˆ ∈ M±1 (QC) ξ ∈ M±1 (QC). Therefore we have ∅ =  M η (QC) ⊆ M±1 (QC) whenever A Now the assertion follows from Proposition 1.1, Proposition η ∈ M±1 (QC). 3.1, and the statements (i)–(iii) above.  A which give rise to the Next we want to characterize those η ∈ M±1 (QC) ∗ A first case. Consider the functionals δλ,τ ∈ QC associated with the moving average (2.2), and define, in analogy to (2.3), A := M (QC) A ∩ clos  ∗ (Λ × {τ }). Mτ0 (QC) QC

(3.2)

We will use this definition for τ ∈ T+ = T+ ∪ {+1, −1}. 0 Theorem 3.3. The map Ψ : ξ → ξˆ is a bijection from M±1 (QC) onto 0 A M±1 (QC).

Proof. Without loss of generality consider the case τ = 1. First of all, Ψ A Indeed, it follows from (2.3) that for any maps M10 (QC) into M10 (QC). 0 A ⊂ QC and ε > 0, there exists λ ∈ Λ ξ ∈ M1 (QC), any q1 , . . . , qk ∈ QC ˆ i ) − δλ,1 (qi )| < ε. such that |ξ(qi ) − δλ,1 (qi )| < ε for all i. But this is just |ξ(q ∗ ˆ ∗ Therefore, ξ lies in the weak- closure of {δλ,1 |QC  }λ∈Λ . Hence, by (3.2), 0 A ˆ ξ ∈ M1 (QC). The injectivity of the map Ψ|M10 (QC) follows from Theorem 3.2 or Proposition 3.1. A By It remains to show that Ψ|M10 (QC) is surjective. Let η ∈ M10 (QC). definition, there exists a net {λω }ω∈Ω (with λω ∈ Λ) such that the net {δλω }ω∈Ω := {δλω ,1 }ω∈Ω converges to η (in the weak-∗ topology of funcA Note that δλ (q) = 0 for any λ ∈ Λ whenever q ∈ QC is an tionals on QC). odd function. Therefore the net {δλω }ω∈Ω (regarded as functionals on QC) converges to the functional ξ ∈ QC ∗ defined by ξ(q) := η(

q + q˜ ), 2

q ∈ QC.

Indeed, we have δλω (q) = 12 δλω (q + q˜) → 12 η(q + q˜) = ξ(q). It follows that ξ ∈ clos QC ∗ (Λ × {1}). Next we show that ξ is multiplicative over QC, i.e., ξ ∈ M (QC). Given arbitrary p, q ∈ QC we can decompose them into even and odd parts as p = pe + po , q = qe + qo . The even part of pq equals pe qe + po qo . Therefore using the definition of ξ in terms of η we get ξ(p)ξ(q) = η(pe )η(qe ) = η(pe qe ),

ξ(pq) = η(pe qe + po qo ).

Hence the multiplicativity of ξ follows if we can show that η(po qo ) = 0. To 0 (QC) = 0 see this we argue as follows. By Lemma 2.3(ac), we have po qo |M±1 and po qo χ+ ∈ QC, and hence by Lemma 2.2 the integral gap    1 δ    γ1 (po qo χ+ ) = lim sup  (po qo )(eix ) dx = 0.   δ δ→+0 0

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In other words, as λ → +∞,  π/λ  λ λ π/λ ix δλ (po qo ) = (po qo )(e ) dx = (po qo )(eix ) dx → 0. 2π −π/λ π 0 A converges to Since the net {δλω }ω∈Ω (regarded as functionals on QC) A η ∈ M (QC), it follows from Lemma 2.4 that λω → +∞. Therefore, δλω (po qo ) → 0

and

δλω (po qo ) → η(po qo ).

We obtain η(po qo ) = 0 and conclude that ξ is multiplicative. Combined with ˆ Hence the map the above this yields ξ ∈ M10 (QC), while clearly η = ξ. 0 0 A Ψ : M1 (QC) → M1 (QC) is surjective.  The previous two theorems imply the following. 0 A A Moreover, Corollary 3.4. M±1 (QC) is a closed subset of M±1 (QC). 0 A 0 (a) if η ∈ M±1 (QC), then M η (QC) = {ξ} with ξ = ξ  ∈ M±1 (QC); 0 A η A (b) if η ∈ M±1 (QC) \ M±1 (QC), then M (QC) = {ξ, ξ  } such that + − − + ξ ∈ M±1 (QC) \ M±1 (QC) and ξ  ∈ M±1 (QC) \ M±1 (QC).

A decomposes into the disjoint union of Note also that M±1 (QC) A \ M 0 (QC) A M±1 (QC) ±1

and

0 A M±1 (QC),

and that the map Ψ is a two-to-one map from M±1 (QC) \ A \ M 0 (QC). A M±1 (QC) ±1

(3.3) 0 M±1 (QC)

onto

A τ ∈ T+ 3.2. Fibers over Mτ (QC), A with τ ∈ T+ . This Now we consider the fibers of M η (QC) over η ∈ Mτ (QC) case is easier than the previous one. Proposition 3.5. If ξˆ1 = ξˆ2 for ξ1 , ξ2 ∈ Mτ (QC) with τ ∈ T+ , then ξ1 = ξ2 . Proof. Otherwise, there exists a q ∈ QC such that ξ1 (q) = 0, ξ2 (q) = 0. Since τ ∈ T+ , one can choose a smooth function cτ such that cτ = 1 in a neighborhood of τ and such that it vanishes on the lower semi-circle. Now, A Note that q −q is continuous at τ and vanishes construct q = qcτ + qc Bτ ∈ QC. A and ξˆ1 = ξˆ2 , there, hence ξ1 (q − q) = ξ2 (q − q) = 0. But then, since q ∈ QC we have 0 = ξ1 (q) = ξ1 (q) = ξ2 (q) = ξ2 (q) = 0, which is a contradiction.



It has been stated in (1.8) that Ψ maps Mτ (QC) ∪ Mτ¯ (QC) onto A Mτ (QC). Taking into account the statements (i)–(iii) at the beginning of this section, the previous proposition implies the following. A Then M η (QC) = {ξ, ξ  } with Corollary 3.6. Let τ ∈ T+ and η ∈ Mτ (QC). some (unique) ξ ∈ Mτ (QC).

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A This corollary implies that Ψ is a bijection from Mτ (QC) onto Mτ (QC) A for τ ∈ T+ . Clearly, Ψ is also a bijection from Mτ¯ (QC) onto Mτ (QC). This suggests to define A := { ξˆ : ξ ∈ Mτ± (QC) }, Mτ± (QC) τ ∈ T+ . (3.4) A by equation (3.2). Recall that we defined Mτ0 (QC) Proposition 3.7. For τ ∈ T+ we have A = M + (QC) A ∪ Mτ− (QC), A Mτ (QC) τ

A = Mτ+ (QC) A ∩ Mτ− (QC). A Mτ0 (QC)

Proof. The first identity is obvious. Regarding the second one, note that by definition and by Proposition 3.5, A ∩ Mτ− (QC) A = { ξˆ : ξ ∈ Mτ0 (QC) }. Mτ+ (QC) A is well defined It suffices to show that the map Ψ : Mτ0 (QC) → Mτ0 (QC) and bijective. Similar to the proof of Theorem 3.3, it can be shown that it is well-defined. Obviously it is injective. It remains to show that it is surjective. A By definition, there exists a net {λω }ω∈Ω , Choose any η ∈ Mτ0 (QC). λω ∈ Λ, such that the net {δλω }ω∈Ω := {δλω ,τ }λω ∈Ω converges to η (in A From Lemma 2.4 it follows that the weak-∗ topology of functionals on QC). λω → +∞. Choose a continuous function cτ such that cτ = 1 in a neighborhood of τ and such that it vanishes on the lower semi-circle. The net {δλω }ω∈Ω (regarded as functionals on QC) converges to the functional ξ ∈ QC ∗ defined by ξ(q) := η(q), q ∈ QC, A Indeed, q −q vanishes on a neighborhood of τ , and where q = qcτ + qc Bτ ∈ QC. hence δλ (q) = δλ (q) for λ sufficiently large. Therefore, δλω (q) − δλω (q) → 0. This together with δλω (q) → η(q) = ξ(q) implies that δλω (q) → ξ(q). It follows that ξ ∈ clos QC ∗ (Λ × {τ }). In order to show that ξ is multiplicative over QC, we write (noting cτ cτ = 0) pq − p · q = pqcτ + pqc Aτ − (pcτ + pc Bτ )(qcτ + qc Bτ ) = pq(cτ − c2τ ) + pq(  cτ − cτ 2 ). This is an even function vanishing in a neighborhood of τ and τ¯. Therefore η(pq − p · q) = 0, which implies ξ(pq) = ξ(p)ξ(q) by definition of ξ. It follows that ξ ∈ M (QC). Therefore, ξ ∈ Mτ0 (QC) by definition (2.3). Since ξˆ = η this implies surjectivity.  A is the disjoint A consequence of the previous proposition is that Mτ (QC) union of A A \ M 0 (QC), A A \ M 0 (QC). A M 0 (QC), M + (QC) and M − (QC) (3.5) τ

τ

τ

τ

τ

Comparing this with (1.9) we obtain that Ψ is a two-to-one map from A \ Mτ0 (QC), A (i) Mτ+ (QC) \ Mτ0 (QC) ∪ Mτ¯− (QC) \ Mτ¯0 (QC) onto Mτ+ (QC) + − 0 0 − A 0 A (ii) Mτ (QC) \ Mτ (QC) ∪ Mτ¯ (QC) \ Mτ¯ (QC) onto Mτ (QC) \ Mτ (QC), A (iii) Mτ0 (QC) ∪ Mτ¯0 (QC) onto Mτ0 (QC).

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A 4. Localization of P QC over QC A This allows Now we are going to identify the fibers M η (P QC) over η ∈ QC. ∗ us to show that certain quotient C -algebras that arise from P QC through localization are isomorphic to concrete C ∗ -algebras. What we precisely mean by the latter is the following. Let A be a commutative C ∗ -algebra and B be a C ∗ -subalgebra, both having the same unit element. For β ∈ M (B) consider the smallest closed ideal of A containing the ideal β, Jβ = clos id A { b ∈ B : β(b) = 0 }. It is known (see, e.g., [1, Lemma 3.65]) that Jβ = { a ∈ A : a|Mβ (A) = 0 }. Therein a is identified with its Gelfand transform. Hence the map a + Jβ ∈ A/Jβ → a|Mβ (A) ∈ C(Mβ (A)) is a well-defined *-isomorphism. In other words, the quotient algebra A/Jβ is isomorphic to C(Mβ (A)). However, it is often more useful to identify this algebra with a more concrete C ∗ -algebra Dβ . This motivates the following definition. A unital *-homomorphism Φβ : A → Dβ is said to localize the algebra A at β ∈ M (B) if it is surjective and if ker Φβ = Jβ . In other words, the induced *-homomorphism a + Jβ ∈ A/Jβ → Φβ (a) ∈ Dβ is a *-isomorphism between A/Jβ and Dβ . A in the above sense. The Our goal is to localize P QC at η ∈ M (QC) corresponding fibers are M η (P QC) = { z ∈ M (P QC) : z|QC  = η} = { z ∈ Mξ (P QC) : ξ ∈ M η (QC) }. Hence they can be obtained from the fibers M η (QC) and Mξ (P QC) (see Theorem 1.2). Recall the identification of z ∈ M (P QC) with (ξ, σ) ∈ M (QC) × {+1, −1} given in (1.2). Furthermore, CN is considered as a C ∗ -algebra with component-wise operations and maximum norm. (It is the N -fold direct product of the C ∗ -algebra C.) Theorem 4.1. 0 A 0 (a) Let η ∈ M±1 (QC) and M η (QC) = {ξ} with ξ ∈ M±1 (QC). Then

M η (P QC) = {(ξ, +1), (ξ, −1)} and Φ : P QC → C2 defined by p ∈ P C → (p(±1 + 0), p(±1 − 0)),

q ∈ QC → (ξ(q), ξ(q))

extends to a localizing *-homomorphism.

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A \ M 0 (QC) A and M η (QC) = {ξ, ξ  } such that (b) Let η ∈ M±1 (QC) ±1 + 0 ξ ∈ M±1 (QC) \ M±1 (QC). Then M η (P QC) = {(ξ, +1), (ξ  , −1)} and Φ : P QC → C2 defined by p ∈ P C → (p(±1 + 0), p(±1 − 0)),

q ∈ QC → (ξ(q), ξ  (q))

extends to a localizing *-homomorphism. A τ ∈ T+ , and M η (QC) = {ξ, ξ  } with ξ ∈ Mτ0 (QC). (c) Let η ∈ Mτ0 (QC), Then M η (P QC) = {(ξ, +1), (ξ, −1), (ξ  , +1), (ξ  , −1)} and Φ : P QC → C4 defined by p ∈ P C → (p(τ + 0), p(τ − 0), p(¯ τ + 0), p(¯ τ − 0)), q ∈ QC → (ξ(q), ξ(q), ξ  (q), ξ  (q)) extends to a localizing *-homomorphism. A \ M 0 (QC), A τ ∈ T+ , and M η (QC) = {ξ, ξ  } with (d) Let η ∈ Mτ± (QC) τ ± 0 ξ ∈ Mτ (QC) \ Mτ (QC). Then M η (P QC) = {(ξ, ±1), (ξ  , ∓1)} and Φ : P QC → C2 defined by p ∈ P C → (p(τ ± 0), p(¯ τ ∓ 0)),

q ∈ QC → (ξ(q), ξ  (q))

extends to a localizing *-homomorphism. A are considered (see (1.7), (3.3), and Proof. Note that all cases of η ∈ M (QC) η (3.5)). The description of M (QC) follows from Corollaries 3.4 and 3.6. Let us consider only one case, say case (c). The other cases can be treated analogously. We can write M η (P QC) = {z ∈ Mξ (P QC) : ξ ∈ M η (QC)}. Since M η (QC) = {ξ, ξ  } we obtain M η (P QC) = Mξ (P QC) ∪ Mξ (P QC). Now use Theorem 1.2 to get the correct description of M η (P QC) as a set of four elements {z1 , z2 , z3 , z4 }. Identifying C(M η (P QC)) = C({z1 , z2 , z3 , z4 }) with C4 , the corresponding localizing homorphism is given by Φ : f ∈ P QC → (z1 (f ), z2 (f ), z3 (f ), z4 (f )) ∈ C4 . Using the identification of z with (ξ, σ) ∈ M (QC)×{+1, −1} as given in (1.2), the above form of the *-homomorphism follows by considering f = p ∈ P C and f = q ∈ QC. 

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References [1] A. B¨ ottcher, B. Silbermann, Analysis of Toeplitz operators, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006, Prepared jointly with Alexei Karlovich. [2] R.G. Douglas, Banach algebra techniques in the theory of Toeplitz operators, Expository Lectures from the CBMS Regional Conference held at the University of Georgia, Athens, Ga., June 12–16, 1972, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 15, American Mathematical Society, Providence, R.I., 1973. [3] T. Ehrhardt, S. Roch, and B. Silbermann, Symbol calculus for singular integrals with operator-valued PQC-coefficients, in: Singular integral operators and related topics (Tel Aviv, 1995), Oper. Theory Adv. Appl., vol. 90, pp. 182–203, Birkh¨ auser, Basel, 1996. [4] T. Ehrhardt, S. Roch, and B. Silbermann, Finite section method for singular integrals with operator-valued PQC-coefficients, in: Singular integral operators and related topics (Tel Aviv, 1995), Oper. Theory Adv. Appl., vol. 90, pp. 204–243, Birkh¨ auser, Basel, 1996. [5] T. Ehrhardt, Z. Zhou, Finite section method for singular integrals with operator-valued PQC-coefficients and a flip, in preparation. [6] I.C. Gohberg and N.Ja. Krupnik, The algebra generated by the Toeplitz matrices, Funkcional. Anal. i Priloˇzen. 3 no. 2 (1969), 46–56. [7] S.C. Power, Hankel operators with PQC symbols and singular integral operators, Proc. London Math. Soc. (3) 41 no. 1 (1980), 45–65. [8] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. [9] D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana Univ. Math. J. 26 no. 5 (1977), 817–838. [10] B. Silbermann, The C ∗ -algebra generated by Toeplitz and Hankel operators with piecewise quasicontinuous symbols, Toeplitz lectures 1987 (Tel-Aviv, 1987), Integral Equations Operator Theory 10 no. 5 (1987), 730–738. Torsten Ehrhardt and Zheng Zhou Department of Mathematics University of California Santa Cruz, CA 95064 USA e-mail: [email protected] [email protected]

Bisection eigenvalue method for Hermitian matrices with quasiseparable representation and a related inverse problem Y. Eidelman and I. Haimovici Dedicated to Rien Kaashoek on the occasion of his 80th birthday

Abstract. We study the bisection method for Hermitian matrices with quasiseparable representations. We extend here our results (published in ETNA, 44, 342–366 (2015)) for quasiseparable matrices of order one to an essentially wider class of matrices with quasiseparable representation of any order. To perform numerical tests with our algorithms we need a set of matrices with prescribed spectrum from which to build their quasiseparable generators, without building the whole matrix. We develop a method to solve this inverse problem. Our algorithm for quasiseparable of Hermitian matrices of any order is used to compute singular values of a matrix A0 , in the general case not a Hermitian one, with given quasiseparable representation via definition, i.e., as the eigenvalues of the Hermitian matrix A = A∗0 A0 . We also show that after the computation of an eigenvalue one can compute the corresponding eigenvector easily. The performance of the developed algorithms is illustrated by a series of numerical tests. Mathematics Subject Classification (2010). Primary 15A18; Secondary 15A29, 15A60, 15B57, 65F15, 65F35. Keywords. Quasiseparable, eigenstructure, Sturm property, bisection, inverse problem.

1. Introduction The bisection method is one of the customary tools to compute all, or selected eigenvalues of a matrix A. The application of this method to Hermitian matrices is based essentially on the Sturm sequence property. This means that for any given real number λ, the number of sign changes in the sequence of the characteristic polynomials of the principal leading submatrices of an © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_7

181

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N × N Hermitian matrix A equals the number of eigenvalues which are less than that λ. We will denote the polynomials in such a sequence by γ0 (λ) ≡ 1, γ1 (λ), γ2 (λ), . . . , γN (λ).

(1.1)

For rank structured matrices there exist fast algorithms to evaluate the polynomials in (1.1) in O(N ) arithmetic operations and then to compute in a fast way the roots of γN , i.e., the eigenvalues of A. The application of this method for symmetric tridiagonal matrix can be found, for instance, in Subsection §8.4.1 of the monograph [10], or in the paper [1]. Related results for semiseparable matrices are contained in Subsection §9.7.2 of [14], where an algorithm using the Sturm property and bisection is devised, based on polynomial values computation as done in [7]. Similarly as it was done in the paper [1] for tridiagonal matrices and in [8], we use here the ratios Dk (λ) =

γk (λ) , γk−1 (λ)

k = 1, 2, . . . , N

(1.2)

instead of polynomials γk (λ), in order to avoid overflow or underflow in the computing process. In [8] we applied this method for order 1 quasiseparable Hermitian matrices. Notice that the recursive relations obtained in [7] for the polynomials (1.1) are not valid for non-scalar generators. At the same time it turns out that the recursions obtained in [8] for the rational functions (1.2) can be extended on quasiseparable generators of any order. This is a basic result of the present paper. Such implementations for tridiagonal symmetric matrices in literature use the LDLT approach with gains in both stability and speed, (see [2], p. 231). If we would apply our results to the particular case of tridiagonal matrices, they would yield precisely the recursions obtained by other means in [1] and presented in [2]. In this paper we will give a complete algorithm for using this method for the much wider class of Hermitian matrices, whenever their quasiseparable generators of any order are known. Because Hermitian matrices have only real eigenvalues and for a given such matrix A and a given real λ, we succeed in evaluating all the sequences (1.2) in O(N ) of entry-wise elementary arithmetical operations, it is entirely feasible to find the eigenvalues of A by Sturm plus bisection. Sturm property with bisection is also most suited when we want only a few eigenvalues, especially if we want certain particular ones such as, for instance, the last (largest) k < N eigenvalues. However, the low complexity compared to the other existing methods for general Hermitian matrices makes the present method appropriate for such matrices even when the complete set of eigenvalues and not only selected ones, are to be computed. As in [8], and for the same reason for which (1.2) are used, we had to scale the matrix, i.e., to divide some of its quasiseparable generators, by a number which is large enough (we chose a number larger than the Frobenius standard norm ||A||F of the matrix), to obtain temporarily smaller eigenvalues, with absolute values in the subunit interval and thus smaller polynomial

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ratios. This latter technique has not been used in [1] and it permits to fight overflow for matrices of any size (we checked up to size 216 × 216 ), while without it, even for matrices of size of tens, the bisection method does not work. The scaling and then the de-scaling of the eigenvalues are linear time operations. If we use instead of the Sturm polynomials from (1.1), ratios (1.2) of consecutive such polynomials and we also scale, the algorithm works well for practically any size of matrices. To check the accuracy of the developed eigenvalue algorithm we need a set of matrices with prescribed eigenvalues from which we obtain the quasiseparable generators, without building the whole matrix. In the present paper we propose an algorithm to solve this inverse problem. These results are used essentially in the numerical part of the paper. In the present paper we derive also an algorithm to compute singular values of a matrix A0 , not Hermitian in the general case, for which we have a given quasiseparable representation. For such a matrix the Hermitian matrix A = A∗0 A0 has quasiseparable generators with twice the orders of those of A0 , which are computed in O(N ) operations. Recall that the element-wise computation of this product costs O(N 3 ) operations. So together with the bisection method to be applied to the Hermitian matrix A, we obtain a fast algorithm to compute the singular values of A0 . Next we show that after the computation of an eigenvalue, one can derive easily the algorithm to compute the corresponding eigenvector. We are aware of the problem that the computation of singular values via definition leads to essential errors for small singular values and the computation of eigenvectors via simplest methods leads to loosing orthogonality and to large errors in eigenvectors for small errors in eigenvalues. That is why in the numerical part we present the computations of separated, not multiple, singular values not too close to zero and of a few eigenvectors with maximal eigenvalues. We carry out numerical experiments in Section §6, which show that the bisection is both fast and very accurate, especially when we found the largest 3 eigenvalues. However, in Subsection §6.1.2 we found that when finding all the eigenvalues, the bisection is slower, but more accurate, than reduction to tridiagonal followed by implicit QR. We checked with Hermitian or, for singular values, with non-Hermitian matrices, out of complex or real order 2 or order 3 generators. The types of checked matrices are: diagonal plus semiseparable, 5-band and 7-band matrices, as well as arbitrary matrices, all these for different sizes N × N , where N is a power of 2 up to 2048 and sometimes 216 . For the Hermitian matrices we check the relative error of the proposed algorithm in finding eigenvalues, while for general matrices we check the relative error in finding singular values. We also check the orthogonality of the eigenvectors that we obtain. We know in advance the true eigenvalues or singular values, as the quasiseparable generators are built to represent matrices which have precisely those prescribed values, using the results in the section on the inverse problem. When we check special matrices, like band matrices, for which the

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eigenvalues are not known a priori, we build the whole matrix out of its quasiseparable generators and then we consider as true values the eigenvalues or singular values given by the proper Matlab function. We also compare in the numerical experiments the bisection method with the reduction to tridiagonal followed by implicit QR. All the algorithms in this paper are done in terms of quasiseparable generators of matrices. As far as we know, for the first time quasiseparable representations of matrices were introduced and studied by I. Gohberg, M.A. Kaashoek and L. Lerer [9]. Various properties of quasiseparable representations and algorithms were discussed in many papers and in the monographs by P.M. Dewilde and A.J. van der Veen [3], by R. Vandebril, M. Van Barel and N. Mastronardi [13, 14], and by I. Gohberg and the authors [5, 6]. The paper consists of seven sections. The first section is the introduction. The second one contains the definition of the quasiseparable representation of the matrix and some important particular cases. In Section 3 we present the basic eigenvalue algorithm and derive a simple algorithm to compute the corresponding eigenvector. Section 4 is devoted to the construction of matrices with given eigenstructure which are used next in the numerical tests. In Section 5 we discuss the computation of singular values. Section 6 contains the results of numerical tests. Section 7 is the conclusion. For an N × N matrix A we denote by Aij or by A(i, j) its element on row 1 ≤ i ≤ N and on column 1 ≤ j ≤ N and by A(i : j, p : q) the submatrix containing the rows between i to j, between columns p to q inclusively. In particular, if i = j then we denote A(i, p : q) and if i < j, p = q we denote A(i : j, p). We denote by col(x(k))N k=1 the column vector of size N , that has the number x(k) as its k th entry, by row(x(k))N k=1 the row vector of size N , that has the number x(k) as its k th entry and by diag(λ1 , λ2 , . . . , λN ) the diagonal N × N matrix D with the numbers D(k, k) = λk , k = 1, . . . , N on its diagonal. If Bj , j = 1, . . . , m are matrices of sizes N1 × N1 , . . . , Nm × Nm respectively and N1 + . . . + Nm = N , then diag(B1 , . . . , Bm ) denotes the N × N matrix with zero entries, except on the diagonal blocks B1 , . . . , Bm .

2. The quasiseparable representation Here we define the basic notions concerning quasiseparable representations of matrices and we discuss some important particular cases. 2.1. The general quasiseparable structure The following definitions and more can be found for instance in the Part I of the book [5]. Let {a(k)} be a family of matrices of sizes rk ×rk−1 . For positive integers > i, j, i > j define the operation a> ij as follows: aij = a(i − 1) · . . . · a(j + 1) for i > j + 1, a> j+1,j = Irj .

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Let {b(k)} be a family of matrices of sizes rk−1 ×rk . For positive integers < i, j, j > i define the operation b< ij as follows: bij = b(i + 1) · . . . · b(j − 1) for j > i + 1, b< i,i+1 = Iri . Let A = {Aij }N i,j=1 be a matrix with scalar entries Aij . Assume that the entries of this matrix are represented in the form ⎧ > ⎨ p(i)aij q(j), 1 ≤ j < i ≤ N, d(i), 1 ≤ i = j ≤ N, Aij = (2.1) ⎩ g(i)b< ij h(j), 1 ≤ i < j ≤ N. Here p(i) (i = 2, . . . , N ), q(j) (j = 1, . . . , N − 1), a(k) (k = 2, . . . , N − 1) L L , rjL × 1, rkL × rk−1 respectively, g(i) (i = are matrices of sizes 1 × ri−1 1, . . . , N − 1), h(j) (j = 2, . . . , N ), b(k) (k = 2, . . . , N − 1) are matriU U ces of sizes 1 × riU , rj−1 × 1, rk−1 × rkU respectively, d(i) (i = 1, . . . , N ) are (possibly complex) numbers. The representation of a matrix A in the form (2.1) is called a quasiseparable representation. The elements p(i) (i = 2, . . . , N ), q(j) (j = 1, . . . , N − 1), a(k) (k = 2, . . . , N − 1); g(i) (i = 1, . . . , N − 1), h(j) (j = 2, . . . , N ), b(k) (k = 2, . . . , N − 1); d(i) (i = 1, . . . , N ) are called quasiseparable generators of the matrix A. The numbers rkL , rkU (k = 1, . . . , N − 1) are called the orders of these generators. The elements p(i) (i = 2, . . . , N ), q(j) (j = 1, . . . , N − 1), a(k) (k = 2, . . . , N − 1) and g(i) (i = 1, . . . , N − 1), h(j) (j = 2, . . . , N ), b(k) (k = 2, . . . , N − 1) are called also lower quasiseparable generators and, respectively, upper quasiseparable generators of the matrix A. In fact the generators p(i), g(i) and q(j), h(j) are rows and columns of the corresponding sizes. For a Hermitian matrix the diagonal entries d(k) (k = 1, . . . , N ) are real and the upper quasiseparable generators could be obtained from the lower ones by taking g(k) = (q(k))∗ , h(k) = (p(k))∗ , b(k) = (a(k))∗ k = 2, . . . , N − 1, g(1) = (q(1))∗ , h(N ) = (p(N ))∗ .

(2.2) (2.3)

See more on the definition and properties of the quasiseparable structure of (Hermitian) matrices in the book [5], starting with §4.2. We can suppose that for an N × N matrix the orders of (the lower) quasiseparable generators are the same, rkL = r (k = 1, . . . , N − 1), since otherwise one can pad the smaller ones with zeroes. It follows that we can ask this as a condition for the theorems below, without loss of generality. 2.2. The diagonal plus small rank representation Here we consider matrices represented in the form A = D + P · Q with a N diagonal matrix D = diag(δ(i))N i=1 , an N × r matrix P = col(p(i))i=1 and an N r × N matrix Q = row(q(i))i=1 . This is a particular case of quasiseparable representation with lower generators p(i) (i = 2, . . . , N ), q(j) (j = 1, . . . , N − 1), a(k) = Ir (k = 2, . . . , N − 1), diagonal entries d(k) = δ(k) + p(k)q(k),

k = 1, . . . , N

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and upper quasiseparable generators g(i) = p(i), i = 2, . . . , N − 1, h(j) = q(j), j = 2, . . . , N, b(k) = Ir , k = 2, . . . , N − 1. In the case of a real D and Q = P ∗ we obtain the Hermitian matrix A = D + P P ∗ with q(j) = p∗ (j), j = 1, . . . , N − 1,

d(k) = δ(k) + p(k)p∗ (k),

k = 1, . . . , N.

2.3. Band matrices Recall that a matrix A = {Aij }N i,j=1 is said to be a (2r + 1)-band matrix if Aij = 0, |i − j| > r. Such a matrix has a quasiseparable representation with order equal r. Quasiseparable representations for any band matrices can be found, e.g., in [5, p. 81]. For instance, for a 7-band quasiseparable matrix the lower quasiseparable generators are ⎞ ⎛ Aj+1,j   p(i) = 1 0 0 , i = 2, . . . , N, q(j) = ⎝ Aj+2,j ⎠ , j = 1, . . . , N − 1; Aj+3,j ⎛ ⎞ 0 1 0 a(k) = ⎝ 0 0 1 ⎠ , k = 2, . . . , N − 1. 0 0 0 Here, AN +1,N −2 , AN +1,N −1 and AN +2,N −1 are taken to be null.

3. The bisection method Here we present the basic algorithm to compute eigenvalues of a Hermitian matrix with a given quasiseparable representation. 3.1. The Sturm sequences property for a Hermitian matrix To get information on the location of the eigenvalues of a Hermitian matrix the well known Sturm property is applied. We use this property in the form presented, for instance, in the monograph [12, p. 296]. Theorem 3.1. Let A be an N × N Hermitian matrix with determinants of leading principal submatrices γk = detA(1 : k, 1 : k), k = 1, . . . , N . Set γ0 = 1. Assume that γk = 0, k = 1, 2, . . . , N. Then the number of negative eigenvalues of A (counting each eigenvalue in accordance with its multiplicity) is equal to the number of alternations of sign in the sequence γ0 , γ1 , . . . , γN −1 , γN . In fact, we will use ratios of consecutive Sturm polynomials γk and we will check whether these ratios are negative, as a method of counting sign changes.

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Corollary 3.2. Let A be an N × N Hermitian matrix, λ be a real number and γ0 (λ) ≡ 1,

γk (λ) = det(A(1 : k, 1 : k) − λI), k = 1, 2, . . . , N.

Assume that γk (λ) = 0,

k = 1, 2, . . . , N , set

Dk (λ) =

γk (λ) , γk−1 (λ)

k = 1, 2, . . . , N.

(3.1)

Then the number of eigenvalues of A which are less than λ equals to the number of negative entries in the sequence (3.1). 3.2. The bisection procedure We perform the bisection method similarly on how it was done in [10, §8.4.1] for symmetric tridiagonal matrices. For a real λ we denote by ν(λ) the number of negative entries in the sequence (3.1). Let bL and bU be lower and upper bounds for the range of the eigenvalues obtained previously, for instance using some matrix norm. To compute λ = λk (A), i.e., the kth largest eigenvalue of A for some prescribed k, one proceeds as follows. We start with z = bU , y = bL then performing repeatedly (until we attain machine precision) the assignment λ = z+y 2 and then z = λ if ν(λ) ≥ k and y = λ otherwise, will produce the desired eigenvalue. 3.3. The basic eigenvalue algorithm The algorithm presented in [5, Theorem 18.2] yields in particular the ratios of determinants of principal leading submatrices of a matrix A with a given quasiseparable representation. Applying this result to the matrix A − λI we obtain recursive relations for the functions in the sequence (3.1). Let λ be a real number and A be a Hermitian matrix with given lower quasiseparable generators and diagonal entries. Using Theorem 3.3 and formulas (2.2), (2.3) we compute the number ν(λ) for any real number for which the conditions (3.2) hold. Theorem 3.3 (findNu(λ,generators)). Let A be an N × N Hermitian matrix and let λ be a real number such that the conditions det(A(1 : k, 1 : k) − λIk ) = 0,

k = 1, 2, . . . , N,

(3.2)

hold. Assume that A has lower quasiseparable generators p(i) (i = 2, . . . , N ), q(j) (j = 1, . . . , N − 1), a(k) (k = 2, . . . , N − 1) of orders rkL (k = 1, . . . , N − 1) and diagonal entries d(k) (k = 1, . . . , N ). Then the number ν(λ) of the eigenvalues of A which are less than λ is obtained via the following algorithm. 1. Compute D1 (λ) = d(1) − λ,

u1 (λ) = q(1)

1 , D1 (λ)

f1 (λ) = u1 (λ)q ∗ (1).

If D1 (λ) < 0 set ν = 1, otherwise set ν = 0.

(3.3)

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2. For k = 2, . . . , N − 1 compute Dk (λ) = d(k) − λ − p(k)fk−1 (λ)p∗ (k), 1 uk (λ) = [q(k) − a(k)fk−1 (λ)p∗ (k)] , Dk (λ) fk (λ) = a(k)fk−1 (λ)a∗ (k) + uk (λ)[q ∗ (k) − p(k)fk−1 (λ)a∗ (k)]. If Dk (λ) < 0 set ν := ν + 1. 3. Compute

(3.4) (3.5) (3.6)

DN (λ) = d(N ) − λ − p(N )fN −1 (λ)p∗ (N ). If DN (λ) < 0 set ν := ν + 1. Set ν(λ) = ν. In order to compute the complexity of the above algorithm, we will count as elementary arithmetical operations addition, subtraction, multiplication, division, conjugation and comparisons performed on scalars. For instance, the multiplication of a row vector of size r with a proper column vector at its right will count as 2r − 1 arithmetical operations, of which r multiplications and r − 1 additions. When optimized, the overall complexity of the algorithm is less than c = (4r3 + 6r + 3)(N − 2) + 5r2 − 2r + 1. 3.4. The computation of eigenvectors Theorem 3.4. Let A be an N ×N Hermitian matrix with lower quasiseparable generators p(i) (i = 2, . . . N ), q(j) (j = 1, . . . , N − 1), a(k) (k = 2, . . . , N − 1) and diagonal entries d(k) (k = 1, . . . , N ) and let λ be one of the eigenvalues of A such that the conditions (3.2) hold. Then an eigenvector col(x(k))N k=1 corresponding to λ can be obtained as follows. 1. Compute auxiliary variables Dk (λ), uk (λ), fk (λ) as in (3.3)–(3.6). 2. Determine the coordinates of the eigenvector x. 2.1. Set x(N ) = 1 and s(N − 1) = p∗ (N ). 2.2. For k = N − 1, . . . , 2 compute x(k) = −u∗k (λ)s(k), ∗

2.3. Set

(3.7) ∗

s(k − 1) = a(k) s(k) + p(k) x(k).

(3.8)

x(1) = −u∗1 (λ)s(1).

(3.9)

Proof. Let x = col(x(k))N to the eigenk=1 be an eigenvector  corresponding  x N −1  value λ. Set x = col(x(k))k=1 , we have x = . xN Step 1 follows from Theorem 3.3. One should justify only Step 2. Notice that the matrix Aλ = A − λIN has lower quasiseparable generators p(i) (i = 2, . . . N ), q(j) (j = 1, . . . , N − 1), a(k) (k = 2, . . . , N − 1) and diagonal entries d(k) − λ (k = 1, . . . , N ). Upper quasiseparable generators

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g(i) (i = 1, . . . , N − 1), h(j) (j = 2, . . . , N ), b(k) (k = 2, . . . , N − 1) of the Hermitian matrix A − λI are given via (2.2), (2.3). Set Ak (λ) = A(1 : k, 1 : k) − λ,

k = 1, . . . , N − 1

and k < k Qk = row(a> k+1,i q(i))i=1 , Gk = col(g(i)bi,k+1 )i=1 ,

k = 1, . . . , N − 1.

Using the formula from [5, p.332] we have   AN −1 (λ) GN −1 h(N ) Aλ = . p(N )QN −1 d(N ) − λ

(3.10)

Using the equation Aλ x = 0 and the partition (3.10) we obtain the linear system   AN −1 (λ)x + GN −1 h(N )x(N ) = 0, (3.11) p(N )QN −1 x + (d(N ) − λ))x(N ) = 0. The condition (3.2) with k = N − 1 implies that the matrix AN −1 (λ) is invertible and therefore (3.11) is equivalent to the system   x = −(AN −1 (λ))−1 GN −1 h(N )x(N ) = 0, (−p(N )QN −1 (AN −1 (λ))−1 GN −1 h(N )x(N ) + (d(N ) − λ))x(N ) = 0. (3.12) The second equation has the form (d(N ) − λ − p(N )QN −1 (AN −1 (λ))−1 GN −1 h(N ))x(N ) = 0. Using the formula (1.60) from Theorem 1.20 in [5, p.23] we get (d(N ) − λ) − (p(N )QN −1 )(AN −1 (λ))−1 (GN −1 h(N )) = DN (λ) =

γN (λ) = 0. γN −1 (λ)

Hence it follows that one take x(N ) = 1 and determine the (N − 1)-dimensional vector x via x = −(AN −1 (λ))−1 (GN −1 h(N )). Set

Uk (λ) = (Ak (λ))−1 Gk ,

k = 1, . . . , N − 1.

We have

x = −UN −1 (λ)h(N ) = −UN −1 (λ)p∗ (N ). In [5, pp. 344–346] it is proved that   Uk−1 (λ)b(1) (k) Uk (λ) = , k = 1, . . . , N − 1, u∗k (λ)

(3.13)

(3.14)

where u∗k (λ) = (Dk (λ))−1 (g(k) − p(k)fk−1 (λ)b(k))

(3.15)

= (Dk (λ))−1 (q ∗ (k) − p(k)fk−1 (λ)a∗ (k)) and

b(1) (k) = b(k) − h(k)u∗k (λ) = a∗ (k) − p∗ (k)u∗k (λ).

(3.16)

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Combining (3.13) and (3.14) together we obtain (3.7), (3.9) with s(k) = ∗ ∗ (b(1) )< k,N p (N ). We have s(N − 1) = p (N ) and using (3.16) we get ∗ (1) ∗ s(k − 1) = (b(1) )< (k)(b(1) )< k−1,N p (N ) = b k,N p (N )

= (a∗ (k) − p∗ (k)u∗k (λ))s(k), 

which implies (3.8).

4. The inverse problem To perform numerical tests with the developed algorithms we need a set of given quasiseparable generators which would build a matrix with prescribed, known spectrum. At first we solve a more general problem to determine quasiseparable generators of a matrix with prescribed singular values. This general result is also used in the corresponding numerical tests in Subsection §6.2. Algorithm 4.1. Let {λ1 , λ2 , . . . , λN } be a set of real numbers and r be a positive integer. Set m1 = m2 = · · · = mN = 1, ν1 = 1 + r, ν2 = ν3 = · · · = νN −r = 1, νN −r+1 = · · · = νN = 0, ρ1 = ρ2 = · · · = ρN −r = r, ρN −r+1 = r − 1, . . . , ρN = 0.

(4.1)

and let Vk and Fk , k = 1, . . . , N be sets of (mk + ρk ) × (mk + ρk ) unitary matrices. Let A be an N × N matrix defined by the formula A = V · Λ · F,

(4.2)

where Λ = diag(λ1 , λ2 , . . . , λN ) is a real diagonal and V, F are unitary matrices given via the products V = V˜N V˜N −1 · · · V˜2 V˜1 ,

F = F˜1 F˜2 · · · F˜N −1 F˜N

(4.3)

with V˜1 = diag{V1 , Iφ1 },

V˜k = diag{Iηk , Vk , Iφk }, k = 2, . . . , N − 1;

(4.4)

V˜N = diag{IηN , VN }, F˜1 = diag{F1 , Iφ1 }, F˜k = diag{Iηk , Fk , Iφk }, k = 2, . . . , N − 1; F˜N = diag{IηN , FN }, k−1 N where ηk = i=1 mi , φk = i=k+1 νi . Then the matrix A has the set of singular values {|λ1 |, |λ2 |, . . . , |λN |} and lower and upper quasiseparable generators of orders ρk (k = 1, . . . , N −1). Moreover a set of such quasiseparable generators p(i) (i = 2, . . . , N ), q(j) (j = 1, . . . , N − 1), a(k) (k = 2, . . . , N − 1) and g(i) (i = 1, . . . , N − 1), h(j) (j = 2, . . . , N ), b(k) (k = 2, . . . , N − 1)

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as well as the diagonal entries d(i) (i = 1, . . . , N ) can be obtained as follows. We determine the generators p(i) (i = 2, . . . , N ), a(k) (k = 2, . . . , N − 1) and b(k) (k = 2, . . . , N − 1), h(j) (j = 2, . . . , N ) from the partitions



p(k) dV (k) h(k) b(k) Vk = , Fk = , k = 2, . . . , N − 1, a(k) qV (k) dF (k) gF (k) (4.5)

  h(N ) VN = p(N ) dV (N ) , FN = (4.6) dF (N ) with the matrices p(k), a(k), dV (k), qV (k) of sizes mk × ρk−1 , ρk × ρk−1 , mk × νk , ρk × νk , respectively, and the matrices p(k), a(k), dV (k), qV (k) of sizes mk ×ρk−1 , ρk ×ρk−1 , mk ×νk , ρk ×νk , respectively, and the matrices h(k), b(k), dF (k), gF (k) of sizes ρk−1 × mk , ρk−1 × ρk , νk × mk , νk × ρk , respectively. Next we set λN +1 , . . . , λN +r to be 0 × 0 empty matrices and compute the generators q(j) (j = 1, . . . , N −1), g(i) (i = 1, . . . , N −1), d(i) (i = 1, . . . , N ) via recursive relations as follows. 1. Set Λ1 = diag(λ1 , λ2 , . . . , λr+1 ) and compute the matrix W 1 = V 1 Λ 1 F1

(4.7)

and determine the matrices d(1), q(1), g(1), β1 of sizes m1 × m1 , m1 × ρ1 , ρ1 × m1 , ρ1 × ρ1 from the partition

d(1) g(1) W1 = . (4.8) q(1) β1 2. Set Λk = λk+r and for k = 2, . . . , N − 1 compute the matrix

βk−1 0 Wk = Vk Fk 0 Λk

(4.9)

and determine the matrices d(k), g(k), q(k), βk of sizes mk × mk , mk × ρk , ρk × mk , ρk × ρk from the partition

d(k) g(k) Wk = (4.10) q(k) βk with the auxiliary variables βk which are ρk × ρk matrices. 3. Set ΛN = λN +r and compute d(N ) = p(N )βN −1 h(N ) + dV (N )ΛN dF (N ).

(4.11)

As an important corollary we obtain the basic results for generating of quasiseparable generators that would give Hermitian matrices with a prescribed spectrum and a prescribed order of quasiseparability. Such matrices are used in the numerical tests in Subsection §6.1. Namely, if we do not compute at all neither the matrices F˜1 , . . . , F˜N from (4.3),(4.4), nor F , but we take in the formulas the matrix V ∗ instead of F and the matrices Vk∗ instead of Fk , k = 1, . . . , N and we consider in formulas (4.8) and (4.10) the generators q ∗ (i), i = 1, . . . , N − 1 instead of the corresponding generators g(i), then the Hermitian matrix A = V λV ∗ that has the such obtained generators has the real spectrum λ1 , . . . , λN .

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5. Singular values of matrices and the standard matrix norm via quasiseparable generators Here we consider the problem of computation of singular values of matrices with quasiseparable representations. We proceed via the definition, i.e., we obtain the singular values of an arbitrary matrix A0 , whenever its quasiseparable generators (of any order) are known, as the eigenvalues of the Hermitian matrix A = (A0 )∗ A0 . To solve the last problem we use the bisection algorithm described in Section 3. Note that this algorithm will produce the singular numbers 0 ≤ s1 (A0 ) ≤ s2 (A0 ) ≤ . . . ≤ sN (A0 )

(5.1)

in ascending order and not in descending order as they are usually enumerated in textbooks. 5.1. The quasiseparable generators of A = (A0 )∗ A0 Formulas for quasiseparable generators of a product of two matrices with given quasiseparable representations have been devised in [4]. See also Section §17.3 in [5]. Here we derive the particular case of the product A = (A0 )∗ A0 . In order to unify notation and to let the indexes k run for 1, . . . , N for all generators alike, let us define the numbers r0L = 0,

L rN = 0 r0U = 0,

U rN =0

and let us also define the generators p(1), q(N ), a(1), a(N ),

h(1), g(N ), b(1), b(N )

to be equal to arbitrary matrices of sizes L L L × 1, r1L × r0L rN × rN 1 × r0L , rN −1 ,

U U U r0U × 1, 1 × rN , r0U × r1U rN −1 × rN

matrices, respectively. Proposition 5.1. Let A0 be a matrix with quasiseparable generators p0 (k), q0 (k), a0 (k); g0 (k), h0 (k), b0 (k); d0 (k) (k = 1, . . . , N ) of orders rkL , rkU (k = 0, . . . , N ). Then the product A = (A0 )∗ A0 is a Hermitian matrix with lower quasiseparable generators p(k), q(k), a(k) (k = 1, . . . , N ) of orders rkU + rkL , (k = 0, . . . , N ) and diagonal entries d(k) (k = 1, . . . , N ). These generators and diagonal entries are determined as follows. Set β0 = 0r0L ×r0U and compute recursively for k = 1, . . . , N   h0 (k) b0 (k) T = , 0 g0 (k)      d (k) ∗ βk−1 0 = T∗ T, (5.2) q˜(k) βk 0 1 L ×r L , and compute recursively next set γN +1 = 0rN N   p0 (k) d0 (k) S= , a0 (k) q0 (k)

Bisection for quasiseparable 

γk p˜(k)

∗ d (k)

 =S

∗



1 0

0

193



γk+1

S, k = N, . . . , 1.

(5.3)

L Here βk , γk are auxiliary variables which are matrices of sizes rkU ×rkU , rk−1 × L rk−1 . Finally, set  ∗   ∗  g0 (k)d0 (k) + q˜(k) p(k) = h0 (k) p˜(k) , q(k) = , (5.4) q0 (k)  ∗  b0 (k) g0∗ (k)p0 (k) a(k) = , (5.5) 0 a0 (k)

d(k) = d (k) + d (k),

k = 1, . . . , N.

(5.6)

Proof. The present proposition is a particular case of the Theorem 17.6 in [5, §17.3].  Note that if the original matrix A0 was quasiseparable of a certain order r, the obtained matrix A = (A0 )∗ A0 with which we will work in the sequel has at most twice that order. In particular, methods to find eigenvalues for order one quasiseparable matrices would not work. Proposition 5.2. The computation (5.2) can be done faster entry-wise in the following way. Set β1 = g0∗ (1)∗ g0 (1) and compute recursively for k = 1, . . . , N − 1 hk = βk h0 (k),

d (k) = h∗0 (k)hk ,

q˜(k) = b∗0 (k)hk , βk+1 = b∗0 (k)βk b0 (k) + g0∗ (k)g0 (k). Finally, set

hN = βN h0 (N ), d (N ) = h∗0 (N )hN . The computation (5.3) can be done faster entry-wise in the following way. Set γN = p∗0 (N )p0 (N ),

p0 (N ) = d0 (N )p0 (N ),

d (N ) = d0 (N )d0 (N )

and compute recursively for k = N − 1, . . . , 1 pk = γk+1 a0 (k),

γk = p∗0 (k)∗ p0 (k) + a∗0 (k)pk ,

p˜(k) = d0 (k)p0 (k) + q0∗ (k)pk , d (k) = q0∗ (k)γk+1 q0 (k) + d0 (k)d0 (k). 5.2. Computation in linear time of the standard matrix norm before SVD Here we show that in the multiplication algorithms in the previous subsections we perform in fact most of the operations which are needed to compute the standard matrix norm via quasiseparable generators. Recall that this = norm is called also the trace norm tr(A0 )∗ A0 , or the Frobenius norm. We have C DN D = ∗ ||A0 ||F = tr((A0 ) A0 ) = E d(k), (5.7) k=1

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were the real nonnegative numbers d(k), k = 1, . . . , N have been computed in the algorithm from Proposition 5.2. Extracting the corresponding parts from this algorithm and using the formula (5.7) we obtain the following procedure. Theorem 5.3. In the conditions of Proposition 5.1 the Frobenius norm n of the matrix A = (A0 )∗ A0 is obtained via the following algorithm. 1. Initialize n = 0, β1 = g ∗ (1)g(1) and compute for k = 2, . . . , N − 1 n = n + h∗ (k)βk−1 h(k),

βk = b∗ (k)βk−1 b(k) + g ∗ (k)g(k).

Finally, set n = n + h∗ (N )βN −1 h(N ). 2. Initialize γN = p∗ (N )p(N ), set n = n + d0 (N )d0 (N ) and compute recursively for k = N − 1, . . . , 1 n = n + q ∗ (k)γk+1 q(k) + d0 (k)d0 (k),

γk = p∗ (k)p(k) + a∗ (k)γk+1 a(k).

Finally, set n=

√

n.

(5.8)

In [8] the Frobenius norm is shown to be most stable with respect to small perturbations of an order one quasiseparable matrix and it is used there as a bound of eigenvalues for the bisection method therein. Theorem 5.3 shows how its computation can be done faster. In order to apply the bisection method briefly described in Section 3, we need a priori a lower and an upper bound for the eigenvalues of the positive definite matrix A. Since √ it has only nonnegative eigenvalues, 0 is a lower bound for them, while trA, which is the same as ||A0 ||F is an upper bound.

6. Numerical experiments All the numerical experiments have been performed on a computer with an i7-5820 microprocessor, 31.9 gigabytes installed memory (RAM) at 3.30GHz and another 4GB in the video card GTX, which is exploited by Matlab as well. The operating system is Windows 10, 64 bits, the least positive number which is used by the machine is 2.2251e-308 as given by the Matlab command realmin and the machine precision is 2.2204e-16, as given by the Matlab command eps.

Bisection for quasiseparable

195

6.1. Results for eigenvalues 10-12

10-12

Real matrix with already known eigvals

Multiple eigvals Random eigvals

10-13

10-13

10-14

10-14

10-15 16 32 64 128 256 512

2048

10-15 16 32 64 128 256 512

2048

Order 3

Order 2

6.1.1. Eigenvalues known in advance. In Figure 1 above we plot the relative error in finding eigenvalues when these are known in advance. We build suited quasiseparable generators such that the matrix would have those prescribed eigenvalues. As small r × r unitary matrices we take the result of the Matlab function qr for a random r × r matrix, where r is the order of the generators. The formula used for plot the errors is the average relative error over the k = 1, . . . , N eigenvalues of each of the 5 considered matrices of each size N × N , N being a power of 2 starting with 16 and finishing with 2048. If we denote by λjT (k) the true known eigenvalue k of one of the j = 1, . . . , 5 considered matrices and by λj (k) the eigenvalue obtained by bisection, then the formula is 5 N 1   |λj (k) − λjT (k)| . 5N j=1 |λjT (k)| k=1

The left figure is for quasiseparable of order 2 real symmetric matrices, while the right figure is for order 3. We use two sets of a-priori given eigenvalues and we compare with them the eigenvalues that the bisection finds. The first set of eigenvalues are chosen as random numbers between −1 and 1. The second set of eigenvalues are of the form λ(k) = 1 + kN/100, k = 1, . . . , N , where N × N is the size of the considered matrix and the natural number k does not end with the digit 1. If it does, we take λ(k) = λ(k + 1). In this way one tenth of the eigenvalues are double eigenvalues, so that we can check wether the algorithm works for multiple eigenvalues too. One can see by comparing the right and the left figure, respectively, that the error for order 3 quasiseparable matrices is not much larger than for order 2. Errors occur after at least 12 exact decimal digits in the larger matrices. We correctly find even eigenvalues which are close to 0. We can see however that the relative error for random eigenvalues is larger, even when the other set of eigenvalues contains also multiple eigenvalues and this is due to the fact that the random eigenvalues could sometimes zero in the way we chose them and, only for the figure plotting purposes, we divide by the true

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eigenvalue in the above formula, while the other set of true eigenvalues could not be even close to zero. For the same reason, in the random case there is an unexpected peak at order 2 and size N = 64. The eigenvalues which are not at random could vary from 1.01 to 21. We also checked one huge matrix of size 216 × 216 and order 2 and it gave an error of only 5 · 10−14 . Its eigenvalues where of course between 1 and 657. Average error of reduction, bisection eigenvals

Average order 3 reduction and bis time

10-13

102

101

10

0

10-1

10-2 16

64 128

512

2048

Reduc order 3 Bisec order 3

10-14

10-15 16

64 128

512

2048

Sizes of matrices are powers of 2

Log of time

6.1.2. Reduction plus QR and bisection: time and error in eigenvalues. In Figure 2 above we compare the reduction to tridiagonal method presented in [6] with bisection and to this end we plot the time and the relative error in finding eigenvalues for order 3 quasiseparable matrices. One can see that the reduction algorithm is faster, while the bisection is more precise. We perform many times bisection again and again until we reach machine precision. If we would not ask from the bisection algorithm results with such low errors, the algorithm would be much faster. 8.5 8

5 band and 7 band bisection error

7.5

10-16 5 band 7 band

7 6.5 6 5.5 5 4.5 4

3.5 16

32

64

128

256

512

1024

2048

Errors on eigenvalues

6.1.3. Eigenvalues of Hermitian band matrices. In Figure 3 above we plot the relative error in finding eigenvalues for 5 and 7 band matrices. We also build the whole matrix out of its quasiseparable generators and we use Matlab function eig() to find Matlab eigenvalues, which are considered now the true

Bisection for quasiseparable

197

ones. The same formula used for computing the errors is used. The errors are particularly small since our algorithm multiplies most of the time with 0 or 1, as most of the scalar entries of the quasiseparable generators of band matrices are equal to 0 or 1. Perturbed real diag matrix bisection error

10-13

rank 2 perturbed rank 3 perturbed

10-14

10-15

16

32

64

128

256

512

1024

2048

Errors on eigenvalues

6.1.4.Eigenvalues of Hermitian diagonal plus small rank matrices. In Figure 4 above we plot the relative error in finding eigenvalues for Hermitian matrices which are two or three rank perturbation of a (real) diagonal matrix. We also build the whole matrix out of its quasiseparable generators and we use Matlab function eig() to find Matlab eigenvalues, which are considered now the true ones. The same formula used for computing the errors is used. We can see also from this figure that errors are not higher for order 3 as compared to order 2. 6.2. Results for singular values

Sing val relative error

10-13

10-14

10-15 Prescribed in advance Random known a priori

16

32

64

128

256

512

1024

2048

Errors in singular values

In Figure 5 above we plot the relative error in finding singular values for Hermitian matrices, where we act on order 2 quasiseparable matrices, when these singular values are known in advance via Theorem 4.1. We choose the singular values sT (k) = 0.5 + 0.3k/N,

k = 1, . . . , N,

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which are between 0.3 and 0.8 and then random singular values between 0 and 1. For the last case, we check only half of the singular values, the larger ones. The same formula used for computing the errors is used, only that the eigenvalues are replaced by singular values. We can see also from this figure that, whenever we do not divide by numbers close to 0 as in the random case and whenever we must not build the whole matrix and then use Matlab, the errors are not too high, even for singular values and even when computed with their definition. Of course we do not multiply matrices, but we only use the quasiseparable generators which correspond to the product matrix A = A∗0 A0 , which are of order 4.

Average scalar product between eigvectors

6.3. Results for eigenvectors 10-12

10-12

10-13

10-13

10-14

10-14

Prescribed eig values Perturb of diagonal

10-15 16 32 64 128 256 512

10-15 16 32 64 128 256 512

2048

2048

for order 2 (left) and 3 (right)

Eigvec orthogonality

We find eigenvectors with Theorem 3.4. In Figure 6 above we plot N 1  6

N 

|

N 

xs (k)xt (k)|,

s=N −3 t=s+1 k=1

i.e., the average of the 6 scalar products between the 4 eigenvectors corresponding to the leading 4 eigenvalues (in increasing order), as they have been determined previously for the case of general order 2 (left) and order 3 (right) quasiseparable matrices with our prescribed eigenvalues, as well as for the cluster of the eigenvectors of the 4 larger eigenvalues of the diagonal plus small rank matrices. If we check the orthogonality X ∗ X − IN of the whole matrix X containing all the N eigenvectors as its columns, the results are fine only up to N = 64.

7. Conclusions and future work We developed the bisection algorithm of eigenvalue computations for Hermitian quasiseparable of any order matrices. We checked that our algorithm is fast and very accurate. Based on this algorithm we obtained the first results for the computation of eigenvectors of Hermitian matrices and the algorithm to compute singular values of general quasiseparable matrices via definition.

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199

We plan to improve our eigenvector algorithm using some well-known methods and to develop new algorithms to compute singular values. Acknowledgements Special thanks to the knowledgeable and detail-oriented referee for his helpful suggestions.

References [1] W. Barth, R.S. Martin, J.H. Wilkinson, Calculations of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection, Numerische Mathematik. 9 (1967), 386–393. [2] J. Demmel, Applied numerical linear algebra, SIAM, 1997. [3] P.M. Dewilde, A.J. van der Veen, Time-varying systems and computations, Kluwer Academic Publishers, New York, 1998. [4] Y. Eidelman, I. Gohberg, On a new class of structured matrices. Integral Equations Operator Theory 34 (1999), 293–324. [5] Y. Eidelman, I. Gohberg, I. Haimovici, Separable type representations of matrices and fast algorithms, Volume I, Basics. completion problems, multiplication and inversion algorithms, Operator Theory, Advances and Applications 234, Birkh¨ auser, Basel, 2013. [6] Y. Eidelman, I. Gohberg, I. Haimovici, Separable type representations of matrices and fast algorithms. Volume II, Eigenvalue method, Operator Theory, Advances and Applications 235, Birkh¨ auser, Basel, 2013. [7] Y. Eidelman, I. Gohberg, V. Olshevsky, Eigenstructure of Order-onequasiseparable matrices. Three-term and two-term recurrence relations, Linear Algebra and its Applications 405 (2005), 1–40. [8] Y. Eidelman, I. Haimovici, The fast bisection eigenvalue method for Hermitian order one quasiseparable matrices and computations of norms, Electronic Transactions on Numerical Analysis (ETNA) 44 (2015), 342–366. [9] I. Gohberg, M.A. Kaashoek, L. Lerer, Minimality and realization of discrete time-varying systems, in: Time-variant systems and interpolation (I. Gohberg, ed.), Operator Theory, Advances and Applications 56, pp. 261–296, Birkh¨ auser, Basel, 1992. [10] G.H. Golub, C.F. Van Loan, Matrix computations, The Johns Hopkins University Press, Baltimore, 1989. [11] R.A. Horn, C.R. Johnson, Norms for vectors and matrices, in: Matrix Analysis, Cambridge University Press, Cambridge, UK, 1990. [12] P. Lancaster, M. Tismenetsky, The theory of matrices, Academic Press, New York, London, 1984. [13] R. Vandebril, M. Van Barel, N. Mastronardi, Matrix computations and semiseparable matrices, Volume I, Linear Systems, The Johns Hopkins University Press, Baltimore, 2008. [14] R. Vandebril, M. Van Barel, N. Mastronardi, Matrix computations and semiseparable matrices, Volume II, Eigenvalue and singular value methods, The Johns Hopkins University Press, Baltimore, 2008.

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Y. Eidelman and I. Haimovici School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University Ramat-Aviv 69978 Israel e-mail: [email protected] [email protected]

A note on inner-outer factorization of wide matrix-valued functions A.E. Frazho and A.C.M. Ran Dedicated to our friend and mentor Rien Kaashoek on the occasion of his eightieth birthday, with gratitude for inspiring and motivating us to work on many interesting problems.

Abstract. In this paper we expand some of the results of [8, 9, 10]. In fact, using the techniques of [8, 9, 10], we provide formulas for the full rank inner-outer factorization of a wide matrix-valued rational function G with H ∞ entries, that is, functions G with more columns than rows. State space formulas are derived for the inner and outer factor of G. Mathematics Subject Classification (2010). Primary 47B35, 47A68; Secondary 30J99 . Keywords. Inner-outer factorization, matrix-valued function, Toeplitz operators, state space representation.

1. Introduction In this note, E, U and Y are finite-dimensional complex vector spaces and dim Y ≤ dim U . We will present a method to compute the inner-outer factorization for certain matrix-valued rational functions G in H ∞ (U , Y), defined on the closure of the unit disc. Computing inner-outer factorizations for the case when dim U ≤ dim Y is well developed and presented in [4, 5, 13] and elsewhere. Recall that a function Gi is inner if Gi is a function in H ∞ (E, Y) and iω Gi (e ) is almost everywhere an isometry. (In particular, dim E ≤ dim Y.) Equivalently (see, e.g., [5, 13]), Gi in H ∞ (E, Y) is an inner function if and only if the Toeplitz operator TGi mapping 2+ (E) into 2+ (Y) is an isometry. A function Go is outer if Go is a function in H ∞ (U , E) and the range of the Toeplitz operator TGo is dense in 2+ (E). Let G be a function in H ∞ (U , Y). Then G admits a unique inner-outer factorization of the form G(λ) = Gi (λ)Go (λ) where Gi is an inner function in H ∞ (E, Y) and Go is an outer function in H ∞ (U , E) for some intermediate space E. Because Gi (eiω ) is almost everywhere an isometry, dim E ≤ dim Y. © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_8

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Since Go is outer, Go (eiω ) is almost everywhere onto E, and thus, dim E ≤ dim U . By unique we mean that if G(λ) = Fi (λ)Fo (λ) is another inner-outer factorization of G where Fi is an inner function in H ∞ (L, Y) and Fo is an outer function in H ∞ (U , L), then there exists an constant unitary operator Ω mapping E onto L such that Gi = Fi Ω and ΩGo = Fo ; see [1, 5, 6, 13, 14, 15] for further details. Throughout we assume that U , E and Y are all finite dimensional. We say that Gi in H ∞ (E, Y) is a square inner function if Gi is an inner function and E and Y have the same dimension, that is, Gi (eiω ) is almost everywhere a unitary operator, or equivalently, Gi is a two-sided inner function. So if Gi Go is an inner-outer factorization of G where Gi is square, then without loss of generality we can assume that E = Y. We say that the inner-outer factorization G = Gi Go is full rank if Gi is a square inner function in H ∞ (Y, Y) and the range of TGo equals 2+ (Y). An inner-outer factorization G = Gi Go is full rank if and only if Gi is a square inner function and the range of TG is closed. If G is a rational function, then G admits a full rank inner-outer factorization if and only if G(eiω )G(eiω )∗ ≥ I

(for all ω ∈ [0, 2π] and some  > 0);

(1.1)

see Lemma 3.1 below. Finally, if G in H ∞ (U , Y) admits a full rank inner-outer factorization, then dim Y ≤ dim U . Here we are interested in computing the inner-outer factorization for full rank rational functions G in H ∞ (U , Y). So throughout we assume that dim Y ≤ dim U . Computing inner-outer factorizations when G does not admit a full rank factorization is numerically sensitive. (In this case, our algebraic Riccati equation may not have a stabilizing solution.) Moreover, if G does not admit a full rank inner-outer factorization, then a small H ∞ perturbation of G does admit such a factorization. (If G in H ∞ (U , Y), does not satisfy (1.1), then a “small random” rational H ∞ perturbation of G will satisfy (1.1).) First we will present necessary and sufficient conditions to determine when G admits a full rank inner-outer factorization. Then we will give a state space algorithm to compute Gi and then Go . Finally, it is emphasized that this note is devoted to finding inner-outer factorizations for wide rational functions G in H ∞ (U , Y) when dim Y ≤ dim U . Finding inner-outer factorizations when dim U ≤ dim Y is well developed and presented in [4, 5] and elsewhere.

2. Preliminaries

∞ Let R = −∞ eiωn Rn be the Fourier series expansion for a function R in L∞ (Y, Y). Then TR is the Toeplitz operator on 2+ (Y) defined by ⎡ ⎤ R0 R−1 R−2 · · · ⎢ R1 R0 R−1 · · · ⎥ ⎢ ⎥ on 2+ (Y). TR = ⎢ R2 R1 (2.2) R0 · · · ⎥ ⎣ ⎦ .. .. .. .. . . . .

A note on inner-outer factorization of wide matrix-valued functions 203 The function R is called the symbol for TR . Recall that the Toeplitz operator TR is strictly positive if and only if there exists an  > 0 such that R(eiω ) ≥ I almost everywhere. The Toeplitz operator TG with symbol G in H ∞ (U , Y), is given by ⎤ ⎡ G0 0 0 ··· ⎢ G 1 G0 0 · · · ⎥ ⎥ ⎢ TG = ⎢ G2 G1 G0 · · · ⎥ : 2+ (U ) → 2+ (Y), (2.3) ⎦ ⎣ .. .. .. .. . . . . ∞ n where G(λ) = 0 λ Gn is the Taylor series expansion for G about the origin. Moreover, if G is in H ∞ (U , Y), then the Hankel operator HG mapping 2+ (U ) into 2+ (Y) is defined by ⎡ ⎤ G1 G 2 G3 · · · ⎢ G2 G 3 G4 · · · ⎥ ⎢ ⎥ HG = ⎢G3 G4 G5 · · ·⎥ : 2+ (U ) → 2+ (Y). (2.4) ⎣ ⎦ .. .. .. .. . . . . Finally, for G in H ∞ (U , Y) it is well know and easy to verify that ∗ TG TG∗ = TGG∗ − HG HG .

(2.5)

3. Inner-outer factorization First a characterization of the existence of a full rank inner-outer factorization is presented. Lemma 3.1. Let G be a rational function in H ∞ (U , Y) where U and Y are finite-dimensional spaces satisfying dim Y ≤ dim U . Then G admits a full rank inner-outer factorization if and only if G(eiω )G(eiω )∗ ≥ I

(for all ω ∈ [0, 2π] and some  > 0),

(3.6)

or equivalently, the Toeplitz operator TGG∗ is strictly positive. Proof. Let G = Gi Go be the inner-outer factorization for G where Gi is an inner function in H ∞ (E, Y) and Go is an outer function in H ∞ (U , E). Clearly, G(eiω )G(eiω )∗ = Gi (eiω )Go (eiω )Go (eiω )∗ Gi (eiω )∗ .

(3.7)

Because Gi is an inner function, G(eiω )G(eiω )∗ and Go (eiω )Go (eiω )∗ have the same nonzero spectrum and rank almost everywhere. The range of TGo equals 2+ (E) if and only if the operator TGo TG∗ o is strictly positive. If TGo TG∗ o ∗ is strictly positive, then TGo G∗o = TGo TG∗ o +HGo HG implies that TGo G∗o is also o 2 strictly positive. So if the range of TGo equals + (E), then Go (eiω )Go (eiω )∗ ≥ IE for some  > 0. In addition, if G = Gi Go is a full rank inner-outer factorization, then Gi (eiω ) is a unitary operator. In this case, equation (3.7) shows that (3.6) holds.

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On the other hand, assume that (3.6) holds, or equivalently, the Toeplitz operator TGG∗ is strictly positive. Because G is rational, the range of HG is ∗ finite dimensional. Using TG TG∗ = TGG∗ − HG HG , we see that TG TG∗ equals a ∗ strictly positive operator TGG∗ minus a finite rank positive operator HG HG . Clearly, TGG∗ is a Fredholm operator with index zero. Since TG TG∗ is a finite rank perturbation of TGG∗ , it follows that TG TG∗ is also a Fredholm operator with index zero. In particular, the range of TG is closed. Hence the range of TGo is also closed. Because G(eiω )G(eiω )∗ and Go (eiω )Go (eiω )∗ have the same rank and dim E ≤ dim Y, equation (3.7) with (3.6) shows that E and Y are of the same dimension. In particular, Gi is a square inner function. Therefore the inner-outer factorization G = Gi Go is of full rank.  Next, we recall some results on the inner-outer factorization in terms of a stable finite-dimensional realization for a rational function G. To this end, let {A on X , B, C, D} be a stable realization for G in H ∞ (U , Y), that is, G(λ) = D + λC (I − λA)

−1

B.

(3.8)

Here A is a stable operator on a finite-dimensional space X and B maps U into X while C maps X into Y and D maps U into Y. By stable we mean that all the eigenvalues for A are inside the open unit disc. Note that {A, B, C, D} is a realization for G if and only if G0 = D and Gn = CAn−1 B (for n ≥ 1) (3.9) ∞ n where G(λ) = 0 λ Gn is the Taylor series expansion for G. Let Wo be the observability operator mapping X into 2+ (Y) and Wc the controllability operator mapping 2+ (U ) into X defined by ⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ Wo = ⎢CA2 ⎥ : X → 2+ (Y), ⎣ ⎦ .. .   Wc = B AB A2 B · · · : 2+ (U ) → X . (3.10)  ∞ Let P = Wc Wc∗ = 0 An BB ∗ A∗n be the controllability Gramian for the pair {A, B}. Then P is the solution to the following Stein equation P = AP A∗ + BB ∗ .

(3.11)

Using (3.9), we see that the Hankel operator HG is equal to HG = W o W c .

(3.12)

In particular, it follows that the Hankel operator HG admits a factorization of the form HG = Wo Wc where Wo is an operator mapping X into 2+ (Y) and Wc is an operator mapping 2+ (U ) into X . Using P = Wc Wc∗ with (2.5), we obtain ∗ = Wo P Wo∗ HG HG

and

TG TG∗ = TGG∗ − Wo P Wo∗ .

(3.13)

A note on inner-outer factorization of wide matrix-valued functions 205 Consider the algebraic Riccati equation Q = A∗ QA + (C − Γ∗ QA)∗ (R0 − Γ∗ QΓ)−1 (C − Γ∗ QA) Γ = BD∗ + AP C ∗

and

R0 = DD∗ + CP C ∗ .

(3.14)

We say that Q is a stabilizing solution to this algebraic Riccati equation if Q is positive, R0 − Γ∗ QΓ is strictly positive, and the following operator Ao on X is stable: Ao = A − Γ(R0 − Γ∗ QΓ)−1 (C − Γ∗ QA). (3.15) Moreover, if the algebraic Riccati equation (3.14) admits a stabilizing solution Q, then the stabilizing solution Q can be computed by Q = lim Qn Qn+1 =

n→∞ A∗ Q n A

(3.16) ∗

∗

∗

+ (C − Γ Qn A) (R0 − Γ Qn Γ)

−1

∗

(C − Γ Qn A)

subject to the initial condition Q0 = 0. In particular, if the limit in (3.16) does not exist or Ao is not stable, then the algebraic Riccati equation (3.14) does not have a stabilizing solution; see [8, 9] for further details. If Θ is an inner function in H ∞ (E, Y), then H(Θ) is the subspace of 2 + (Y) defined by H(Θ) = 2+ (Y) $ TΘ 2+ (E) = ker TΘ∗ .

(3.17)

Because TΘ is an isometry, I − TΘ TΘ∗ is the orthogonal projection onto H(Θ). It is noted that H(Θ) is an invariant subspace for the backward shift SY∗ on 2+ (Y). According to the Beurling–Lax–Halmos Theorem if H is any invariant subspace for the backward shift, then there exists a unique inner function Θ in H ∞ (E, Y) such that H = H(Θ). By unique we mean that if H = H(Ψ) where Ψ is an inner function in H ∞ (L, Y), then there exists a constant unitary operator Ω from E onto L such that Θ = ΨΩ; see [5, 11, 12, 13, 14, 15] for further details. By combining Lemma 3.1 with the results in [9], we obtain the following result. (For part (v) compare also Lemma 4.1 below.) Theorem 3.2. Let {A on X , B, C, D} be a minimal realization for a rational function G in H ∞ (U , Y) where dim Y ≤ dim U . Let R be the function in L∞ (Y, Y) defined by R(eiω ) = G(eiω )G(eiω )∗ . Let P the unique solution to the Stein equation P = AP A∗ + BB ∗ . Then the following statements are equivalent. (i) The function G admits a full rank inner-outer factorization; (ii) the Toeplitz operator TR is invertible; (iii) there exists a stabilizing solution Q to the algebraic Riccati equation (3.14). In this case, Q = Wo∗ TR−1 Wo and the following holds. (iv) The eigenvalues of QP are real numbers contained in the interval [0, 1]. (v) If Gi is the inner factor of G, then the dimension of H(Gi ) is given by dim H(Gi ) = dim ker TG∗ i = dim ker TG∗ = dim ker(I − QP ).

(3.18)

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(vi) The McMillan degree of Gi is given by δ(Gi ) = dim H(Gi ) = dim ker(I − QP ).

(3.19)

In particular, the McMillan degree of Gi is less than or equal to the McMillan degree of G. (vii) The operator TR−1 Wo is given by ⎡ ⎤ Co ⎢ C o Ao ⎥ ⎢ ⎥ TR−1 Wo = ⎢Co A2 ⎥ : X → 2+ (Y), o⎦ ⎣ .. . Co = (R0 − Γ∗ QΓ)−1 (C − Γ∗ QA) : X → Y.

(3.20)

Finally, because {C, A} is observable, TR−1 Wo is one-to-one and {Co , Ao } is a stable observable pair. Let us present the following classical result; see Theorem 7.1 in [7], Sections 4.2 and 4.3 in [5] and Section XXVIII.7 in [11]. Lemma 3.3. Let Θ be an inner function in H ∞ (Y, Y) where Y is finite dimensional. Then the Hankel operator HΘ is a partial isometry whose range equals H(Θ), that is, ∗ PH(Θ) = HΘ HΘ (3.21) where PH(Θ) denotes the orthogonal projection onto H(Θ). Furthermore, the following holds. (i) The subspace H(Θ) is finite dimensional if and only if Θ is rational. (ii) The dimension of H(Θ) equals the McMillan degree of Θ. Proof. For completeness a proof is given. By replacing G by Θ in (2.5), we see that ∗ . TΘ TΘ∗ = TΘΘ∗ − HΘ HΘ Because Θ is a square inner function, Θ(eiω )Θ(eiω )∗ = I almost everywhere on the unit circle. Hence TΘΘ∗ = I. This readily implies that ∗ HΘ HΘ = I − TΘ TΘ∗ = PH(Θ) .

Therefore (3.21) holds and HΘ is a partial isometry whose range equals H(Θ). It is well know that the range of a Hankel operator HF is finite dimensional if and only if its symbol F is rational. Moreover, the dimension of the range of the Hankel operator HF equals the McMillan degree of F . Therefore parts (i) and (ii) follow from the fact that H(Θ) = ran HΘ .  Let {Ai on Xi , Bi , Ci , Di } be a minimal state space realization for a rational function Θ in H ∞ (Y, Y). It is well known (see, e.g., [7], Section III.7) that Θ is a square inner function if and only if ∗





Qi 0 Ai Ci∗ Qi 0 Ai Bi (3.22) = Bi∗ Di∗ 0 I Ci Di 0 I

A note on inner-outer factorization of wide matrix-valued functions 207 where Qi = A∗i Qi Ai + Ci∗ Ci . Moreover, in this case, H(Θ) = ran HΘ = ran Wi where Wi is the observability operator for {Ci , Ai } defined by ⎡ ⎤ Ci ⎢ C i Ai ⎥ ⎢ ⎥ Wi = ⎢Ci A2 ⎥ : Xi → 2+ (Y). i ⎣ ⎦ .. .

(3.23)

It is noted that SY∗ Wi = Wi Ai . So the range of Wi is a finite-dimensional invariant subspace for the backward shift SY∗ . Finally, Qi = Wi∗ Wi . On the other hand, if {Ci , Ai on Xi } is a stable observable pair where Xi is finite dimensional, then there exists operators Bi mapping Y into Xi and Di on Y such that Θ(λ) = Di + λCi (I − λAi )−1 Bi

(3.24)

is a square inner function in H ∞ (Y, Y). Moreover, H(Θ) = ran Wi and (3.22) holds. The Beurling–Lax–Halmos Theorem guarantees that the inner function Θ is unique up to a unitary constant on the right. The operators Bi and Di are called the complementary operators for the pair {Ci , Ai }. To compute the complementary operators Bi and Di explicitly, let



E1 Xi :Y→ (3.25) E2 Y $ % 1 be an isometry from Y onto the kernel of A∗i Qi2 Ci∗ . Then set −1

B i = Q i 2 E1

and

D i = E2 .

(3.26)

Because the pair {Ci , Ai } is observable, the operator Wi defined in (3.23) is one to one, and the complementary operators Bi and Di together with Ai and Ci form a minimal realization {Ai , Bi , Ci , Di } for a square inner function Θ such that ran Wi = H(Θ). For further details see Lemma XXVIII7.7 in [11] and Sections 4.2 and 4.3 in [5]. We are now in a position to present our main result. The proof is given in Section 5. Let {A, B, C, D} be a minimal realization for a rational function G in H ∞ (U , Y) where dim Y ≤ dim U . To compute a full rank inner-outer factorization G = Gi Go for G, let P be the controllability Gramian for the pair {A, B} (see (3.11)) and Q the stabilizing solution to the algebraic Riccati equation (3.14). If this algebraic Riccati equation does not admit a stabilizing solution, then G does not have a full rank inner-outer factorization. Theorem 3.4. Let {A, B, C, D} be a minimal realization for a rational function G in H ∞ (U , Y) where dim Y ≤ dim U . Assume there exists a stabilizing solution Q to the algebraic Riccati equation (3.14).

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Let Xi be any space isomorphic to the kernel of I − QP . Let U be any isometry from Xi onto the kernel of I − QP . In particular, U = QP U . Let Ai on Xi and Ci mapping Xi into Y be the operators computed by Ai = U ∗ QAo P U

and

Ci = Co P U.

(3.27)

Then {Ci , Ai } is a stable observable pair. Let Bi and Di be the complementary operators for {Ci , Ai } as constructed in (3.25) and (3.26). Then the square inner factor Gi for G is given by Gi (λ) = Di + λCi (I − λAi )−1 Bi .

(3.28)

The outer factor Go for G is given by Go (λ) = Di∗ D + Bi∗ U ∗ B + λ (Di∗ C + Bi∗ U ∗ A) (I − λA)−1 B.

(3.29)

4. An auxiliary lemma To prove that the inner-outer factorization of G = Gi Go is indeed given by (3.28) and (3.29), let us begin with the following auxiliary result. Lemma 4.1. Let T be a strictly positive operator on H and P a strictly positive operator on X . Let W be an operator mapping X into H and set Q = W ∗ T −1 W . Then the following two assertions hold. (i) Let X and H be the subspaces defined by X = ker(I − QP )

and

H = ker (T − W P W ∗ ) .

(4.1)

and

Λ2 = T −1 W P |X : X → H

(4.2)

Then the operators Λ1 = W ∗ |H : H → X

are both well defined and invertible. Moreover, Λ−1 1 = Λ2 . (ii) The operator T − W P W ∗ is positive if and only if P −1 − Q is positive, 1 1 or equivalently, P 2 QP 2 is a contraction. In this case, the spectrum of QP is contained in [0, 1]. In particular, if X is finite dimensional, then the eigenvalues for QP are contained in [0, 1]. Proof. The proof is based on some ideas involving Schur complements; see [2] and Section 2.2 in [3]. Consider the operator matrix







T W I WP T − WPW∗ 0 I 0 M= = W ∗ P −1 0 I 0 P −1 P W ∗ I





I 0 T 0 I T −1 W = . W ∗ T −1 I 0 P −1 − Q 0 I From this we conclude several things: first, by the fact that both T and P are strictly positive, the congruences above imply that T − W ∗ P W is positive if 1 1 and only if P −1 − Q is positive, or equivalently, P 2 QP 2 is a contraction. In particular, if T − W ∗ P W is positive, then the spectrum of QP is contained in the interval [0, 1]. This proves part (ii) of the lemma.

A note on inner-outer factorization of wide matrix-valued functions 209 To prove part (i) observe that we can describe ker M in two different ways. Based on the first factorization we have 

< h ∗ | h ∈ ker(T − W P W ) = H . ker M = −P W ∗ h The second factorization yields

<  −T −1 W y −1 | y ∈ ker(P − Q) = P X ker M = y  −1

< T WPx ker M = | x ∈ ker(I − QP ) = X . −P x Together these equalities prove the first assertion in Lemma 4.1. Indeed,

I : H → ker M Φ1 = −P W ∗ is a one-to-one operator from H onto ker M . Likewise, −1

T WP : X → ker M Φ2 = −P is a one-to-one operator from X onto ker M . Because the first component of Φ1 is the identity operator on H, we see that T −1 W P maps X onto H. Since the second component of Φ2 is −P and P is invertible, W ∗ maps H onto X. Therefore the operators Λ1 and Λ2 in (4.2) are well defined. If x is in X, then Φ2 x = Φ1 h for some unique h in H, that is,

−1





Λ2 x T WPx h Λ2 x = = Φ2 x = Φ1 h = = . −P x −P x −P W ∗ h −P W ∗ Λ2 x The last equality follows from the fact that h = Λ2 x. The second component of the previous equation shows that x = W ∗ Λ2 x, and thus, Λ1 = W ∗ |H is the left inverse of Λ2 . On the other hand, if h is in H, then Φ1 h = Φ2 x for some unique x in X, that is,

−1



h T WPx Λ2 x = Φ1 h = Φ2 x = = . −P W ∗ h −P x −P x By consulting the second component, we have Λ1 h = W ∗ h = x. Substituting x = Λ1 h into the first component, yields h = Λ2 Λ1 h. Therefore Λ1 is the right inverse of Λ2 and Λ−1  1 = Λ2 .

5. Proof of the inner-outer factorization Proof. Assume that the algebraic Riccati equation (3.14) admits a stabilizing solution Q. In other words, assume that TR is strictly positive, or equivalently, G admits a full rank inner-outer factorization G = Gi Go . Using P = Wc Wc∗ with HG = Wo Wc , we have ∗ TG TG∗ = TR − HG HG = TR − Wo P Wo∗ .

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Recall that the subspace H(Gi ) = 2+ (Y) $ TGi 2+ (Y). Then H(Gi ) = ker TG∗ i = ker TG∗ = ker (TR − Wo P Wo∗ ) . It is noted that H(Gi ) is an invariant subspace for the backward shift SY∗ on 2+ (Y). Recall that Q = Wo∗ TR−1 Wo . Let k = dim ker(I − QP ), and put Xi = Ck . Let U be an isometry from Xi onto ker(I − QP ). According to Lemma 4.1, the operator ⎡ ⎤ Co ⎢ C o Ao ⎥ ⎢ ⎥ 2⎥ ⎢ Λ2 = TR−1 Wo P U = ⎢Co Ao3 ⎥ P U : Xi → H(Gi ) ⎢ C o Ao ⎥ ⎣ ⎦ .. . is invertible, where we also use (3.20). In particular, the dimension of H(Gi ) equals dim Xi . Since P is invertible and U is an isometry, the operator P U from Xi into X is one to one. Because H(Gi ) is an invariant subspace for the backward shift SY∗ , there exists an operator Ai on Xi = Ck such that SY∗ TR−1 Wo P U = TR−1 Wo P U Ai . TR−1 Wo P U

(5.1)

SY∗n

Since is one to one and converges to zero pointwise, Ai is stable. Now observe that ⎡ ⎤ ⎡ ⎤ Co Co ⎢ C o Ao ⎥ ⎢ C o Ao ⎥ ⎢ ⎢ ⎥ ⎥ ⎢Co A2o ⎥ Ao P U = SY∗ TR−1 Wo P U = ⎢Co A2o ⎥ P U Ai . ⎣ ⎣ ⎦ ⎦ .. .. . . Since the observability matrix for {Co , Ao } is one to one, Ao P U = P U Ai . Because P U is one to one, the spectrum of Ai is contained in the spectrum of Ao . Multiplying Ao P U = P U Ai by U ∗ Q on the left and using QP U = U shows that Ao P U = P U Ai and Ai = U ∗ QAo P U. (5.2) Setting Ci = Co P U and using Ajo P U = P U Aji for all positive integers j, we obtain Co Ajo P U = Co P U Aji = Ci Aji (for all integers j ≥ 0). (5.3) In particular,

⎡ ⎤ ⎤ Co Ci ⎢ C i Ai ⎥ ⎢ C o Ao ⎥ ⎢ ⎢ ⎥ ⎥ TR−1 Wo P U = ⎢Co A2 ⎥ P U = ⎢Ci A2 ⎥ . o⎦ i⎦ ⎣ ⎣ .. .. . . ⎡

Since TR−1 Wo P U is one to one, {Ci , Ai } is a stable observable pair. Let Bi mapping Y into Xi = Ck and Di on Y be the complementary operators for the

A note on inner-outer factorization of wide matrix-valued functions 211 pair {Ci , Ai }. Since H(Gi ) equals the range of TR−1 Wo P U , the inner function Gi (up to a unitary constant on the right) is given by Gi (λ) = Di + λCi (I − λAi )−1 Bi . To find the outer factor Go , first notice that ⎡  Q = Wo∗ TR−1 Wo = Co∗

A∗o Co∗

⎤ C ∞ ⎥ ⎢ ⎢ CA2 ⎥  ∗j ∗ ∗ A∗2 C · · · = A C CAj . ⎢ ⎥ o o ⎣CA ⎦ j=0 o o .. .

The second equality follows from (3.20). In other words, Q satisfies the Stein equation Q = A∗o QA + Co∗ C. (5.4) Now note that U ∗ P Co∗ C = Ci∗ C, so that Ci∗ C = U ∗ P (Q−A∗o QA). Moreover, U ∗ P A∗o = A∗i U ∗ P and U ∗ P Q = U ∗ . Hence Ci∗ C = U ∗ − A∗i U ∗ P QA = U ∗ − A∗i U ∗ A. (5.5)  ∞ ∗ j It follows that U ∗ = j=0 A∗j i Ci CA . Next observe that TG = TGi Go = TGi TGo . Multiplying by TG∗ i on the left, with the fact that TGi is an isometry, we have TG∗ i TG = TGo . Using this ∞ ∗ j with U ∗ = j=0 A∗j i Ci CA , we see that the first column of TGo is given by ⎡ ∗ ⎤⎡ ⎤ ∗ ⎤ ⎡ Di Bi∗ Ci∗ Bi∗ A∗i Ci∗ Bi∗ A∗2 D ··· i Ci D ⎢ ⎢ ⎥ Di∗ Bi∗ Ci∗ Bi∗ A∗i Ci∗ · · ·⎥ ⎢ CB ⎥ ⎢ 0 ⎥ ⎢ CB ⎥ ∗ ∗ ∗ ⎢ ⎢ ⎥ ⎢ ⎥ 0 Di Bi Ci · · ·⎥ ⎢ CAB ⎥ TG∗ i ⎢CAB ⎥ = ⎢ 0 ⎥ ∗ 2 ⎥ ⎢ ⎦ ⎢ 0 ⎣ 0 0 Di · · ·⎥ ⎣ ⎦ ⎣CA B ⎦ .. .. .. .. .. .. . . . . . ··· . ⎡ ⎤ ∗ ∗ ∗ D i D + Bi U B ⎢ (Di∗ C + Bi∗ U ∗ A)B ⎥ ⎢ ∗ ⎥ ∗ ∗ ⎢ ⎥ = ⎢ (D∗i C + B∗i U∗ A)AB ⎥. ⎢(Di C + Bi U A)A2 B ⎥ ⎣ ⎦ .. . By taking the Fourier transform of the first column of TGo , we obtain the following state space formula: Go (λ) = Di∗ D + Bi∗ U ∗ B + λ (Di∗ C + Bi∗ U ∗ A) (I − λA)−1 B. This completes the proof.



For completeness we shall also provide a slightly different derivation of the last part of the proof, that is, the formula for Go . The idea is similar in nature but slightly different in execution: for |λ| = 1 we compute Go (λ) = Gi (λ)∗ G(λ) using the realization formulas for Gi and G. This leads to

212

A.E. Frazho and A.C.M. Ran Go (λ) = Gi (λ)∗ G(λ) = (Di∗ + λBi∗ (I − λA∗i )−1 Ci∗ )(D + λC(I − λA)−1 B) = Di∗ D + λBi∗ (I − λA∗i )−1 Ci∗ D + λDi∗ C(I − λA)−1 B + |λ|2 Bi∗ (I − λA∗i )−1 Ci∗ C(I − λA)−1 B.

Since we consider |λ| = 1 this is equal to 1 ∗ 1 Bi (I − A∗i )−1 Ci∗ D + λDi∗ C(I − λA)−1 B λ λ 1 ∗ −1 ∗ ∗ + Bi (I − Ai ) Ci C(I − λA)−1 B. λ

Go (λ) = Di∗ D +

Consider the Stein equation Ci∗ C = U ∗ − A∗i U ∗ A; see (5.5). This may be used to compute 1 ∗ −1 ∗ A ) Ci C(I − λA)−1 λ i   1 1 = (I − A∗i )−1 U ∗ − A∗i U ∗ (λA) (I − λA)−1 λ λ  1 ∗ −1  ∗ 1 = (I − Ai ) U (I − λA) + (I − A∗i )U ∗ (λA) (I − λA)−1 λ λ 1 ∗ −1 ∗ ∗ −1 = (I − Ai ) U + λU A(I − λA) . λ

(I −

Inserting this in the formula for Go (λ) we obtain 1 ∗ 1 B (I − A∗i )−1 Ci∗ D + λDi∗ C(I − λA)−1 B+ λ i λ 1 + Bi∗ (I − A∗i )−1 U ∗ B + λBi∗ U ∗ A(I − λA)−1 B λ = Di∗ D + Bi∗ (λI − A∗i )−1 Ci∗ D   1 + Bi∗ I + (I − A∗i )−1 − I U ∗ B λ

Go (λ) = Di∗ D +

+ λ(Di∗ C + Bi∗ U ∗ A)(I − λA)−1 B = Di∗ D + Bi∗ U ∗ B + Bi∗ (λI − A∗i )−1 (Ci∗ D + A∗i U ∗ B) + λ(Di∗ C + Bi∗ U ∗ A)(I − λA)−1 B. Because Go is analytic in the open unit disc, we know that the term Bi∗ (λI − A∗i )−1 (Ci∗ D + A∗i U ∗ B) must be zero. Let us give a direct proof of this fact. This turns out to be an easy consequence of formula (3.23) in [9]. Indeed, this formula states that C1∗ C1 = (Q − QP Q) − A∗0 (Q − QP Q)A0 , where C1 = D∗ C0 + B ∗ QA0 . Multiplying the above formula with P U on the right and U ∗ P on the left, we obtain U ∗ P C1∗ C1 P U = U ∗ P (Q − QP Q)P U − U ∗ P A∗0 (Q − QP Q)A0 P U.

A note on inner-outer factorization of wide matrix-valued functions 213 Since QP U = U it follows that (Q − QP Q)P U = 0, so the first term on the right hand side is zero. Further, since A0 P U = P U Ai it follows that also the second term on the right hand side is zero. Hence C1 P U = 0, which means 0 = (D∗ C0 + B ∗ QA0 )P U = D∗ Ci + B ∗ QP U Ai = D∗ Ci + B ∗ U Ai . Thus the formula for Go can also be established by a direct computation using the realizations of Gi and G.

References [1] D.Z. Arov and H. Dym, J-Contractive matrix valued functions and related topics, Encl. of Math. and its Appl. 116, Cambridge Univ. Press, 2008. [2] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Schur complements and state space realizations. Linear Algebra Appl. 399 (2005), 203–224. [3] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of matrix and operator functions: the state space method, OT 178, Birkh¨ auser Verlag, Basel, 2008. [4] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, A state space approach to canonical factorization: convolution equations and mathematical systems, OT 200, Birkh¨ auser Verlag, Basel, 2010. [5] A.E. Frazho and W. Bosri, An operator perspective on signals and systems, OT 204, Birkh¨ auser Verlag, Basel, 2010. [6] C. Foias and A. Frazho, The commutant lifting approach to interpolation problems, OT 44, Birkh¨ auser Verlag, Basel, 1990. [7] C. Foias, A. Frazho, I. Gohberg, and M.A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, OT 100, Birkh¨ auser Verlag, Basel, 1998. [8] A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, The non-symmetric discrete algebraic Riccati equation and canonical factorization of rational matrix functions on the unit circle, Integral Equations and Operator Theory 66 (2010), 215–229. [9] A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, I. The least squares solution, Integral Equations and Operator Theory 70 (2011), 395–418. [10] A.E. Frazho, M.A. Kaashoek, and A.C.M. Ran, Right invertible multiplication operators and stable rational matrix solutions to an associate Bezout equation, II: Description of all solutions, Operators and Matrices 6 (2012), 833–857. [11] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Volume II, OT 63, Birkh¨ auser Verlag, Basel, 1993. [12] N.K. Nikol’skii, Treatise on the shift operator, Grundlehren 273, Springer Verlag, Berlin, 1986. [13] M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford Univ. Press, 1985. [14] B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Co., Amsterdam–Budapest, 1970.

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[15] B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kerchy, Harmonic Analysis of Operators on Hilbert Space, Springer Verlag, New York, 2010. A.E. Frazho Department of Aeronautics and Astronautics, Purdue University West Lafayette, IN 47907 USA e-mail: [email protected] A.C.M. Ran Department of Mathematics, Faculty of Science, VU Amsterdam De Boelelaan 1081a, 1081 HV Amsterdam The Netherlands and Unit for BMI, North-West University Potchefstroom South Africa e-mail: [email protected]

An application of the Schur complement to truncated matricial power moment problems Bernd Fritzsche, Bernd Kirstein and Conrad Mädler Dedicated to the 80th birthday of M. A. Kaashoek

Abstract. The main goal of this paper is to reconsider a phenomenon which was treated in earlier work of the authors’ on several truncated matricial moment problems. Using a special kind of Schur complement we obtain a more transparent insight into the nature of this phenomenon. In particular, a concrete general principle to describe it is obtained. This unifies an important aspect connected with truncated matricial moment problems. Mathematics Subject Classification (2010). Primary 44A60; Secondary 47A57. Keywords. Truncated matricial Hamburger moment problems, truncated matricial α-Stieltjes moment problems, Schur complement.

1. Introduction In this paper, we reconsider a phenomenon which was touched in our joint research with Yu.M. Dyukarev on truncated matricial power moment problems (see [8], for the Hamburger case and [7] for the α-Stieltjes case). In this introduction, we restrict our considerations to the description of the Hamburger case. In order to describe more concretely the central topics studied in this paper, we give some notation. Throughout this paper, let p and q be positive integers. Let N, N0 , Z, R, and C be the set of all positive integers, the set of all non-negative integers, the set of all integers, the set of all real numbers, and the set of all complex numbers, respectively. For every choice of ρ, κ ∈ R ∪ {−∞, ∞}, let Zρ,κ := {k ∈ Z : ρ ≤ k ≤ q×q q×q κ}. We will write Cp×q , Cq×q for the set of all complex H , C≥ , and C> p × q matrices, the set of all Hermitian complex q × q matrices, the set of all non-negative Hermitian complex q × q matrices, and the set of all positive © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_9

215

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B. Fritzsche, B. Kirstein and C. Mädler

Hermitian complex q × q matrices, respectively. We will use BR to denote the σ-algebra of all Borel subsets of R. For all Ω ∈ BR \ {∅}, let BΩ := BR ∩ Ω. Furthermore, we will write M≥ q (Ω) to designate the set of all non-negative Hermitian q × q measures defined on BΩ , i. e., the set of σ-additive mappings μ : BΩ → Cq×q ≥ . We will use the integration theory with respect to nonnegative Hermitian q × q measures which was worked out independently by I. S. Kats [12] and M. Rosenberg [13]. For all j ∈ N0 , we will use Mq≥,j (Ω) to denote the set of all σ ∈ M≥ q (Ω) such that the integral  [σ] sj := xj σ(dx) Ω

exists. Obviously, if k,  ∈ N0 with k < , then it can be verified, as in the scalar case, that the inclusion Mq≥, (Ω) ⊆ Mq≥,k (Ω) holds true. Now we formulate two related versions of matricial moment problems. (The Hamburger moment problem corresponds to Ω = R.) Problem MP[Ω; (sj )κj=0 , =]. Let κ ∈ N0 ∪ {∞} and let (sj )κj=0 be a sequence of complex q × q matrices. Describe the set Mq≥ [Ω; (sj )κj=0 , =] of all σ ∈ [σ]

Mq≥,κ (Ω) for which sj = sj for all j ∈ Z0,κ . The just formulated moment problem is closely related to the following: m Problem MP[Ω; (sj )m j=0 , ≤]. Let m ∈ N0 and let (sj )j=0 be a sequence of comq m plex q × q matrices. Describe the set M≥ [Ω; (sj )j=0 , ≤] of all σ ∈ Mq≥,m (Ω) [σ]

for which sm − sm is non-negative Hermitian and, in the case m ∈ N moreover [σ] sj = sj for all j ∈ Z0,m−1 . Remark 1.1. If m ∈ N0 and (sj )m j=0 is a sequence of complex q × q matrices, q then Mq≥ [Ω; (sj )m , =] ⊆ M [Ω; (sj )m j=0 j=0 , ≤]. ≥ m Remark 1.2. If m ∈ N0 and (sj )m j=0 , (tj )j=0 are two sequences of complex q × q matrices satisfying sm − tm ∈ Cq×q and sj = tj for all j ∈ Z0,m−1 , then ≥ q m Mq≥ [Ω; (tj )m , ≤] ⊆ M [Ω; (s ) , ≤]. j j=0 j=0 ≥

In order to state a necessary and sufficient condition for the solvability of each of the above formulated moment problems in the case Ω = R, we have to recall the notion of two types of sequences of matrices. If n ∈ N0 2n and if (sj )2n j=0 is a sequence of complex q × q matrices, then (sj )j=0 is called Hankel non-negative definite if the block Hankel matrix Hn := [sj+k ]nj,k=0 is ≥ non-negative Hermitian. For all n ∈ N0 , we will write Hq,2n for the set of all Hankel non-negative definite sequences (sj )2n j=0 of complex q × q matrices. ≥,e Furthermore, for all n ∈ N0 , let Hq,2n be the set of all sequences (sj )2n j=0 of complex q × q matrices for which there exist complex q × q matrices s2n+1 and 2(n+1) ≥,e ≥ s2n+2 such that (sj )j=0 ∈ Hq,2(n+1) and denote by Hq,2n+1 the set of all 2n+1 sequences (sj )j=0 of complex q × q matrices for which there exists a complex 2(n+1)

q × q matrix s2n+2 such that the sequence (sj )j=0

≥ belongs to Hq,2(n+1) .

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≥,e For each m ∈ N0 , the elements of the set Hq,m are called Hankel non-negative ≥ definite extendable sequences. We use Hq,∞ for the set of all sequences (sj )∞ j=0 ≥ of complex q × q matrices satisfying (sj )2n ∈ H for all n ∈ N . For 0 j=0 q,2n ≥,e := ≥ ≥ technical reasons we set Hq,∞ Hq,∞ . The sequences belonging to Hq,∞ ≥,e (resp. Hq,∞ ) are also said to be Hankel non-negative definite (resp. Hankel non-negative definite extendable). Now we can characterize the situations that the mentioned problems have a solution:

Theorem 1.3 ([5, Theorem 3.2]). Let n ∈ N0 and let (sj )2n j=0 be a sequence of complex q × q matrices. Then Mq≥ [R; (sj )2n , ≤] =

∅ if and only if (sj )2n j=0 j=0 ∈ ≥ Hq,2n . In addition to Theorem 1.3, one can show that, in the case (sj )2n j=0 ∈ Hermitian measure belongs to

≥ Hq,2n , a distinguished molecular non-negative Mq≥ [R; (sj )2n j=0 , ≤] (see, [8, Theorem 4.16]).

Now we characterize the solvability of Problem MP[R; (sj )κj=0 , =].

Theorem 1.4 ([9, Theorem 6.6]). Let κ ∈ N0 ∪ {∞} and let (sj )κj=0 be a sequence of complex q × q matrices. Then Mq≥ [R; (sj )κj=0 , =] = ∅ if and only ≥,e if (sj )κj=0 ∈ Hq,κ . The following result is the starting point of our subsequent considerations: ≥ Theorem 1.5 ([8, Theorem 7.3]). Let n ∈ N0 and let (sj )2n j=0 ∈ Hq,2n . Then ≥,e there exists a unique sequence (˜ sj )2n j=0 ∈ Hq,2n such that q 2n Mq≥ [R; (˜ sj )2n j=0 , ≤] = M≥ [R; (sj )j=0 , ≤].

(1.1)

Theorem 1.5 was very essential for the considerations in [8]. Following [8], we sketch now some essential features of the history of ≥,e Theorem 1.5. The existence of a sequence (˜ sj )2n j=0 ∈ Hq,2n satisfying (1.1) was already formulated by V.A. Bolotnikov (see [4, Lemma 2.12] and [2, Lemma 2.12]). However it was shown in [8, pp. 804–805] by constructing a counterexample that the proof in [4] is not correct. However, in the subsequent considerations of [8, Section 7] it was then shown that the result formulated by V.A. Bolotnikov is correct and that the sequence (˜ sj )2n j=0 in Theorem 1.5 is unique. The proof of Theorem 1.5 given in [8] is constructive and does not yield a nice formula. The main goal of this paper is to present a general purely matrix theoretical object which yields applied to a special case the explicit ≥,e construction of the desired sequence (˜ sj )2n j=0 ∈ Hq,2n . Furthermore, we will see that another application of our construction yields the answer to a similar situation connected with truncated matricial α-Stieltjes moment problems (see Section 6). The discovery of the above mentioned object of matrix theory was inspired by investigations of T. Ando [1] in the context of Schur complements and its applications to matrix inequalities. (It should be mentioned that Ando’s view on the Schur complement is the content of Chapter 5 in the

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book [14] which is devoted to several aspects of the Schur complement.) Given a non-negative Hermitian q × q matrix A and a linear subspace V of Cq , we introduce a particular non-negative Hermitian q × q matrix GA,V which turns out to possess several extremal properties. Appropriate choices of A and V lead to the construction of the desired sequences connected to the truncated matricial moment problems under consideration.

2. On a special kind of Schur complement Against to the background of application to matrix inequalities T. Ando [1] presents an operator theoretic approach to Schur complements. In particular, T. Ando generalized the notion of Schur complement of a block matrix by considering block partitions with respect to an arbitrary fixed linear subspace. For the case of a given non-negative Hermitian q × q matrix A and a fixed linear subspace V the construction by T. Ando produces a non-negative Hermitian q × q matrix GA,V having several interesting extremal properties. For our purposes it is more convenient to choose a more matrix theoretical view as used by T. Ando [1]. For this reason we use a different starting point to the main object of this section (see Definition 2.2). This leads us to a self-contained approach to several results due to Ando. In the sequel, Cp is short for Cp×1 . Let Op×q be the zero matrix from Cp×q . Sometimes, if the size of the zero matrix is clear from the context, we will omit the indices and write O. We denote by N (A) := {x ∈ Cq : Ax = Op×1 } the null space of a complex p × q matrix A. Remark 2.1. If M ∈ Cq×p and V is a linear subspace of Cq , then ΦM (V) := {x ∈ Cp : M x ∈ V} is a linear subspace of Cp . Obviously, ΦM ({Oq×1 }) = N (M ) and ΦM (Cq ) = Cp . We write A∗ for the conjugate transpose and R(A) := {Ax : x ∈ Cq } for the column space of a complex p × q matrix A, resp. With the Euclidean scalar product ·, · E : Cq × Cq → C given by x, y E := y ∗ x, which is C-linear in its first argument, the vector space Cq over the field C becomes a unitary space. Let U be an arbitrary non-empty subset of Cq . The orthogonal complement U ⊥ := {v ∈ Cq : v, u E = 0 for all u ∈ U} of U is a linear subspace of the unitary space Cq . If U is a linear subspace itself, the unitary space Cq is the orthogonal sum of U and U ⊥ . In this case, we write PU for the transformation matrix corresponding to the orthogonal projection onto U with respect to the standard basis of Cq , i. e., PU is the uniquely determined matrix P ∈ Cq×q satisfying P 2 = P = P ∗ and R(P ) = U. For each matrix A ∈ Cq×q ≥ , there 2 exists a uniquely determined matrix Q ∈ Cq×q with Q = A called the ≥ √ non-negative Hermitian square root Q = A of A. As a starting point of our subsequent considerations, we choose a matrix which will turn out to coincide with a matrix which was introduced in an alternate way in [14, see Equation (5.1.11) and Theorem 5.8].

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Definition 2.2. Let A ∈ Cq×q and let V be a linear subspace of Cq . Then we call ≥ the matrix QA,V := PΦ√A (V) the orthogonal projection corresponding to (A, V) √ √ and the matrix GA,V := AQA,V A is said to be the Schur complement associated to A and V. Let Iq := [δjk ]qj,k=1 be the identity matrix from Cq×q , where δjk is the Kronecker delta. Sometimes, we will omit the indices and write I. Remark 2.3. If A ∈ Cq×q ≥ , then QA,{Oq×1 } = PN (A) , GA,{Oq×1 } = Oq×q , QA,Cq = Iq , and GA,Cq = A. Remark 2.4. Let A ∈ Cq×q and let V be a linear subspace of Cq . Then the ≥ matrices QA,V and GA,V are both Hermitian. Remark 2.5. If V is a linear subspace of Cq , then QIq ,V = PV , GIq ,V = PV , QPV ,V = Iq , and GPV ,V = PV . We write rank A for the rank of a complex p × q matrix A. The set Cq×q H of Hermitian matrices from Cq×q is a partially ordered vector space over the field R with positive cone Cq×q ≥ . For two complex q × q matrices A and B, we write A ≤ B or B ≥ A if A, B ∈ Cq×q and B − A ∈ Cq×q are fulfilled. For a ≥ H complex q × q matrix A, we have obviously A ≥ O if and only if A ∈ Cq×q ≥ . The above mentioned partial order ≤ on the set of Hermitian matrices is sometimes called Löwner semi-ordering. Parts of the following proposition coincide with results stated in [14, Theorems 5.3 and 5.6 in combination with Theorem 5.8]. Proposition 2.6. If A ∈ Cq×q and V is a linear subspace of Cq , then: ≥ √ √ (a) R(GA,V ) = R( AQA,V ), N (GA,V ) = N (QA,V A), and Oq×q ≤ GA,V ≤ A. (b) R(GA,V ) = R(A) ∩ V, N (GA,V ) = N (A) + V ⊥ , and in particular rank GA,V ≤ min{rank A, dim V}. (c) The following statements are equivalent: (i) GA,V = A. (ii) R(GA,V ) = R(A). (iii) R(A) ⊆ V. (d) R(GA,V ) = V if and only if V ⊆ R(A). (e) GA,V ∈ Cq×q if and only if A ∈ Cq×q and V = Cq . > > q×q q×q (f) If V = Cq , then GA,V ∈ C≥ \ C> . Proof. (a) Use Oq×q ≤ QA,V ≤ Iq and Q2A,V = QA,V . (b) First y ∈ R(GA,V ). According to (a), we √ we consider an arbitrary q have y = AQA,V z with some z ∈ C . Obviously, x := QA,V z belongs to √ ΦA (V) and fulfills y = Ax. In particular, y belongs to √ R(A) ∩ V. Conversely, now assume that y belongs to R(A) ∩ V. Then y ∈ R( A), i. e., there exists √ √ q an x ∈ C with y = Ax. Consequently, Ax ∈ V, i. e., x ∈ ΦA (V). This √ implies QA,V x = x. Hence, y ∈ R( AQA,V ). Taking (a) into account, then

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y ∈ R(GA,V ) follows. Thus, R(GA,V ) = R(A) ∩ V is proved. Therefore, we get ⊥

N (GA,V ) = R(G∗A,V )⊥ = R(GA,V )⊥ = [R(A) ∩ V]

= R(A)⊥ + V ⊥ = N (A∗ ) + V ⊥ = N (A) + V ⊥ . (c) Condition (i) is obviously sufficient for (ii). According to (b) statements (ii) and (iii) are equivalent. If (iii) holds true then Φ√A (V) = Cq and, consequently, Q√A,V = Iq . This means that (iii) implies (i). (d) This equivalence follows from (b). (e) This is an immediate consequence of the definition of GA,V . (f) This follows from (a) and (e).  Remark 2.7. If A ∈ Cq×q and V is a linear subspace of Cq , then Proposi> tion 2.6(d) shows that R(GA,V ) = V. In the sequel the Moore–Penrose inverse plays an essential role. For this reason, we recall this notion. For each matrix A ∈ Cp×q , there exists a uniquely determined matrix X ∈ Cq×p , satisfying the four equations AXA = A,

XAX = X,

(AX)∗ = AX,

and

(XA)∗ = XA.

This matrix X is called the Moore–Penrose inverse of A and is denoted by A† . Given n ∈ N and arbitrary rectangular complex matrices A1 , A2 , . . . , An , we write diag (Aj )nj=1 or diag(A1 , A2 , . . . , An ) for the block diagonal matrix with matrices A1 , A2 , . . . , An on its block diagonal. The following lemma yields essential insights into the structure of the Schur complement with respect to the Schur complement associated to A and V. Lemma 2.8. Assume q ≥ 2, let d ∈ Z1,q−1 , and let V be a linear subspace of Cq with dim V = d. Let u1 , u2 , . . . , uq be an orthonormal basis of Cq such that u1 , u2 , . . . , ud is an orthonormal basis of V and let U := [u1 , . . . , uq ]. Let A ∈ Cq×q and let B = [Bjk ]2j,k=1 be the block representation of B := U ∗ AU ≥ with d × d block B11 . Then † U ∗ GA,V U = diag(B11 − B12 B22 B21 , O(q−d)×(q−d) ). (2.1) √   Proof. Let R := U ∗ AU and let R =  E F be the  block representation of R ∗ EF ∗ . Consequently, with d × q block E. Then B = RR∗ = EE ∗ ∗ FE FF   † ∗ ∗ † (2.2) B11 − B12 B22 B21 = E Iq − F (F F ) F E ∗ = E(Iq − F † F )E ∗ .

We set U1 := [u1 , . . . , ud ] and W := Φ√A (V). Because of U ∗ U = Iq , we have √ √ U ∗ W = {U ∗ x : x ∈ Cq and Ax ∈ V} = {y ∈ Cq : AU y ∈ V}     √ √    z = y ∈ Cq : AU y ∈ R(U1 ) = y ∈ Cq  ∃z ∈ Cd : AU y = U O(q−d)×1   √ = y ∈ Cq : [O(q−d)×d , Iq−d ]U ∗ AU y = O(q−d)×1 = N (F ).

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Hence, PU ∗ W = Iq − F † F . Since U ∗ PW U is an idempotent and Hermitian complex matrix fulfilling R(U ∗ PW U ) = U ∗ W, we get then U ∗ PW U = Iq − F † F . Thus, in view of PW = QA,V , we obtain √ √ U ∗ GA,V U = (U ∗ AU )(U ∗ PW U )(U ∗ AU )∗ = R(Iq − F † F )R∗ ⎡ ⎤ E(Iq − F † F )E ∗ E(Iq − F † F )F ∗ ⎦. =⎣ F (Iq − F † F )E ∗ F (Iq − F † F )F ∗ Using (2.2), F (Iq − F † F ) = O, and F † F = (F † F )∗ , then (2.1) follows.



The following observation makes clear why in Definition 2.2 the terminology “Schur complement associated to A and V” was chosen. Remark 2.9. Assume q ≥ 2, let d ∈ Z1,q−1 , let A = [Ajk ]2j,k=1 be the block representation of a matrix A ∈ Cq×q with d × d block A11 , and ≥  Id  let V := R( O(q−d)×d ). In view of Lemma 2.8, then GA,V = diag(A11 −

A12 A†22 A21 , O(q−d)×(q−d) ).

The following result shows in combination with [14, Theorem 5.1] that the construction introduced in Definition 2.2 coincides with the matrix introduced by Ando [1, Formula (5.1.11)]. Proposition 2.10. Let A ∈ Cq×q and let V be a linear subspace of Cq . For ≥ q all x ∈ C , then x∗ GA,V x = min (x − y)∗ A(x − y). (2.3) y∈V ⊥

Proof. Set d := dim V. We consider an arbitrary x ∈ Cq . If d = 0, then Proposition 2.6(b) yields GA,V = Oq×q and, because of A ∈ Cq×q ≥ , therefore (2.3). If d = q, then V ⊥ = {O} and Proposition 2.6(c) yields GA,V = A, which implies (2.3). Now we consider the case 1 ≤ d ≤ q − 1. Let u1 , u2 , . . . , uq be an orthonormal basis of Cq such that u1 , u2 , . . . , ud is an orthonormal basis of V. Let U1 := [u1 , . . . , ud ] and let U2 := [ud+1 , . . . , uq ]. Then U := [U1 , U2 ] is unitary. Let B = [Bjk ]2j,k=1 be the block representation of B := U ∗ AU † with d × d block B11 . Setting S := B11 − B12 B22 B21 , from Lemma 2.8 we get U ∗ GA,V U = diag(S, O(q−d)×(q−d) ). Let f1 := U1∗ x and let f2 := U2∗ x. Then   f := U ∗ x admits the block representation f = ff12 . Thus, we get   x∗ GA,V x = f ∗ diag(S, O(q−d)×(q−d) ) f = f1∗ Sf1 . (2.4) We consider an arbitrary y ∈ V ⊥ . Setting g1 :=U1∗ y and g2 := U2∗ y, we see that g := U ∗ y admits the block representation g = gg12 and that g1 = Od×1 . Thus,   h := f − g can be represented via h = hh12 , where h1 := f1 and h2 := f2 − g2 . The matrices B and B22 are obviously non-negative Hermitian. Therefore, using a well-known factorization formula (see, e. g., [6, Lemmata 1.1.9

and 1.1.7]), Id Od×(q−d) ∗ we have B = R [diag(S, B22 )]R where R := B † B . It is easily I 22

21

q−d

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 1  † checked that Rh = r2f−g where r2 := B22 B21 f1 + f2 . Applying x − y = U h 2 and (2.4) we conclude then  ∗   f1 f1 (x − y)∗ A(x − y) = h∗ Bh = [diag(S, B22 )] r2 − g2 r2 − g2 = f1∗ Sf1 − (r2 − g2 )∗ B22 (r2 − g2 ) = x∗ GA,V x − (r2 − g2 )∗ B22 (r2 − g2 ). Consequently, (x − y)∗ A(x − y) ≥ x∗ GA,V x for all y ∈ V ⊥ with equality if and only if r2 − g2 ∈ N (B22 ). Obviously, the particular vector y := U2 r2 belongs to V ⊥ and we have g2 = r2 . Thus, y := U2 r2 fulfills B22 (r2 − g2 ) = O.  For two Hermitian q × q matrices A and B with A ≤ B, the (closed) matricial interval [[A, B]] := {X ∈ Cq×q H : A ≤ X ≤ B} is non-empty. Notation 2.11. If A ∈ Cq×q and V is a linear subspace of Cq , then let HA,V := ≥ {X ∈ [[Oq×q , A]] : R(X) ⊆ V}. Theorem 2.12 (cf. [14, Theorem 5.3]). If A ∈ Cq×q and V is a linear subspace ≥ of Cq , then GA,V ∈ HA,V and GA,V ≥ X for all X ∈ HA,V . Proof. For the convenience of the reader, we reproduce the proof given in [14, Theorem 5.3]: From parts (a) and (b) of Proposition 2.6 we infer GA,V ∈ HA,V . Now consider an arbitrary X ∈ HA,V . Furthermore, consider an arbitrary x ∈ Cq and an arbitrary y0 ∈ V ⊥ . Because of R(X) ⊆ V and X ∈ Cq×q ≥ , then y0 ∈ R(X)⊥ = N (X ∗ ) = N (X), implying (x − y0 )∗ X(x − y0 ) = x∗ Xx. Because of A ≥ X, moreover (x − y0 )∗ A(x − y0 ) ≥ (x − y0 )∗ X(x − y0 ). Taking additionally into account Proposition 2.10, we thus obtain (2.3) and, hence, x∗ GA,V x ≥ x∗ Xx. Consequently, GA,V ≥ X.  Remark 2.13. Let A, B ∈ Cq×q ≥ . Then R(B) ⊆ R(A) if and only if B ≤ γA for some γ ∈ (0, ∞). In this case, γ0 := sup{(x∗ Bx)(x∗ Ax)−1 : x ∈ Cq \ N (A)} fulfills γ0 ∈ [0, ∞) and B ≤ γ0 A. Lemma 2.14 (cf. [14, Equivalence (5.0.7)]). If A, B ∈ Cq×q ≥ , then R(A) ∩ R(B) = {Oq×1 } if and only if [[Oq×q , A]] ∩ [[Oq×q , B]] = {Oq×q }. Proof. Let A, B ∈ Cq×q ≥ . Then Oq×q ∈ [[Oq×q , A]] ∩ [[Oq×q , B]]. First assume R(A) ∩ R(B) = {Oq×1 }. We consider an arbitrary X ∈ [[Oq×q , A]] ∩ [[Oq×q , B]]. Then R(X) ⊆ R(A) ∩ R(B), by virtue of Remark 2.13. Consequently, X = Oq×q . Conversely, assume that R(A)∩R(B) = {Oq×1 }. Then P := PR(A)∩R(B) fulfills P ∈ Cq×q \ {Oq×q } and R(P ) = R(A) ∩ R(B). From Remark 2.13 we ≥ can conclude then the existence α, β ∈ (0, ∞) with P ≤ αA and P ≤ βB. Then γ := min{1/α, 1/β} fulfills γ ∈ (0, ∞) and γP ∈ ([[Oq×q , A]] ∩ [[Oq×q , B]]) \ {Oq×q }.  Theorem 2.15 (cf. [14, Theorem 5.7]). Let A ∈ Cq×q and let V be a linear ≥ subspace of Cq . Then: (a) R(A − GA,V ) ∩ V = {Oq×1 }.

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(b) Let X, Y ∈ Cq×q be such that X + Y = A. Then the following statements ≥ are equivalent: (i) R(X) ⊆ V and R(Y ) ∩ V = {Oq×1 }. (ii) X = GA,V and Y = A − GA,V . Proof. For the convenience of the reader, we reproduce the proof given in [14, Theorem 5.7]: First observe that Oq×q ≤ GA,V ≤ A and R(GA,V ) ⊆ V, by virtue of parts (a) and (b) of Proposition 2.6. (a) We have {A − GA,V , PV } ⊆ Cq×q ≥ . Hence, Oq×q ∈ [[Oq×q , A − GA,V ]] ∩ [[Oq×q , PV ]]. Consider an arbitrary X ∈ [[Oq×q , A − GA,V ]] ∩ [[Oq×q , PV ]]. Then X ∈ Cq×q and, moreover, X ≤ A − GA,V and X ≤ PV . In particular, ≥ R(X) ⊆ R(PV ) = V, by virtue of Remark 2.13. Taking additionally into account GA,V ∈ Cq×q and R(GA,V ) ⊆ V, we thus obtain GA,V + X ∈ HA,V . ≥ Hence, Theorem 2.12 yields Oq×q ≤ GA,V + X ≤ GA,V . This implies X = Oq×q . Consequently, [[Oq×q , A − GA,V ]] ∩ [[Oq×q , PV ]] = {Oq×q }. According to Lemma 2.14, then R(A − GA,V ) ∩ R(PV ) = {Oq×1 } follows. In view of R(PV ) = V, the proof of part (a) is complete. (b) Obviously Oq×q ≤ X ≤ A. First suppose (i). Then X ∈ HA,V . Theorem 2.12 yields then GA,V ≥ X. Taking into account A − GA,V ∈ Cq×q ≥ , thus Oq×q ≤ GA,V − X ≤ A − X. According to Remark 2.13 and Y = A − X, then R(GA,V − X) ⊆ R(Y ). Because of X ∈ Cq×q ≥ , we can furthermore conclude Oq×q ≤ GA,V −X ≤ GA,V . By virtue of Remark 2.13 and R(GA,V ) ⊆ V, then R(GA,V −X) ⊆ R(GA,V ) ⊆ V. Taking additionally into account (i), we thus obtain R(GA,V − X) ⊆ R(Y ) ∩ V = {Oq×1 }. Consequently, GA,V = X. Thus, (ii) holds true. If we conversely suppose (ii), then (i) follows from R(GA,V ) ⊆ V and (a). 

3. On a restricted extension problem for a finite Hankel non-negative definite extendable sequence This section is written against to the background of applying the results of Section 2 to the truncated matricial Hamburger moment problems formulated in the introduction for Ω = R. In the heart of our strategy lies the treatment of a special restricted extension problem for matrices. The complete answer to this problem is contained in Theorem 3.17, which is the central result of this section. Using [11, Lemma 3.2], we can conclude: q×q ≥ Remark 3.1. If κ ∈ N0 ∪ {∞} and if (sj )2κ for all j=0 ∈ Hq,2κ , then sj ∈ CH j ∈ Z0,2κ and s2k ∈ Cq×q for all k ∈ Z . 0,κ ≥

Given n ∈ N arbitrary rectangular complex matrices A1 , A2 , . . . , An , we write col (Aj )nj=1 = col(A1 , A2 , . . . , An ) (resp., row (Aj )nj=1 := [A1 , A2 , . . . , An ]) for the block column (resp., block row) build from the matrices A1 , A2 , . . . , An if their numbers of columns (resp., rows) are all equal.

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Notation 3.2. Let κ ∈ N0 ∪ {∞} and let (sj )κj=0 be a sequence of complex p × q matrices. m (a) Let y,m := col (sj )m j= and z,m := row (sj )j= for all , m ∈ N0 with  ≤ m ≤ κ. † (b) Let Θ0 := Op×q and let Θn := zn,2n−1 Hn−1 yn,2n−1 for all n ∈ N with 2n − 1 ≤ κ. (c) Let Ln := s2n − Θn for all n ∈ N0 with 2n ≤ κ. ≥ Remark 3.3. If κ ∈ N0 ∪ {∞} and (sj )2κ j=0 ∈ Hq,2κ , then [8, Remark 2.1(a)] q×q ≥ shows that (sj )2n for all n ∈ Z0,κ . j=0 ∈ Hq,2n and Ln ∈ C≥ ≥,e ≥ Remark 3.4. Remark 3.3 shows that Hq,2κ ⊆ Hq,2κ for all κ ∈ N0 ∪ {∞}. ≥,e Remark 3.5. If κ ∈ N0 ∪ {∞} and (sj )κj=0 ∈ Hq,κ , then, because of Req×q marks 3.3 and 3.1, we have Θn ∈ C≥ for all n ∈ Z0,κ .

Notation 3.6. Let n ∈ N and let (sj )2n−1 j=0 be a sequence of complex q × q ma2n−1 trices. Then let H≥ [(sj )j=0 ] be the set of all s2n ∈ Cq×q for which (sj )2n j=0 be≥,e ≥ 2n−1 longs to Hq,2n . Obviously, H≥ [(sj )2n−1 ] =

∅ if and only if (s ) ∈ H j j=0 q,2n−1 . j=0 Proposition 3.7 (cf. [8, Proposition 2.22(a)]). Let n ∈ N, let (sj )2n−1 ∈ j=0 ≥,e 2n−1 q×q Hq,2n−1 , and let s2n ∈ C . Then s2n ∈ H≥ [(sj )j=0 ] if and only if Ln ∈ Cq×q ≥ . Notation 3.8. Let n ∈ N, let (sj )2n−1 j=0 be a sequence of complex q × q matrices, and let Y ∈ Cq×q . Then denote by H≥ [(sj )2n−1 j=0 , Y ] the set of all X ∈ H q×q 2n−1 H≥ [(sj )j=0 ] satisfying Y − X ∈ C≥ . ≥ Lemma 3.9. Let κ ∈ N0 ∪ {∞} and let (sj )2κ j=0 ∈ Hq,2κ . Then Oq×q ≤ Θn ≤ s2n for all n ∈ Z0,κ .

Proof. Remark 3.1 shows that Oq×q ≤ Θ0 ≤ s0 . Now assume κ ≥ 1 and consider an arbitrary n ∈ Z1,κ . Remark 3.1 yields s2n ∈ Cq×q H . According ≥ 2n to Remark 3.3, furthermore Ln ∈ Cq×q and (s ) ∈ H j j=0 q,2n . In particular, ≥ ≥,e 2n−1 (sj )j=0 ∈ Hq,2n−1 . Thus, Remark 3.5 and Notation 3.2(c) yield Oq×q ≤ Θn ≤ s2n .  To indicate that a certain (block) matrix X is built from a sequence (sj )κj=0 , we sometimes write X (s) for X. ≥ Proposition 3.10. If n ∈ N and (sj )2n j=0 ∈ Hq,2n , then Oq×q ≤ Θn ≤ s2n and H≥ [(sj )2n−1 j=0 , s2n ] = [[Θn , s2n ]].

Proof. By virtue of Lemma 3.9, we have Oq×q ≤ Θn ≤ s2n . In particular, the matrices Θn and s2n are Hermitian . Let the sequence (tj )2n−1 j=0 be (t)

given by tj := sj for each j ∈ Z0,2n−1 . Obviously, then Θn = Θn and ≥,e 2n−1 q×q {(sj )2n−1 we have j=0 , (tj )j=0 } ⊆ Hq,2n−1 . For any t2n ∈ C (t) t2n − Θn = t2n − Θ(t) n = Ln .

(3.1)

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2n We first consider an arbitrary t2n ∈ H≥ [(sj )2n−1 j=0 , s2n ]. Then (tj )j=0 ∈ q×q q×q ≥ Hq,2n and s2n − t2n ∈ C≥ . In particular, t2n ∈ CH . Since Remark 3.3 (t)

shows that Ln is non-negative Hermitian, (3.1) yields Θn ≤ t2n ≤ s2n . Conversely, we consider now an arbitrary t2n ∈ [[Θn , s2n ]]. Then s2n − (t) q×q t2n ∈ Cq×q and t2n − Θn ∈ Cq×q ≥ ≥ . Thus, (3.1) yields Ln ∈ C≥ . According to ≥ 2n−1 Proposition 3.7, thus (tj )2n j=0 ∈ Hq,2n . Consequently, t2n ∈ H≥ [(sj )j=0 , s2n ].  [σ]

≥ Corollary 3.11. If n ∈ N0 and (sj )2n j=0 ∈ Hq,2n , then {s2n : σ ∈ Mq≥ [R; (sj )2n j=0 , ≤]} ⊆ [[Θn , s2n ]].

Proof. In view of Θ0 = Oq×q , the case n = 0 is obvious. If n ≥ 1, combine Theorem 1.3, Remark 3.4, and Proposition 3.10.  Notation 3.12. If n ∈ N and (sj )2n−1 j=0 is a sequence of complex q × q matrices. Then let H≥,e [(sj )2n−1 ] be the set of all complex q × q matrices s2n such that j=0 ≥,e 2n (sj )j=0 ∈ Hq,2n . From [8, Proposition 2.22(a), (b)] we know that H≥,e [(sj )2n−1 j=0 ] = ∅ if ≥,e and only if (sj )2n−1 belongs to H . q,2n−1 j=0 Proposition 3.13 (cf. [8, Proposition 2.22(b)]). Let n ∈ N, let (sj )2n−1 j=0 ∈ ≥,e 2n−1 q×q Hq,2n−1 , and let s2n ∈ C . Then s2n ∈ H≥,e [(sj )j=0 ] if and only if Ln ∈ Cq×q and R(Ln ) ⊆ R(Ln−1 ). ≥ Notation 3.14. Let n ∈ N, let (sj )2n−1 j=0 be a sequence of complex q × q matrices, q×q and let Y ∈ CH . Then denote by H≥,e [(sj )2n−1 j=0 , Y ] the set of all X ∈ q×q 2n−1 H≥,e [(sj )j=0 ] satisfying Y − X ∈ C≥ . Observe that the following construction is well defined, due to Remark 3.3 and Definition 2.2: ≥ Notation 3.15. Let κ ∈ N0 ∪ {∞} and let (sj )2κ j=0 ∈ Hq,2κ . Then let Ξ0 := s0 . If n ∈ Z1,κ , then let Ξn := Θn + GLn ,R(Ln−1 ) . ≥ Lemma 3.16. Let κ ∈ N0 ∪ {∞} and let (sj )2κ j=0 ∈ Hq,2κ . For all n ∈ Z0,κ , then Oq×q ≤ Θn ≤ Ξn ≤ s2n . (3.2)

Proof. According to Lemma 3.9, we have Oq×q ≤ Θn ≤ s2n for all n ∈ Z0,κ . In particular, the matrices Θn and s2n are Hermitian for all n ∈ Z0,κ . By virtue of Ξ0 = s0 , we get (3.2) for n = 0. Now assume κ ≥ 1 and n ∈ Z1,κ . According to Remark 3.3, we have Ln ∈ Cq×q ≥ . From Proposition 2.6(a) we obtain Oq×q ≤ GLn ,R(Ln−1 ) ≤ Ln , implying, by virtue of Notation 3.15 and Notation 3.2(c) then Oq×q ≤ Θn ≤ Ξn ≤ s2n .  ≥ Theorem 3.17. Let n ∈ N and let (sj )2n j=0 ∈ Hq,2n . Then:

(a) (3.2) and H≥,e [(sj )2n−1 j=0 , s2n ] = [[Θn , Ξn ]] hold true.

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≥,e (b) Ξn = s2n if and only if (sj )2n j=0 ∈ Hq,2n .

Proof. (a) From Lemma 3.16 we get (3.2). In particular, the matrices Θn , Ξn , and s2n are Hermitian. Remark 3.3 yields Ln ∈ Cq×q ≥ . With A := Ln and V := R(Ln−1 ) we have Ξn = Θn + GA,V and s2n = Θn + A. Let the sequence (t) (tj )2n−1 j=0 be given by tj := sj for all j ∈ Z0,2n−1 . Then Ln−1 = Ln−1 and (t)

(t)

2n−1 Θn = Θn . In particular, V = R(Ln−1 ). Furthermore, {(sj )2n−1 j=0 , (tj )j=0 } ⊆ ≥,e Hq,2n−1 . 2n First consider now an arbitrary t2n ∈ H≥,e [(sj )2n−1 j=0 , s2n ]. Then (tj )j=0 ∈ (t)

≥,e q×q Hq,2n and s2n − t2n ∈ Cq×q ≥ . In particular, t2n ∈ CH . Let X := Ln . q×q According to Proposition 3.13, then X ∈ C≥ and R(X) ⊆ V. In view of (t)

Notation 3.2(c), we have X = t2n −Θn = t2n −Θn . Hence, X ≤ s2n −Θn = A. Consequently, X ∈ HA,V . Theorem 2.12 yields then GA,V ≥ X. In view of Notation 3.2(c), thus Θn ≤ t2n ≤ Ξn . Conversely, let t2n ∈ [[Θn , Ξn ]]. Then X := t2n − Θn is Hermitian and fulfills Oq×q ≤ X ≤ GA,V . Hence, R(X) ⊆ R(GA,V ) and, according to Proposition 2.6, furthermore R(GA,V ) = R(A) ∩ V and GA,V ≤ A. Consequently, R(X) ⊆ V and Oq×q ≤ X ≤ A. In view of Notation 3.2(c), (t) (t) (t) furthermore Ln = t2n − Θn = t2n − Θn = X. Therefore, Ln ∈ Cq×q and ≥ (t)

(t)

≥,e R(Ln ) ⊆ R(Ln−1 ). According to Proposition 3.13, thus (tj )2n j=0 ∈ Hq,2n . Moreover, because of s2n − t2n = A − X, we get t2n ≤ s2n . Consequently, t2n ∈ H≥,e [(sj )2n−1 j=0 , s2n ]. ≥,e ≥ 2n−1 (b) In view of (sj )2n ∈ Hq,2n−1 and j=0 ∈ Hq,2n , we have (sj )j=0 q×q 2n−1 s2n ∈ H≥ [(sj )j=0 ]. Proposition 3.7 yields then Ln ∈ C≥ . By virtue of Notation 3.15 and Notation 3.2(c) we have Ξn = s2n if and only if Ln = GLn ,R(Ln−1 ) . According to Proposition 2.6(c), the latter is equiva≥,e q×q lent to R(Ln ) ⊆ R(Ln−1 ). In view of (sj )2n−1 j=0 ∈ Hq,2n−1 and Ln ∈ C≥ , Proposition 3.13 shows the equivalence of R(Ln ) ⊆ R(Ln−1 ) and s2n ∈ H≥,e [(sj )2n−1  j=0 ]. Hence, (b) follows.

Theorem 3.17 leads us now quickly in an alternative way to one of the main results of [8]. ≥ Theorem 3.18 (cf. [8, Theorem 7.8]). If n ∈ N0 and (sj )2n j=0 ∈ Hq,2n , then [σ]

{s2n : σ ∈ Mq≥ [R; (sj )2n j=0 , ≤]} = [[Θn , Ξn ]]. Proof. In view of Θ0 = Oq×q and Ξ0 = s0 , the case n = 0 is obvious. If n ≥ 1, combine Theorems 1.3 and 3.17. 

4. On equivalence classes of truncated matricial moment problems of type MP[R; (sj )2n j=0 , ≤] This section contains an aspect of our considerations in [8]. We are striving for a natural classification of the set of truncated matricial Hamburger moment

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problems of type “≤”. From Theorem 1.3 we see that these problems have a solution if and only if the sequence of data is Hankel non-negative definite. ≥ This leads us to the following relation in the set Hq,2n . ≥ 2n Notation 4.1. If n ∈ N0 and if {(sj )2n j=0 , (tj )j=0 } ⊆ Hq,2n , then we write q q 2n 2n 2n (sj )2n j=0 ∼R (tj )j=0 if M≥ [R; (sj )j=0 , ≤] = M≥ [R; (tj )j=0 , ≤].

Remark 4.2. Let n ∈ N0 . Then the relation ∼R is an equivalence relation on ≥ the set Hq,2n . ≥ 2n 2n Let n ∈ N0 . If (sj )2n j=0 ∈ Hq,2n , then let (sj )j=0 R := {(tj )j=0 ∈ ≥ ≥ 2n Hq,2n : (tj )2n j=0 ∼R (sj )j=0 }. Furthermore, if S is a subset of Hq,2n , then let 2n 2n S R := {(sj )j=0 R : (sj )j=0 ∈ S}. Looking back to Theorem 1.5 we see that each equivalence class contains ≥,e a unique representative belonging to Hq,2n . The considerations of Section 3 provide us now not only detailed insights into the explicit structure of this distinguished representative but even an alternative approach. The following notion is the central object of this section. ≥ Definition 4.3. If n ∈ N0 and (sj )2n sj )2n j=0 ∈ Hq,2n , then the sequence (˜ j=0 given by s˜2n := Ξn , where Ξn is given in Notation 3.15, and by s˜j := sj for all j ∈ Z0,2n−1 is called the Hankel non-negative definite extendable sequence equivalent to (sj )2n j=0 .

Now we derive the announced sharpened version of Theorem 1.5. In the following, we will use the notation given in Definition 4.3. ≥,e ≥ sj )2n Proposition 4.4. Let n ∈ N0 and let (sj )2n j=0 ∈ Hq,2n . Then (˜ j=0 ∈ Hq,2n 2n 2n and (˜ sj )j=0 ∼R (sj )j=0 . ≥,e ≥ Proof. In the case n = 0, we have s˜0 = Ξ0 = s0 and Hq,0 = Hq,0 . Now assume n ≥ 1. According to Theorem 3.17, we have s˜2n ≤ s2n and, in view of Definition 4.3, furthermore, s˜j = sj for all j ∈ Z0,2n−1 . From Remark 1.2, q 2n we get then Mq≥ [R; (˜ sj )2n j=0 , ≤] ⊆ M≥ [R; (sj )j=0 , ≤]. Conversely, we consider q 2n now an arbitrary σ ∈ M≥ [R; (sj )j=0 , ≤]. Let the sequence (uj )2n j=0 be given  j q×q by uj := R x σ(dx). Then s2n − u2n ∈ C≥ and uj = sj for all j ∈ Z0,2n−1 . ≥,e 2n Furthermore, σ ∈ Mq≥ [R; (uj )2n j=0 , =], implying (uj )j=0 ∈ Hq,2n , by virtue of Theorem 1.4. Consequently, u2n ∈ H≥,e [(sj )2n−1 j=0 , s2n ]. According to Theorem 3.17, thus u2n ∈ [[Θn , s˜2n ]]. In particular, s˜2n − u2n ∈ Cq×q ≥ . Since uj = sj = s˜j for all j ∈ Z0,2n−1 , then Remark 1.2 yields Mq≥ [R; (uj )2n j=0 , ≤] ⊆ Mq≥ [R; (˜ sj )2n , ≤]. Taking additionally into account Remark 1.1, we can conj=0 q q q 2n 2n sj )j=0 , ≤]. Hence, M≥ [R; (sj )j=0 , ≤] ⊆ M≥ [R; (˜ sj )2n clude σ ∈ M≥ [R; (˜ j=0 , ≤]. q q 2n 2n Consequently, M≥ [R; (˜ sj )j=0 , ≤] = M≥ [R; (sj )j=0 , ≤], implying (˜ sj )2n j=0 ∼R (sj )2n .  j=0

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≥,e ≥ 2n Proposition 4.5. Let n ∈ N0 and let (sj )2n j=0 ∈ Hq,2n . If (tj )j=0 ∈ Hq,2n 2n 2n satisfies (tj )2n j=0 ∼R (sj )j=0 , then (tj )j=0 coincides with the Hankel nonnegative definite extendable sequence equivalent to (sj )2n j=0 . ≥,e 2n 2n Proof. Let (tj )2n j=0 ∈ Hq,2n be such that (tj )j=0 ∼R (sj )j=0 . Observe that ≥ (tj )2n j=0 ∈ Hq,2n , by virtue of Remark 3.4. In view of Remark 4.2, we inq 2n fer from Proposition 4.4 then (tj )2n sj )2n j=0 ∼R (˜ j=0 , i. e., M≥ [R; (tj )j=0 , ≤] = q M≥ [R; (˜ sj )2n j=0 , ≤]. According to Theorem 1.4, we can chose a measure q q τ ∈ M≥ [R; (tj )2n [R; (t )2n , ≤]. j=0 , =]. By virtue of Remark 1.1, then τ ∈ M  ≥2n j j=0 q 2n Thus, τ ∈ M≥ [R; (˜ sj )j=0 , ≤]. Consequently, we have t2n = R x τ (dx) ≤ s˜2n  j ≥,e and tj = R x τ (dx) = s˜j for all j ∈ Z0,2n−1 . Since (˜ sj )2n j=0 belongs to Hq,2n , according to Proposition 4.4, we can conclude in a similar way s˜2n ≤ t2n . Hence, t2n = s˜2n follows. 

Now we state the main result of this section, which sharpens Theorem 1.5. ≥,e ≥ 2n Theorem 4.6. Let n ∈ N0 and let (sj )2n j=0 ∈ Hq,2n . Then (sj )j=0 R ∩ Hq,2n = {(˜ sj )2n j=0 }.



Proof. Combine Propositions 4.4 and 4.5.

Our next aim can be described as follows. Let n ∈ N and let (sj )2n j=0 ∈ Then an appropriate application of Theorem 2.15 leads us to the determination of all sequences (rj )2n j=0 which are contained in the equivalence 2n class (sj )j=0 R . ≥ Hq,2n .

≥ 2n Proposition 4.7. Let n ∈ N and let (sj )2n j=0 ∈ Hq,2n . Then (sj )j=0 R coincides 2n with the set of all sequences (rj )j=0 of complex q × q matrices fulfilling R(r2n − Ξn ) ∩ R(Ln−1 ) = {Oq×1 }, r2n − Ξn ∈ Cq×q ≥ , and rj = sj for all j ∈ Z0,2n−1 .

Proof. Let (tj )2n j=0 be the Hankel non-negative definite extendable sequence ≥,e 2n equivalent to (sj )2n j=0 . By virtue of Proposition 4.4, we have (tj )j=0 ∈ Hq,2n 2n and (tj )2n j=0 ∼R (sj )j=0 . According to Definition 4.3, furthermore t2n = Ξn (t)

(t)

and tj = sj for all j ∈ Z0,2n−1 . In particular, Ln−1 = Ln−1 and Θn = Θn . ≥ 2n 2n Consider now an arbitrary (rj )2n j=0 ∈ (sj )j=0 R , i. e., (rj )j=0 ∈ Hq,2n 2n 2n 2n 2n with (rj )j=0 ∼R (sj )j=0 . In particular, (tj )j=0 ∼R (rj )j=0 , by virtue of Remark 4.2. From Proposition 4.5 we can conclude then that (tj )2n j=0 coincides with the Hankel non-negative definite extendable sequence equivalent to (r) (rj )2n j=0 . In view of Definition 4.3, consequently t2n = Ξn and tj = rj for all (r)

j ∈ Z0,2n−1 . Hence, Ξn = Ξn and sj = rj for all j ∈ Z0,2n−1 . In particular, (r) (r) (r) Ln−1 = Ln−1 and Θn = Θn . From Remark 3.3 we know that Ln ∈ Cq×q ≥ . (r) (r) Setting A := Ln and V := R(Ln−1 ), we obtain, in view of Notation 3.15 and Notation 3.2(c), then (r) (r) r2n − Ξn = r2n − Ξ(r) n = r2n − Θn − GA,V = Ln − GA,V = A − GA,V .

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Because of R(Ln−1 ) = V and Theorem 2.15(a), thus R(r2n −Ξn )∩R(Ln−1 ) = {Oq×1 }. Furthermore, Proposition 2.6(a) yields r2n − Ξn ∈ Cq×q ≥ . Conversely, we consider now an arbitrary sequence (rj )2n j=0 of complex q × q matrices fulfilling R(r2n − Ξn ) ∩ R(Ln−1 ) = {Oq×1 }, r2n − Ξn ∈ Cq×q ≥ , (r)

(r)

and rj = sj for all j ∈ Z0,2n−1 . Then Ln−1 = Ln−1 and Θn = Θn . Because ≥,e ≥ 2n−1 of (sj )2n j=0 ∈ Hq,2n , we have (rj )j=0 ∈ Hq,2n−1 . From Lemma 3.16 we infer that Ξ∗n = Ξn with Θn ≤ Ξn . Consequently, r2n ∈ Cq×q and Θn ≤ Ξn ≤ r2n . H (r) (r) q×q Hence, Θn ≤ r2n , i. e., Ln ∈ C≥ . By virtue of Proposition 3.7, we obtain ≥ then (rj )2n rj )2n j=0 ∈ Hq,2n . Denote by (˜ j=0 the Hankel non-negative definite

(r) (r) extendable sequence equivalent to (rj )2n j=0 and let A := Ln and V := R(Ln−1 ). (r)

By Definition 4.3 and Notation 3.15, then r˜2n = Θn + GA,V and r˜j = rj for all j ∈ Z0,2n−1 . Consequently, r˜j = rj = sj = tj for all j ∈ Z0,2n−1 . Setting (r) X := t2n −Θn and Y := r2n −t2n , we have, by virtue of Notation 3.2(c), then (r) (r) (t) (t) X + Y = r2n − Θn = Ln = A and X = t2n − Θn = t2n − Θn = Ln and, furthermore, Y = r2n − Ξn . By assumption, then R(Y ) ∩ R(Ln−1 ) = {Oq×1 } ≥ 2n and Y ∈ Cq×q ≥ . From Remark 3.4 we infer (tj )j=0 ∈ Hq,2n . In particular, ≥,e 2n−1 2n−1 (tj )j=0 ∈ Hq,2n−1 and t2n ∈ H≥ [(sj )j=0 ]. Proposition 3.13 yields then (t)

(t)

(t)

Ln ∈ Cq×q and R(Ln ) ⊆ R(Ln−1 ). Hence, X ∈ Cq×q ≥ ≥ . Taking into account (t)

(r)

Ln−1 = Ln−1 = Ln−1 , we see furthermore R(X) ⊆ V and R(Y )∩V = {Oq×1 }. (r) From Theorem 2.15(b) we get then X = GA,V . Hence, r˜2n = Θn + GA,V = 2n 2n t2n . Thus, the sequences (˜ rj )j=0 and (tj )j=0 coincide. Using Proposition 4.4 2n 2n and Remark 4.2, we get then (rj )2n rj )2n j=0 ∼R (˜ j=0 ∼R (tj )j=0 ∼R (sj )j=0 . 2n Consequently, (rj )2n  j=0 ∈ (sj )j=0 R .

5. On truncated matricial [α, ∞)-Stieltjes moment problems In our following considerations, let α be a real number. In order to state a necessary and sufficient condition for the solvability of each of the moment m problems MP[[α, ∞); (sj )m j=0 , ≤] and MP[[α, ∞); (sj )j=0 , =], we have to recall the notion of two types of sequences of matrices. Let κ ∈ N ∪ {∞} and let (sj )κj=0 be a sequence of complex p × q matrices. For each n ∈ N0 with 2n + 1 ≤ κ, let the block Hankel matrix Kn be given by Kn := [sj+k+1 ]nj,k=0 . Furthermore, let the sequence (aj )κ−1 j=0 be given by

(s) aj := −αsj + sj+1 . For each matrix Xk = Xk built from the sequence (a) (sj )κj=0 , denote (if possible) by Xα,k := Xk the corresponding matrix built κ from the sequence (aj )κ−1 j=0 instead of (sj )j=0 . In particular, we have then Hα,n = −αHn + Kn for all n ∈ N0 with 2n + 1 ≤ κ. In the classical case α = 0, we see that aj = sj+1 for all j ∈ Z0,κ−1 . ≥ ≥ ≥ := Hq,0 Let Kq,0,α . For each n ∈ N, denote by Kq,2n,α the set of all 2n sequences (sj )j=0 of complex q × q matrices for which the block Hankel matrices Hn and Hα,n−1 are both non-negative Hermitian. For each n ∈ N0 ,

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≥ denote by Kq,2n+1,α the set of all sequences (sj )2n+1 j=0 of complex q × q matrices for which the block Hankel matrices Hn and Hα,n are both non-negative ≥ Hermitian. Furthermore, denote by Kq,∞,α the set of all sequences (sj )∞ j=0 m ≥ of complex q × q matrices satisfying (sj )j=0 ∈ Kq,m,α for all m ∈ N0 . The ≥ ≥ ≥ ≥ sequences belonging to Kq,0,α , Kq,2n,α , Kq,2n+1,α , or Kq,∞,α are said to be α-Stieltjes right-sided non-negative definite. Now we can characterize the situations that the mentioned problems have a solution:

Theorem 5.1 ([7, Theorem 1.4]). Let m ∈ N0 and let (sj )m j=0 be a sequence of complex q × q matrices. Then Mq≥ [[α, ∞); (sj )m , ≤]

= ∅ if and only j=0 ≥ if (sj )m ∈ K . q,m,α j=0 ≥,e For each m ∈ N0 , denote by Kq,m,α the set of all sequences (sj )m j=0 of complex q × q matrices for which there exists a complex q × q matrix ≥ sm+1 such that the sequence (sj )m+1 j=0 belongs to Kq,m+1,α . Furthermore, let ≥,e ≥ ≥,e ≥,e := Kq,∞,α Kq,∞,α . The sequences belonging to Kq,m,α or Kq,∞,α are said to be α-Stieltjes right-sided non-negative definite extendable. Now we characterize the solvability of Problem MP[[α, ∞); (sj )κj=0 , =].

Theorem 5.2 ([10, Theorem 1.6]). Let κ ∈ N0 ∪ {∞} and let (sj )κj=0 be a sequence of complex q × q matrices. Then Mq≥ [[α, ∞); (sj )κj=0 , =] = ∅ if and ≥,e only if (sj )κj=0 ∈ Kq,κ,α . The following result is the starting point of our subsequent considerations: ≥ Theorem 5.3 ([7, Theorem 5.2]). Let m ∈ N0 and let (sj )m j=0 ∈ Kq,m,α . Then m ≥,e there exists a unique sequence (ˆ sj )j=0 ∈ Kq,m,α such that q m Mq≥ [[α, ∞); (ˆ sj )m j=0 , ≤] = M≥ [[α, ∞); (sj )j=0 , ≤].

(5.1)

Theorem 5.3 was very essential for the considerations in [7]. The main goal of the rest of this paper is to derive this result by use of an appropriate application of the machinery developed in Section 2. This will lead us to an explicit formula for the desired sequence (ˆ sj )m j=0 . Following [7] we sketch now some essential features of the history of ≥,e Theorem 5.3. In the case α = 0 the existence of a sequence (ˆ sj )m j=0 ∈ Kq,m,α satisfying (5.1) was already formulated by V. A. Bolotnikov [3, Theorem 1.5, Lemma 1.6]. This result is true. However, it was shown in [7, Example 5.1] that the concrete sequence (ˆ sj )m j=0 constructed in [3, Lemmata 2.7 and 6.3] does not produce a moment problem equivalent to MP[[0, ∞); (sj )m j=0 , ≤].

6. On a restricted extension problem for a finite α-Stieltjes right-sided non-negative definite extendable sequence This section is written against to the background of applying the results of Section 2 to the truncated matricial [α, ∞)-Stieltjes moment problems formulated in the introduction for Ω = [α, ∞). In the heart of our strategy

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lies the treatment of a special restricted extension problem for matrices. The complete answer to this problem is contained in Theorem 6.19 which is the central result of this section. Using Remark 3.1 we can conclude: ≥ Remark 6.1. Let κ ∈ N0 ∪ {∞} and let (sj )κj=0 ∈ Kq,κ,α . Then sj ∈ Cq×q for H q×q all j ∈ Z0,κ and s2k ∈ C≥ for all k ∈ N0 with 2k ≤ κ. If κ ≥ 1, furthermore aj ∈ Cq×q for all j ∈ Z0,κ−1 and a2k ∈ Cq×q for all k ∈ N0 with 2k ≤ κ − 1. ≥ H

Definition 6.2 (cf. [10, Definition 4.2]). If (sj )κj=0 is a sequence of complex p × q matrices, then the sequence (Qj )κj=0 given by Q2k := s2k − Θk for all k ∈ N0 with 2k ≤ κ and by Q2k+1 := a2k −Θα,k for all k ∈ N0 with 2k +1 ≤ κ is called the right-sided α-Stieltjes parametrization of (sj )κj=0 . ≥ Remark 6.3. If κ ∈ N0 ∪ {∞} and (sj )κj=0 ∈ Kq,κ,α , then one can easily see q×q ≥ from [10, Theorem 4.12(b)] that (sj )m ∈ K for all q,m,α and Qm ∈ C≥ j=0 m ∈ Z0,κ .

Notation 6.4. If m ∈ N0 and (sj )m j=0 is a sequence of complex q × q matrices, m then denote by K≥,α [(sj )j=0 ] the set of all complex q × q matrices sm+1 such ≥ that (sj )m+1 j=0 ∈ Kq,m+1,α . Obviously, if m ∈ N0 and if (sj )m j=0 is a sequence of complex q × q mam ≥,e trices, then K≥,α [(sj )j=0 ] = ∅ if and only if (sj )m j=0 ∈ Kq,m,α ≥,e ≥ Remark 6.5. From Remark 6.3 one can easily see that Kq,κ,α ⊆ Kq,κ,α for all κ ∈ N0 ∪ {∞}.

Proposition 6.6 (cf. [10, Theorem 4.12(b), (c)]). Let m ∈ N0 , let (sj )m j=0 ∈ ≥,e Kq,m,α , and let sm+1 ∈ Cq×q . Then sm+1 ∈ K≥,α [(sj )m ] if and only if j=0 q×q Qm+1 ∈ C≥ . Notation 6.7. If κ ∈ N0 ∪ {∞} and (sj )κj=0 is a sequence of complex p × q matrices, then let a2k−1 := Θk for all k ∈ N0 with 2k − 1 ≤ κ and let a2k := αs2k + Θα,k for all k ∈ N0 with 2k ≤ κ. Remark 6.8. If κ ∈ N0 ∪ {∞} and (sj )κj=0 is a sequence of complex p × q matrices, then Definition 6.2 shows that Qj = sj − aj−1 for all j ∈ Z0,κ . ≥,e Remark 6.9. If κ ∈ N0 ∪ {∞} and (sj )κj=0 ∈ Kq,κ,α , then Remarks 6.3 and 6.1 show that a∗m = am for all m ∈ Z−1,κ .

Notation 6.10. Let m ∈ N0 , let (sj )m j=0 be a sequence of complex q × q maq×q trices, and let Y ∈ CH . Then denote by K≥,α [(sj )m j=0 , Y ] the set of all q×q X ∈ K≥,α [(sj )m ] satisfying Y − X ∈ C . j=0 ≥ ≥ Lemma 6.11. If κ ∈ N0 ∪ {∞} and (sj )κj=0 ∈ Kq,κ,α , then am−1 ≤ sm for all m ∈ Z0,κ .

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Proof. Obviously, a−1 = Oq×q ≤ s0 by virtue of Remark 6.1. Now assume κ ≥ 1 and we consider an arbitrary m ∈ Z1,κ . Remark 6.1 yields sm ∈ Cq×q H . q×q Remark 6.3 shows that Qm ∈ Cq×q . In view of Remark 6.8, then a ∈ C m−1 ≥ H and am−1 ≤ sm .  ≥ Theorem 6.12. If m ∈ N and (sj )m j=0 ∈ Kq,m,α , then am−1 ≤ sm and m−1 K≥,α [(sj )j=0 , sm ] = [[am−1 , sm ]].

Proof. By virtue of Lemma 6.11, we have {am−1 , sm } ⊆ Cq×q and am−1 ≤ sm . H m−1 Let the sequence (tj )j=0 be given by tj := sj for all j ∈ Z0,m−1 . Obviously, (t)

≥,e m−1 then am−1 = am−1 . Furthermore, {(sj )m−1 j=0 , (tj )j=0 } ⊆ Kq,m−1,α . m First consider now an arbitrary tm ∈ K≥,α [(sj )m−1 j=0 , sm ]. Then (tj )j=0 ∈ q×q q×q ≥ Kq,m,α and sm − tm ∈ C≥ . In particular, tm ∈ CH . In view of Remark 6.8, (t)

(t)

tm −am−1 = tm −am−1 = Qm , and Remark 6.3, we get then am−1 ≤ tm ≤ sm . Conversely, let tm ∈ [[am−1 , sm ]]. Then sm − tm ∈ Cq×q and, in view of ≥ (t)

(t)

(t)

Remark 6.8, furthermore Qm = tm − am−1 = tm − am−1 . In particular, Qm is ≥ non-negative Hermitian. According to Proposition 6.6, thus (tj )m j=0 ∈ Kq,m,α . m−1 Consequently, tm ∈ K≥,α [(sj )j=0 , sm ].  [σ]

≥ Corollary 6.13. If m ∈ N0 and (sj )m j=0 ∈ Kq,m,α , then {sm : σ ∈ q m M≥ [[α, ∞); (sj )j=0 , ≤]} ⊆ [[am−1 , sm ]].

Proof. In view of a−1 = Θ0 = Oq×q , the case m = 0 is obvious. If m ≥ 1, combine Theorem 5.1, Remark 6.5, and Theorem 6.12.  Notation 6.14. If m ∈ N0 and (sj )m j=0 is a sequence of complex q × q matrices, then denote by K≥,e,α [(sj )m ] the set of all complex q × q matrices sm+1 j=0 ≥,e m+1 such that (sj )j=0 ∈ Kq,m+1,α . The following result is essential for the realization of our concept of a new approach to Theorem 5.3. ≥,e Theorem 6.15 (cf. [10, Theorem 4.12(c)]). Let m ∈ N0 , let (sj )m j=0 ∈ Kq,m,α , q×q m and let sm+1 ∈ C . Then sm+1 ∈ K≥,e,α [(sj )j=0 ] if and only if Qm+1 ∈ Cq×q and R(Qm+1 ) ⊆ R(Qm ). ≥

Notation 6.16. Let m ∈ N0 , let (sj )m j=0 be a sequence of complex q × q maq×q trices, and let Y ∈ CH . Then denote by K≥,e,α [(sj )m j=0 , Y ] the set of all q×q X ∈ K≥,e,α [(sj )m ] satisfying Y − X ∈ C . j=0 ≥ Observe that the following construction is well defined due to Remark 6.3: ≥ Notation 6.17. If κ ∈ N0 ∪ {∞} and (sj )κj=0 ∈ Kq,κ,α , then let Γ0 := s0 and let Γm := am−1 + GQm ,R(Qm−1 ) for all m ∈ Z1,κ . ≥ . Then Γ∗m = Γm Lemma 6.18. Let κ ∈ N0 ∪ {∞} and let (sj )κj=0 ∈ Kq,κ,α and am−1 ≤ Γm ≤ sm for all m ∈ Z0,κ .

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Proof. Let m ∈ Z0,κ . From Lemma 6.11 we see that the matrices am−1 and sm are Hermitian and that am−1 ≤ sm . Remark 2.4 shows that Γ∗m = Γm . By virtue of Γ0 = s0 , we get in particular a−1 ≤ Γ0 ≤ s0 . Now assume κ ≥ 1 and consider an arbitrary m ∈ Z1,κ . According to Remark 6.3, we have Qm ∈ Cq×q ≥ . From Remark 2.4 and Proposition 2.6(a) we obtain Oq×q ≤ GQm ,R(Qm−1 ) ≤ Qm , implying, by virtue of Notation 6.17 and Remark 6.8, then am−1 ≤ Γm ≤ sm .  ≥ Theorem 6.19. Let m ∈ N and let (sj )m j=0 ∈ Kq,m,α . Then:

(a) am−1 ≤ Γm ≤ sm and K≥,e,α [(sj )m−1 j=0 , sm ] = [[am−1 , Γm ]]. ≥,e (b) Γm = sm if and only if (sj )m ∈ Kq,m,α . j=0 Proof. (a) From Lemma 6.18, we see that the matrices am−1 , Γm , and sm are Hermitian and that am−1 ≤ Γm ≤ sm . In view of Remark 6.3, we get Qm ∈ Cq×q ≥ . Setting A := Qm and V := R(Qm−1 ), we have Γm = am−1 +GA,V and sm = am−1 + A, by virtue of Remark 6.8. Let the sequence (tj )m−1 j=0 be

(t) (t) given by tj := sj . Then am−1 = am−1 and Qm−1 = Qm−1 . In particular, (t) ≥,e m−1 V = R(Qm−1 ). Furthermore, {(sj )m−1 j=0 , (tj )j=0 } ⊆ Kq,m−1,α . m−1 First consider an arbitrary tm ∈ K≥,e,α [(sj )j=0 , sm ]. Then (tj )m j=0 ∈

(t) q×q ≥,e Kq,m,α , tm ∈ Cq×q H , and sm − tm ∈ C≥ . Let X := Qm . According to Theorem 6.15, then X ∈ Cq×q and R(X) ⊆ V. In view of Remark 6.8, ≥ (t)

we have X = tm − am−1 = tm − am−1 and, hence, X ≤ sm − am−1 = A follows. Consequently, X ∈ HA,V . The application of Theorem 2.12 yields then GA,V ≥ X. In view of Notation 6.17, thus am−1 ≤ tm ≤ Γm . Conversely, let tm ∈ [[am−1 , Γm ]], i. e., tm ∈ Cq×q with am−1 ≤ tm ≤ Γm . H Then X := tm − am−1 is Hermitian and fulfills Oq×q ≤ X ≤ GA,V . By virtue of Remark 2.13, then R(X) ⊆ R(GA,V ) and, according to Proposition 2.6, furthermore R(GA,V ) = R(A) ∩ V and GA,V ≤ A. Consequently, R(X) ⊆ V (t) (t) and Oq×q ≤ X ≤ A. In view of Remark 6.8, furthermore Qm = tm − am−1 = (t) (t) (t) tm − am−1 = X. Therefore, Qm ∈ Cq×q and R(Qm ) ⊆ R(Qm−1 ). According ≥ m ≥,e to Theorem 6.15, thus tm ∈ K≥,e,α [(tj )m−1 j=0 ], i. e., (tj )j=0 ∈ Kq,m,α . Moreover, since Remark 6.8 shows that sm − tm = A − X we see that sm − tm ∈ Cq×q ≥ . m−1 Consequently, tm ∈ K≥,e,α [(sj )j=0 , sm ]. ≥,e m−1 ≥ (b) In view of (sj )m j=0 ∈ Kq,m,α , we have (sj )j=0 ∈ Kq,m−1,α and sm ∈ q×q K≥,α [(sj )m−1 j=0 ]. Proposition 6.6 yields then Qm ∈ C≥ . By virtue of Notation 6.17 and Remark 6.8 we have Γm = sm if and only if Qm = GQm ,R(Qm−1 ) . According to Proposition 2.6(c), the latter is equivalent to R(Qm ) ⊆ R(Qm−1 ). ≥,e q×q In view of (sj )m−1 j=0 ∈ Kq,m−1,α and Qm ∈ C≥ , Theorem 6.15 shows the equivalence of R(Qm ) ⊆ R(Qm−1 ) and sm ∈ K≥,e,α [(sj )m−1 j=0 ]. Hence, (b) follows.  Theorem 6.19 leads us now quickly in an alternative way to one of the main results of [7].

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≥ Theorem 6.20 (cf. [7, Theorem 5.4]). If m ∈ N0 and (sj )m j=0 ∈ Kq,m,α , then [σ]

{sm : σ ∈ Mq≥ [[α, ∞); (sj )m j=0 , ≤]} = [[am−1 , Γm ]]. Proof. In view of a−1 = Θ0 = Oq×q and Γ0 = s0 , the case m = 0 is obvious. If m ≥ 1, combine Theorems 5.1 and 6.19. 

7. On equivalence classes of truncated matricial moment problems of type MP[[α, ∞); (sj )m j=0 , ≤] This section contains an aspect of our considerations in [7]. We are striving for a natural classification of the set of truncated matricial [α, ∞)-Stieltjes moment problems of type “≤”. From Theorem 5.1 we see that these problems have a solution if and only if the sequence of data is α-Stieltjes right-sided ≥ non-negative definite. This leads us to the following relation in the set Kq,m,α . m m If m ∈ N0 and (sj )j=0 , (tj )j=0 are two sequences belonging to q ≥ m m Kq,m,α , then we write (sj )m j=0 ∼[α,∞) (tj )j=0 if M≥ [[α, ∞); (sj )j=0 , ≤] = q M≥ [[α, ∞); (tj )m j=0 , ≤] holds true. Remark 7.1. It is readily checked that ∼[α,∞) is an equivalence relation on ≥ the set Kq,m,α . ≥ m Notation 7.2. Let m ∈ N0 . If (sj )m j=0 ∈ Kq,m,α , then let (sj )j=0 [α,∞) := m ≥ m m {(tj )j=0 ∈ Kq,m,α : (tj )j=0 ∼[α,∞) (sj )j=0 }. Furthermore, if S is a subset of ≥ m Kq,m,α , then let S [α,∞) := {(sj )m j=0 [α,∞) : (sj )j=0 ∈ S}.

Regarding [7, Theorem 5.2] we see that each equivalence class contains ≥,e a unique representative belonging to Kq,m,α . The considerations of Section 6 provide us now not only detailed insights into the explicit structure of this distinguished representative but even an alternative approach. The following notion is the central object of this section. ≥ Definition 7.3. If m ∈ N0 and (sj )m sj )m j=0 ∈ Kq,m,α , then the sequence (ˆ j=0 given by sˆm := Γm , where Γm is given in Notation 6.17, and by sˆj := sj for all j ∈ Z0,m−1 is called the α-Stieltjes right-sided non-negative definite extendable sequence equivalent to (sj )m j=0 .

Now we derive a sharpened version of [7, Theorem 5.2]. ≥ Proposition 7.4. Let m ∈ N0 and let (sj )m j=0 ∈ Kq,m,α . Then the α-Stieltjes right-sided non-negative definite extendable sequence (ˆ sj )m j=0 equivalent to m ≥,e m m (sj )j=0 belongs to Kq,m,α and (ˆ sj )j=0 ∼[α,∞) (sj )j=0 . ≥ Proof. Let m ∈ N0 and let (sj )m j=0 ∈ Kq,m,α . First assume m = 0. We ≥,e ≥ have sˆ0 = Γ0 = s0 . In view of Kq,0,α = Kq,0,α , then the assertions follow. Now assume m ≥ 1. According to Theorem 6.19, we have sˆm ≤ sm and, in view of Definition 7.3, furthermore sˆj = sj for all j ∈ Z0,m−1 . From q m Remark 1.2, we get then Mq≥ [[α, ∞); (ˆ sj )m j=0 , ≤] ⊆ M≥ [[α, ∞); (sj )j=0 , ≤]. q m Conversely, let σ ∈ M≥ [[α, ∞); (sj )m j=0 , ≤]. Let the sequence (uj )j=0 be

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 given by uj := [α,∞) xj σ(dx). Then sm − um ∈ Cq×q and uj = sj for all ≥ q m j ∈ Z0,m−1 . Furthermore, σ ∈ M≥ [[α, ∞); (uj )j=0 , =], implying (uj )m j=0 ∈ ≥,e Kq,m,α , by virtue of Theorem 5.2. Consequently, um ∈ K≥,e,α [(sj )m−1 j=0 , sm ]. According to Theorem 6.19, thus um ∈ [[am−1 , sˆm ]]. In particular, sˆm − um ∈ Cq×q ˆj for all j ∈ Z0,m−1 , then Remark 1.2 yields ≥ . Since uj = sj = s q Mq≥ [[α, ∞); (uj )m , ≤] ⊆ M sj )m j=0 j=0 , ≤]. Taking additionally into ≥ [[α, ∞); (ˆ account Remark 1.1, we can conclude σ ∈ Mq≥ [[α, ∞); (ˆ sj )m j=0 , ≤]. Hence, we q q m m have shown M≥ [[α, ∞); (sj )j=0 , ≤] ⊆ M≥ [[α, ∞); (ˆ sj )j=0 , ≤]. Consequently, q m Mq≥ [[α, ∞); (ˆ sj )m sj )m j=0 , ≤] = M≥ [[α, ∞); (sj )j=0 , ≤], implying (ˆ j=0 ∼[α,∞) m (sj )j=0 .  ≥ m ≥,e Proposition 7.5. Let m ∈ N0 and let (sj )m j=0 ∈ Kq,m,α . If (tj )j=0 ∈ Kq,m,α m m m satisfies (tj )j=0 ∼[α,∞) (sj )j=0 , then (tj )j=0 coincides with the α-Stieltjes right-sided non-negative definite extendable sequence equivalent to (sj )m j=0 . ≥,e m m Proof. Let (tj )m j=0 ∈ Kq,m,α be such that (tj )j=0 ∼[α,∞) (sj )j=0 . Ob≥ serve that (tj )m j=0 ∈ Kq,m,α , by virtue of Remark 6.5. In view of Remark 7.1, we infer from Proposition 7.4 then (tj )m sj )m j=0 ∼[α,∞) (ˆ j=0 , i. e., q q m m M≥ [[α, ∞); (tj )j=0 , ≤] = M≥ [[α, ∞); (ˆ sj )j=0 , ≤]. According to Theorem 5.2, we can chose a measure τ ∈ Mq≥ [[α, ∞); (tj )m j=0 , =]. By virtue of Remark 1.1, then τ ∈ Mq≥ [[α, ∞); (tj )m , ≤]. Thus, τ ∈ Mq≥ [[α, ∞); (ˆ sj )m j=0 , ≤]. Conse j=0 m  j quently, we have tm = [α,∞) x τ (dx) ≤ sˆm and tj = [α,∞) x τ (dx) = sˆj for ≥,e all j ∈ Z0,m−1 . Since (ˆ sj )m j=0 belongs to Kq,m,α , according to Proposition 7.4, we can conclude in a similar way sˆm ≤ tm . Hence, tm = sˆm follows.  ≥ m ≥,e Theorem 7.6. If m ∈ N0 and (sj )m j=0 ∈ Kq,m,α , then (sj )j=0 [α,∞) ∩Kq,m,α = m {(ˆ sj )j=0 }.

Proof. Combine Propositions 7.4 and 7.5.



Our next aim can be described as follows. Let m ∈ N and let (sj )m j=0 ∈ ≥ Kq,m,α . Then an appropriate application of Theorem 2.15 leads us to the determination of all sequences (rj )m j=0 which are contained in the equivalence class (sj )m j=0 [α,∞) . ≥ m Proposition 7.7. Let m ∈ N and let (sj )m j=0 ∈ Kq,m,α . Then (sj )j=0 [α,∞) m coincides with the set of all sequences (rj )j=0 of complex q × q matrices fulfilling R(rm − Γm ) ∩ R(Qm−1 ) = {Oq×1 }, rm − Γm ∈ Cq×q ≥ , and rj = sj for all j ∈ Z0,m−1 .

Proof. Denote by (tj )m j=0 the α-Stieltjes right-sided non-negative definite extendable sequence equivalent to (sj )m j=0 . By virtue of Proposition 7.4, ≥,e m m ∈ K and (t we have then (tj )m j )j=0 ∼[α,∞) (sj )j=0 . According to q,m,α j=0 Definition 7.3, furthermore tm = Γm and tj = sj for all j ∈ Z0,m−1 . In (t) (t) particular, Qm−1 = Qm−1 and am−1 = am−1 .

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m m ≥ Consider an arbitrary (rj )m j=0 ∈ (sj )j=0 [α,∞) , i. e., (rj )j=0 ∈ Kq,m,α m m m m with (rj )j=0 ∼[α,∞) (sj )j=0 . In particular, (tj )j=0 ∼[α,∞) (rj )j=0 , by virtue of Remark 7.1. From Proposition 7.5 we can conclude then that (tj )m j=0 coincides with the α-Stieltjes right-sided non-negative definite extendable sequence (r) equivalent to (rj )m j=0 . In view of Definition 7.3, consequently tm = Γm and (r)

tj = rj for all j ∈ Z0,m−1 . Hence, Γm = Γm and sj = rj for all j ∈ Z0,m−1 (r) (r) follow. In particular, Qm−1 = Qm−1 and am−1 = am−1 . From Remark 6.3, (r) (r) (r) we see furthermore Qm ∈ Cq×q ≥ . Setting A := Qm and V := R(Qm−1 ) we obtain, in view of Notation 6.17 and Remark 6.8, then (r)

(r) rm − Γm = rm − Γ(r) m = rm − am−1 − GA,V = Qm − GA,V = A − GA,V .

Because of R(Qm−1 ) = V and Theorem 2.15(a), thus R(rm − Γm ) ∩ R(Qm−1 ) = {Oq×1 }. Furthermore, Proposition 2.6(a) yields rm − Γm ∈ Cq×q ≥ . Conversely, consider an arbitrary sequence (rj )m of complex q × q maj=0 q×q trices fulfilling R(rm − Γm ) ∩ R(Qm−1 ) = {Oq×1 }, rm − Γm ∈ C≥ , (r)

and rj = sj for all j ∈ Z0,m−1 . In particular, then Qm−1 = Qm−1 and (r) ≥,e m−1 ≥ am−1 = am−1 . Because of (sj )m j=0 ∈ Kq,m,α , we have (sj )j=0 ∈ Kq,m−1,α . ≥,e q×q Thus, (rj )m−1 j=0 ∈ Kq,m−1,α . From Lemma 6.18 we infer that Γm ∈ CH q×q and am−1 ≤ Γm . Consequently, we can conclude rm ∈ CH and am−1 ≤ (r) (r) Γm ≤ rm . Hence, am−1 ≤ rm , implying Qm ∈ Cq×q ≥ , by virtue of Remark 6.8. Using Proposition 6.6, we obtain then rm ∈ K≥,α [(rj )m−1 j=0 ], i. e., ≥ m (rj )m ∈ K . Denote by (ˆ r ) the α-Stieltjes right-sided non-negative j q,m,α j=0 j=0 (r) definite extendable sequence equivalent to (rj )m j=0 and let A := Qm and (r)

(r)

V := R(Qm−1 ). By Definition 7.3, then rˆm = am−1 + GA,V and rˆj = rj for all j ∈ Z0,m−1 . Consequently, rˆj = rj = sj = tj for all j ∈ Z0,m−1 . Setting (r) X := tm − am−1 and Y := rm − tm , we have, by virtue of Remark 6.8, then (r) (r) (t) (t) X + Y = rm − am−1 = Qm = A and X = tm − am−1 = tm − am−1 = Qm and furthermore Y = rm − Γm . In particular, R(Y ) ∩ R(Qm−1 ) = {Oq×1 } ≥ and Y ∈ Cq×q by assumption. From Remark 6.5 we infer (tj )m j=0 ∈ Kq,m,α ≥ ≥,e m−1 and, consequently, tm ∈ K≥,α [(tj )m−1 j=0 ]. In particular, (tj )j=0 ∈ Kq,m−1,α . (t)

(t)

(t)

Theorem 6.15 yields then Qm ∈ Cq×q and R(Qm ) ⊆ R(Qm−1 ). Hence, ≥ (t) Qm−1

(r) Qm−1 ,

X ∈ Cq×q = Qm−1 = we see furthermore ≥ . Taking into account R(X) ⊆ V and R(Y ) ∩ V = {Oq×1 }. From Theorem 2.15(b) we get then (r) X = GA,V . Hence, rˆm = am−1 + GA,V = tm follows. Thus, the sequences m (ˆ rj )m j=0 and (tj )j=0 coincide. Using Proposition 7.4 and Remark 7.1, we get m m m then (rj )j=0 ∼[α,∞) (ˆ rj )m j=0 ∼[α,∞) (tj )j=0 ∼[α,∞) (sj )j=0 . Consequently, m m (rj )j=0 ∈ (sj )j=0 [α,∞) .  Acknowledgment The authors thank the referee for careful reading and valuable hints.

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References [1] T. Ando, Schur complements and matrix inequalities, in: The Schur complement and its applications (F. Zhang, ed.), vol. 4 of Numerical Methods and Algorithms, ch. 5, pp. 137–162, Springer-Verlag, New York, 2005. [2] V.A. Bolotnikov, http://www.math.wm.edu/ vladi/dhmp.pdf, Revised version of [4]. [3] V.A. Bolotnikov, Degenerate Stieltjes moment problem and associated J-inner polynomials, Z. Anal. Anwendungen 14 no. 3 (1995), 441–468. [4] V.A. Bolotnikov, On degenerate Hamburger moment problem and extensions of nonnegative Hankel block matrices, Integral Equations Operator Theory 25 no. 3 (1996), 253–276. [5] G.N. Chen and Y.J. Hu, The truncated Hamburger matrix moment problems in the nondegenerate and degenerate cases, and matrix continued fractions, Linear Algebra Appl. 277 no. 1–3 (1998), 199–236. [6] V.K. Dubovoj, B. Fritzsche, and B. Kirstein, Matricial version of the classical Schur problem, vol. 129, Teubner-Texte zur Mathematik, B.G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992. With German, French and Russian summaries. [7] Yu.M. Dyukarev, B. Fritzsche, B. Kirstein, and C. Mädler, On truncated matricial Stieltjes type moment problems, Complex Anal. Oper. Theory 4 no. 4 (2010), 905–951. [8] Yu.M. Dyukarev, B. Fritzsche, B. Kirstein, C. Mädler, and H.C. Thiele, On distinguished solutions of truncated matricial Hamburger moment problems, Complex Anal. Oper. Theory 3 no. 4 (2009), 759–834. [9] B. Fritzsche, B. Kirstein, and C. Mädler, On Hankel nonnegative definite sequences, the canonical Hankel parametrization, and orthogonal matrix polynomials, Complex Anal. Oper. Theory 5 no. 2 (2011), 447–511. [10] B. Fritzsche, B. Kirstein,and C. Mädler, On a special parametrization of matricial α-Stieltjes one-sided non-negative definite sequences, in: Interpolation, Schur functions and moment problems. II, vol. 226 of Oper. Theory Adv. Appl., pp. 211–250, Birkhäuser/Springer Basel AG, Basel, 2012. [11] B. Fritzsche, B. Kirstein, C. Mädler, and T. Schwarz, On a Schur-type algorithm for sequences of complex p × q-matrices and its interrelations with the canonical Hankel parametrization, in: Interpolation, Schur functions and moment problems. II, vol. 226 of Oper. Theory Adv. Appl., pp. 117–192, Birkhäuser/Springer Basel AG, Basel, 2012. [12] I.S. Kats, On Hilbert spaces generated by monotone Hermitian matrix-functions, Harkov Gos. Univ. Uč. Zap. 34 = Zap. Mat. Otd. Fiz.-Mat. Fak. i Harkov. Mat. Obšč. (4) 22 (1951), 95–113, 1950. [13] M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure, Duke Math. J. 31 (1964), 291–298. [14] F. Zhang (ed.), The Schur complement and its applications, vol. 4 of Numerical Methods and Algorithms, Springer-Verlag, New York, 2005.

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Bernd Fritzsche, Bernd Kirstein and Conrad Mädler Mathematisches Institut, Universität Leipzig Augustusplatz 10/11, 04109 Leipzig Germany e-mail: [email protected] [email protected] [email protected]

A Toeplitz-like operator with rational symbol having poles on the unit circle I: Fredholm properties G.J. Groenewald, S. ter Horst, J. Jaftha and A.C.M. Ran Dedicated to our mentor and friend Rien Kaashoek on the occasion of his eightieth birthday.

Abstract. In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed. Mathematics Subject Classification (2010). Primary 47B35, 47A53; Secondary 47A68. Keywords. Toeplitz operators, unbounded operators, Fredholm properties.

1. Introduction The Toeplitz operator Tω on H p = H p (D), 1 < p < ∞, over the unit disc D with rational symbol ω having no poles on the unit circle T is the bounded linear operator defined by Tω : H p → H p ,

Tω f = Pωf (f ∈ H p ),

with P the Riesz projection of Lp = Lp (T) onto H p . This operator, and many of its variations, has been extensively studied in the literature, cf., [1, 3, 5, 16] and the references given there. This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 90670 and 93406). Part of the research was done during a sabbatical of the third author, in which time several research visits to VU Amsterdam and North-West University were made. Support from University of Cape Town and the Department of Mathematics, VU Amsterdam is gratefully acknowledged.

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_10

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In this paper the case where ω is allowed to have poles on the unit circle is considered. Let Rat denote the space of rational complex functions, and Rat0 the subspace of strictly proper rational complex functions. We will also need the subspaces Rat(T) and Rat0 (T) of Rat consisting of the rational functions in Rat with all poles on T and the strictly proper rational functions in Rat with all poles on T, respectively. For ω ∈ Rat, possibly having poles on T, we define a Toeplitz-like operator Tω (H p → H p ), for 1 < p < ∞, as follows: Dom(Tω ) = {g ∈ H p | ωg = f + ρ with f ∈ Lp, ρ ∈ Rat0 (T)} , Tω g = Pf. (1.1) Note that in case ω has no poles on T, then ω ∈ L∞ and the Toeplitz-like operator Tω defined above coincides with the classical Toeplitz operator Tω on H p . In general, for ω ∈ Rat, the operator Tω is a well-defined, closed, densely defined linear operator. By the Euclidean division algorithm, one easily verifies that all polynomials are contained in Dom(Tω ). Moreover, it can be verified that Dom(Tω ) is invariant under the forward shift operator Tz and that the following classical result holds: Tz−1 Tω Tz f = Tω f,

f ∈ Dom(Tω ).

These basic properties are derived in Section 2. This definition is somewhat different from earlier definitions of unbounded Toeplitz-like operators, as discussed in more detail in a separate part, later in this introduction. The fact that all polynomials are contained in Dom(Tω ), which is not the case in several of the definitions in earlier publications, enables us to determine a matrix representation with respect to the standard basis of H p and derive results on the convergence behaviour of the matrix entries; see Theorem 1.3 below. In this paper we are specifically interested in the Fredholm properties of Tω . For the case that ω has no poles on T, when Tω is a classical Toeplitz operator, the operator Tω is Fredholm if and only if ω has no zeroes on T, a result of R. Douglas; cf., Theorem 2.65 in [1] and Theorem 10 in [17]. This result remains true in case ω ∈ Rat. We use the standard definitions of Fredholmness and Fredholm index for an unbounded operator, as given in [4], Section IV.2: a closed linear operator which has a finite dimensional kernel and for which the range has a finite dimensional complement is called a Fredholm operator, and the index is defined by the difference of the dimension of the kernel and the dimension of the complement of the range. Note that a closed Fredholm operator in a Banach space necessarily has a closed range ([4], Corollary IV.1.13).The main results on unbounded Fredholm operators can be found in [4], Chapters IV and V. Theorem 1.1. Let ω ∈ Rat. Then Tω is Fredholm if and only if ω has no zeroes on T. Moreover, in that case the index of Tω is given by <  <  zeroes of ω in D multi. poles of ω in D multi. − . Index(Tω ) =  taken into account taken into account

Toeplitz-like operator with symbol having poles on the unit circle I 241 It should be noted that when we talk of poles and zeroes of ω these do not include the poles or zeroes at infinity. The result of Theorem 1.1 may also be expressed in terms of the winding number as follows: Index(Tω ) = − limr↓1 wind (ω|rT). In the case where ω is continuous on the unit circle and has no zeroes there, it is well-known that the index of the Fredholm operator Tω is given by the negative of the winding number of the curve ω(T) with respect to zero (see, e.g., [1], or [6], Theorem XVI.2.4). However, if ω has poles on the unit circle, the limit limr↓1 cannot be replaced by either limr→1 or limr↑1 in this formula. The proof of Theorem 1.1 is given in Section 5. It relies heavily on the following analogue of Wiener–Hopf factorization given in Lemma 5.1: for ω ∈ Rat we can write ω(z) = ω− (z)(z κ ω0 (z))ω+ (z) where κ is the difference between the number of zeroes of ω in D and the number of poles of ω in D, ω− has no poles or zeroes outside D, ω+ has no poles or zeroes inside D and ω0 has all its poles and zeroes on T. Based on the choice of the domain as in (1.1) it can then be shown that Tω = Tω− Tzκ ω0 Tω+ . This factorization eventually allows us to reduce the proof of Theorem 1.1 to the case where ω has only poles on T. It also allows us to characterize invertibility of Tω and to give a formula for the inverse of Tω in case it exists. If ω has only poles on T, i.e., ω ∈ Rat(T), then we have a more complete description of Tω in case it is a Fredholm operator. Here and in the remainder of the paper, we let P denote the space of complex polynomials in z, i.e., P = C[z], and Pn ⊂ P the subspace of polynomials of degree at most n. Theorem 1.2. Let ω ∈ Rat(T), say ω = s/q with s, q ∈ P co-prime. Then Tω is Fredholm if and only if ω has no zeroes on T. Assume s has no roots on T and factor s as s = s− s+ with s− and s+ having roots only inside and outside T, respectively. Then  Dom(Tω ) = qH p + Pdeg(q)−1 , Ran(Tω ) = sH p + P,  < (1.2) r0 | deg(r0 ) < deg(q) − deg(s− ) . Ker(Tω ) = s+  is the subspace of Pdeg(s)−1 given by Here P  = {r ∈ P | rq = r1 s + r2 for r1 , r2 ∈ Pdeg(q)−1 }. P Moreover, a complement of Ran(Tω ) in H p is given by Pdeg(s− )−deg(q)−1 (to be interpreted as {0} in case deg(s− ) ≤ deg(q)). In particular, Tω is either injective or surjective, and both injective and surjective if and only if deg(s− ) = deg(q), and the Fredholm index of Tω is given by  <  < poles of ω multi. zeroes of ω in D multi. Index(Tω ) =  − . taken into account taken into account The proof of Theorem 1.2 is given in Section 4. In case ω has zeroes on T, so that Tω is not Fredholm, part of the claims of Theorem 1.2 remain valid, after slight reformulation. For instance, the formula for Ker(Tω ) holds provided that the roots of s on T are included in s+ (see Lemma 4.1) and of

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the identities for Dom(Tω ) and Ran(Tω ) only one-sided inclusions are proved in case zeroes on T are present (see Proposition 4.5 for further detail). Since all polynomials are in the domain of Tω we can write down the matrix representation of Tω with respect to the standard basis of H p . It turns out that this matrix representation has the form of a Toeplitz matrix. In addition, there is an assertion on the growth of the coefficients in the upper triangular part of the matrix. Theorem 1.3. Let ω ∈ Rat possibly with poles on T. Then we can write the matrix representation [Tω ] of Tω with respect to the standard basis {z n }∞ n=0 of H p as ⎛ ⎞ a0 a−1 a−2 a−3 a−4 · · · ⎜ a1 a0 a−1 a−2 a−3 · · · ⎟ ⎜ ⎟ [Tω ] = ⎜ a2 a1 ⎟. a a a · · · 0 −1 −2 ⎝ ⎠ .. .. . . In addition, a−j = O(j M −1 ) for j ≥ 1 where M is the largest order of the 2 poles of ω in T and (aj )∞ j=0 ∈  . In subsequent papers we will discuss further properties of the class of Toeplitz operators given by (1.1). In particular, in [7] the spectral properties of such operators are discussed. In further subsequent papers a formula for the adjoint will be given, and several properties of the adjoint will be presented, and the matrix case will be discussed. Connections to earlier work on unbounded Toeplitz operators. Several authors have considered unbounded Toeplitz operators before. In the following we shall distinguish between several definitions by using superscripts. For ω : T → C the Toeplitz operator is defined usually by Tω f = Pωf with domain given by Dom(Tω ) = {f ∈ H p | ωf ∈ Lp }, see, e.g., [9]. Note that for ω rational with a pole on T this is a smaller set than in our definition (1.1). To distinguish between the two operators, we denote the classical operator by Tωcl . Hartman and Wintner have shown in [9] that the Toeplitz operator Tωcl is bounded if and only if its symbol is in L∞ , as was established earlier by Otto Toeplitz in the case of symmetric operators. Hartman, in [8], investigated unbounded Toeplitz operators on 2 (equivalently on H 2 ) with L2 -symbols. The operator in [8] is given by Dom(TωHr ) = {f ∈ H 2 | ωf = g1 + g2 ∈ L1 , g1 ∈ H 2 , g2 ∈ zH 1 },

TωHr f = g1 .

Observe the similarity with the definition (1.1). These operators are not bounded, unless ω ∈ L∞ . Note that the class of symbols discussed in the current paper does not fall into this category, as a rational function with a pole on T is not in L2 . The Toeplitz operator TωHr with L2 -symbol is necessarily densely defined as its domain would contain the polynomials. The operator TωHr is an adjoint operator and so it is closed. Necessary and sufficient conditions for invertibility have been established for the case where ω is real valued on T in terms of ω ± i. Of course, TωHr is symmetric in this case.

Toeplitz-like operator with symbol having poles on the unit circle I 243 In [14] Rovnyak considered a Toeplitz operator in H 2 with real-valued L2 symbol W such that log(W ) ∈ L1 . The operator is symmetric and densely defined via a construction of a resolvent involving a Reproducing Kernel Hilbert Space. This leads to a self-adjoint operator and clearly, the construction is very different from the approach taken in the current paper. Janas, in [11], considered Toeplitz operators in the Bargmann–Segal space B of Gaussian square integrable entire functions in Cn . The Bargmann– Segal space is also referred to as the Fock space or the Fisher space in the literature. The symbol of the operator is a measurable function. A Toeplitzlike operator, TωJ , is introduced as  J Dom(Tω ) = {f ∈ B | ωf = h+r, h ∈ B, rp dμ = 0, for all p ∈ P}, TωJ f = h. Again, observe the similarity with the definition (1.1). Consider also the operator Tωcl,B on the domain {f ∈ B | ωf ∈ L2 (μ)} with Tωcl,B f = Pωf . It is shown in [11] that 1. Tωcl,B ⊂ TωJ , i.e. TωJ is an extension of the Toeplitz operator Tωcl,B , 2. TωJ is closed, 3. Tωcl,B is closable whenever Dom(Tωcl,B ) is dense in B, 4. if P ⊂ Dom(Tωcl,B ) and ω is an entire function then Tωcl,B = TωJ . Let N + be the Smirnov class of holomorphic functions in D that consists of quotients of functions in H ∞ with the denominator an outer function. Note that a nonzero function ω ∈ N + can always be written uniquely as ω = ab where a and b are in the unit ball of H ∞ , a an outer function, a(0) > 0 and |a|2 + |b|2 = 1 on T, see [15, Proposition 3.1]. This is called the canonical representation of ω ∈ N + . For ω ∈ N + the Toeplitz operator TωHe on H 2 is defined by Helson in [10] and Sarason in [15] as the multiplication operator with domain Dom(TωHe ) = {f ∈ H 2 | ωf ∈ H 2 } and so this is a closed operator. Note that although a rational function with poles only on the unit circle is in the Smirnov class, the definition of the domain in (1.1) is different from the one used in [15]. In fact, for ω ∈ Rat(T), the operator (1.1) is an extension of the operator TωHe , i.e., TωHe ⊂ Tω . In [15] it is shown that if Dom(TωHe ) is dense in H 2 then ω ∈ N + . Also, if ω has canonical representation ω = ab then Dom(TωHe ) = aH 2 ; compare with (1.2) to see the difference. By extending our domain as in (1.1), our Toeplitzlike operator Tω is densely defined for any ω ∈ Rat, i.e., poles inside D are allowed. Helson in [10] studied TωHe in H 2 where ω ∈ N + with ω real valued on T. In this case TωHe is symmetric, and Helson showed among other things that TωHe has finite deficiency indices if and only if ω is a rational function. Overview. The paper consists of six sections, including the current introduction. In Section 2 we prove several basic results concerning the Toeplitz-like operator Tω . In the following section, Section 3, we look at division with remainder by a polynomial in H p . The results in this section form the basis

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of many of the proofs in subsequent sections, and may be of independent interest. Section 4 is devoted to the case where ω is in Rat(T). Here we prove Theorem 1.2. In Section 5 we prove the Fredholm result for general ω ∈ Rat, Theorem 1.1, and in Section 6 we prove Theorem 1.3 on the matrix representation of Tω . Finally, in Section 7 we discuss three examples that illustrate the main results of the paper. Notation. We shall use the following notation, most of which is standard: P is the space of polynomials (of any degree) in one variable; Pn is the subspace of polynomials of degree at most n. Throughout, K p denotes the standard complement of H p in Lp ; W + denotes the analytic Wiener algebra ∞ n on D, that is, power series f (z) = with absolutely summable n=0 fn z Taylor coefficients, hence analytic on D and continuous on D. In particular, P ⊂ W+ ⊂ Lp for each p.

2. Basic properties of Tω In this section we derive some basic properties of the Toeplitz-like operator Tω as defined in (1.1). The main result is the following proposition. Proposition 2.1. Let ω ∈ Rat, possibly having poles on T. Then Tω is a well-defined closed linear operator on H p with a dense domain which is invariant under the forward shift operator Tz . More specifically, the subspace P of polynomials is contained in Dom(Tω ). Moreover, Tz−1 Tω Tz f = Tω f for f ∈ Dom(Tω ). The proof of the well-definedness relies on the following well-known result. Lemma 2.2. Let ψ ∈ Rat have a pole on T. Then ψ ∈ Lp . In particular, the intersection of Rat0 (T) and Lp consists of the zero function only. −n Indeed, if ψ ∈ Rat has a pole at α > ∈ T of porder n, then |ψ(z)| ∼ |z−α| as z → α, and therefore the integral T |ψ(z)| dz diverges.

Proof of well-definedness claim of Proposition 2.1. Let g ∈ Dom(Tω ) and assume f1 , f2 ∈ Lp and ρ1 , ρ2 ∈ Rat0 (T) such that f1 +ρ1 = ωg = f2 +ρ2 . Then f1 −f2 = ρ2 −ρ1 ∈ Lp ∩Rat0 (T). By Lemma 2.2 we have f1 −f2 = ρ2 −ρ1 = 0, i.e., f1 = f2 and ρ1 = ρ2 . Hence f and ρ in the definition of Dom(Tω ) are uniquely determined. From this and the definition of Tω it is clear that Tω is a well-defined linear operator.  In order to show that Tω is a closed operator, we need the following alternative formula for Dom(Tω ) for the case where ω ∈ Rat(T). Lemma 2.3. Let ω ∈ Rat(T), say ω = s/q with s, q ∈ P co-prime. Then Dom(Tω ) = {g ∈ H p : ωg = h+r/q, h ∈ H p , r ∈ Pdeg(q)−1 }, Tω g = h. (2.1) Moreover, Dom(Tω ) is invariant under the forward shift operator Tz and Tz−1 Tω Tz f = Tω f,

f ∈ Dom(Tω ).

Toeplitz-like operator with symbol having poles on the unit circle I 245 Proof. Assume g ∈ H p with ωg = h + r/q, where h ∈ H p and r ∈ Pdeg(q)−1 . Since H p ⊂ Lp and r/q ∈ Rat0 (T), clearly g ∈ Dom(Tω ) and Tω g = Ph = h. Thus it remains to prove the reverse implication. Assume g ∈ Dom(Tω ), say ωg = f + ρ, where f ∈ Lp and ρ ∈ Rat0 (T). Since ρ ∈ Rat0 (T), we can write qρ as qρ = r0 + ρ0 with r0 ∈ Pdeg(q)−1 and ρ0 ∈ Rat0 (T). Then sg = qωg = qf + qρ = qf + r0 + ρ0 ,

i.e.,

ρ0 = sg − qf − r0 ∈ Lp .

By Lemma 2.2 we find that ρ0 ≡ 0. Thus sg = qf + r0 . Next write f = h + k with h ∈ H p and k ∈ K p . Then qk has the form qk = r1 + k1 with r1 ∈ Pdeg(q)−1 and k1 ∈ K p . Thus sg = qh + qk + r0 = qh + k1 + r1 + r0 ,

i.e.,

k1 = sg − qh − r1 − r0 ∈ H p .

Since also k1 ∈ K p , this shows that k1 ≡ 0, and we find that sg = qh + r with r = r0 + r1 ∈ Pdeg(q)−1 . Dividing by q gives ωg = h + r/q with h ∈ H p as claimed. Finally, we prove that Dom(Tω ) is invariant under Tz . Let f ∈ Dom(Tω ), say sf = qh + r with h ∈ H p and r ∈ Pdeg(q)−1 . Then szf = qzh + zr. Now write zr = cq + r0 with c ∈ C and r0 ∈ Pdeg(q)−1 . Then szf = q(zh + c) + r0 is in qH p + Pdeg(q)−1 . Thus zf ∈ Dom(Tω ), and Tω Tz f = zh + c. Hence Tz−1 Tω Tz f = h = Tω f as claimed.  Lemma 2.4. Let ω ∈ Rat. Then ω = ω0 + ω1 with ω0 ∈ Rat0 (T) and ω1 ∈ Rat with no poles on T. Moreover, ω0 and ω1 are uniquely determined by ω and the poles of ω0 and ω1 correspond to the poles of ω on and off T, respectively. Proof. The existence of the decomposition follows from the partial fraction decomposition of ω into the sum of a polynomial and elementary fractions of the form c/(z − zk )n . To obtain the uniqueness, split ω1 into the sum of a strictly proper rational function ν1 and a polynomial p1 . Assume also ω = ω0 + ν1 + p1 with ω0 in Rat0 (T), ν1 ∈ Rat0 with no poles on T and p1 a polynomial. Then (ω0 − ω0 ) + (ν1 − ν1 ) = p1 − p1 is in Rat0 ∩ P, and hence is zero. So p1 = p1 . Then ω0 − ω0 = ν1 − ν1 is in Rat0 and has no poles on C, and hence it is the zero function.  Proof of closedness claim of Proposition 2.1. By Lemma 2.4, ω ∈ Rat can be written as ω = ω0 + ω1 with ω0 ∈ Rat0 (T) and ω1 ∈ Rat with no poles on T, hence ω1 ∈ L∞ . Then Tω = Tω0 + Tω1 and Tω1 is bounded on H p . It follows that Tω is closed if and only if Tω0 is closed. Hence, without loss of generality we may assume ω ∈ Rat0 (T), which we will do in the remainder of the proof. Say ω = s/q with s, q ∈ P co-prime, q having roots only on T and deg(s) < deg(q). Let g1 , g2 , . . . be a sequence in Dom(Tω ) such that in H p we have gn → g ∈ H p and Tω gn → h ∈ H p as n → ∞. (2.2) We have to prove that g ∈ Dom(Tω ) and Tω g = h. Applying Lemma 2.3 above, we know that ωgn = hn + rn /q with hn ∈ H p and rn ∈ Pdeg(q)−1 .

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Moreover hn = Tω gn → h. Using (2.2) it follows that rn = sgn − qhn → sg − qh =: r

as n → ∞, with convergence in H p .

Since deg(rn ) < deg(q) for each n, it follows that r = limn→∞ rn is also a polynomial with deg(r) < deg(q). Thus r/q ∈ Rat0 (T), and r = sg − qh implies that ωg = h + r/q. Thus g ∈ Dom(Tω ) and Tω g = h. We conclude that Tω is closed.  Proof of Proposition 2.1. In the preceding two parts of the proof we showed all claims except that Dom(Tω ) contains P and is invariant under Tz . Again write ω as ω = ω0 + ω1 with ω0 ∈ Rat0 (T) and ω1 ∈ Rat with no poles on T. Let r ∈ P. Then ωr = ω0 r+ω1 r. We have ω1 r ∈ Rat with no poles on T, hence ω1 r ∈ Lp . By Euclidean division, ω0 r = ψ + r0 with ψ ∈ Rat0 (T) (having the same denominator as ω0 ) and r0 ∈ P ⊂ Lp . Hence ωr ∈ Lp + Rat0 (T), so that r ∈ Dom(Tω ). This shows P ⊂ Dom(Tω ). Finally, we have Dom(Tω ) = Dom(Tω0 ) and it follows by the last claim of Lemma 2.3 that Dom(Tω0 ) is invariant under Tz . 

3. Intermezzo: Division with remainder by a polynomial in H p Let s ∈ P, s ≡ 0. The Euclidean division algorithm says that for any v ∈ P there exist unique u, r ∈ P with v = us + r and deg(r) < deg(s). If deg(v) ≥ deg(s), then deg(v) = deg(s) + deg(u). We can reformulate this as: ˙ deg(s)−1 and Pn = sPn−deg(s) +P ˙ deg(s)−1 , n ≥ deg(s), P = sP +P ˙ indicating direct sum. What happens when P is replaced with a class with + of analytic functions, say by H p , 1 < p < ∞? That is, for s ∈ P, s ≡ 0, when do we have H p = sH p + Pdeg(s)−1 ? (3.1) p ˙ deg(s)−1 ⊂ sH p + Pdeg(s)−1 . Hence Since P ⊂ H , we know that P = sP +P sH p + Pdeg(s)−1 contains a dense (non-closed) subspace of H p . Thus question (3.1) is equivalent to asking whether sH p + Pdeg(s)−1 is closed. The following theorem provides a full answer to the above question. Theorem 3.1. Let s ∈ P, s ≡ 0. Then H p = sH p + Pdeg(s)−1 if and only if s has no roots on the unit circle T. Another question is, even if s has no roots on T, whether sH p +Pdeg(s)−1 is a direct sum. This does not have to be the case. In fact, if s has only roots outside T, then 1/s ∈ H ∞ and sH p = H p , so that sH p + Pdeg(s)−1 is not a direct sum, unless if s is constant. Clearly, a similar phenomenon occurs if only part of the roots of s are outside T. In case all roots of s are inside T, then the sum is a direct sum. Proposition 3.2. Let s ∈ P, s ≡ 0 and having no roots on T. Write s = s− s+ with s− , s+ ∈ P having roots inside and outside T, respectively. Then H p = sH p + Pdeg(s− )−1 is a direct sum decomposition of H p . In particular, sH p + Pdeg(s)−1 is a direct sum if and only if s has all its roots inside T.

Toeplitz-like operator with symbol having poles on the unit circle I 247 We also consider the question whether there are functions in H p that are not in sH p + Pdeg(s)−1 and that can be divided by another polynomial q. This turns out to be the case precisely when s has a root on T which is not a root of q. Theorem 3.3. Let s, q ∈ P, s, q ≡ 0. Then there exists a f ∈ qH p which is not in sH p + Pdeg(s)−1 if and only if s has a root on T which is not a root of q. In order to prove the above results we first prove a few lemmas. Lemma 3.4. Let s ∈ P and α ∈ C a root of s. Then sH p + Pdeg(s)−1 ⊂ (z − α)H p + C. Proof. Since s(α) = 0, we have s(z) = (z−α)s0 (z) for some s0 ∈ P, deg(s0 ) = deg(s) − 1. Let f = sg + r ∈ sH p + Pdeg(s)−1 . Then r(z) = (z − α)r0 (z) + c for a r0 ∈ P and c ∈ C. This yields f (z) = s(z)g(z) + r(z) = (z − α)(s0 (z)g(z) + r0 (z)) + c ∈ (z − α)H p + C.  Lemma 3.5. Let α ∈ T. Then there exists a f ∈ W+ such that f ∈ (z − α)H p + C. Proof. By rotational symmetry we may assume ∞ without loss of generality that α = 1. Let hn ↓ 0 such that h(z) = n=0 hn z n is analytic on D but h ∈ H p . Define f0 , f1 , . . . recursively by f0 = −h0 ,

fn+1 = (hn − hn+1 ), n ≥ 0. N Then f (z) = (z − 1)h(z) ∞and k=0 |fk | = 2h0 − hN → 2h0 . Hence the Taylor coefficients of f (z) = k=0 fk z k are absolutely summable and thus f ∈ W+ . Now assume f ∈ (z − 1)H p + C, say f = (z − 1)g + c for g ∈ H p and c ∈ C. Then h = g + c/(z − 1). Since the Taylor coefficients of c/(z − 1) have to go to zero, we obtain c = 0 and h = g, which contradicts the assumption h∈ / H p.  Proof of Theorem 3.1. Assume s has no roots on T. Since s ∈ P ⊂ H ∞ , we know from Theorem 8 of [17] that the range of the multiplication operator of s on H p is closed (i.e., sH p closed in H p ) if and only if |s| is bounded away from zero on T. Since s is a polynomial, the latter is equivalent to s having no roots on T. Hence sH p is closed. Since Pdeg(s)−1 is a finite-dimensional subspace of H p , and thus closed, we obtain that sH p + Pdeg(s)−1 is closed [2, Chapter 3, Proposition 4.3]. Also, sH p + Pdeg(s)−1 contains the dense subspace P of H p , therefore sH p + Pdeg(s)−1 = H p . Conversely, assume s has a root α ∈ T. Then by Lemmas 3.4 and 3.5 we know sH p + Pdeg(s)−1 ⊂ (z − α)H p + C = H p .  Proof of Proposition 3.2. Assume s ∈ P has no roots on T. Write s = s− s+ with s− , s+ ∈ P, s− having only roots inside T and s+ having only roots outside T. Assume s has roots outside T, i.e., deg(s+ ) > 0. Then 1/s+ is

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in H ∞ and s+ H p = H p and hence sH p = s− H p . Using Theorem 3.1, this implies that H p = s− H p + Pdeg(s− )−1 = sH p + Pdeg(s− )−1 . Next we show that sH p + Pdeg(s− )−1 is a direct sum. Let f = sh1 + r1 = sh2 + r2 ∈ sH p + Pdeg(s− )−1 with h1 , h2 ∈ H p , r1 , r2 ∈ Pdeg(s− )−1 . Then r1 − r2 = s(h2 − h1 ). Clearly, each root α of s− with multiplicity n, is also a root of s with multiplicity n. Evaluate both sides of r1 − r2 = s(h2 − h1 ) at α, possible since α ∈ D, as well as the identities obtained by taking derivatives on dm both sides up to order n−1, this yields dz m (r1 −r2 )(α) = 0 for m = 0 . . . n−1. Since deg(r1 − r2 ) < deg(s− ), this can only occur when r1 − r2 ≡ 0, i.e., r1 = r2 . We thus arrive at s(h2 − h1 ) ≡ 0. Since s has no roots on T, we have 1/s ∈ L∞ so that h2 − h1 = s−1 s(h2 − h1 ) ≡ 0 as a function in Lp . Hence h1 = h2 in Lp , but then also h1 = h2 in H p . Hence we have shown sH p + Pdeg(s− )−1 is a direct sum. In case s has all its roots inside T, we have s = s− and thus Pdeg(s)−1 = Pdeg(s− )−1 so that sH p + Pdeg(s)−1 is a direct sum. Conversely, if s has a root outside T, we have deg(s− ) < deg(s) and the identity sH p + Pdeg(s)−1 = sH p + Pdeg(s− )−1 shows that any r ∈ Pdeg(s)−1 with deg(r) ≥ deg(s− ) can be written as r = 0 + r ∈ sH p + Pdeg(s)−1 and as r = sh + r ∈ sH p + Pdeg(s)−1 with deg(r ) < deg(s− ) and h ∈ H p , h ≡ 0. Hence sH p + Pdeg(s)−1 is not a direct sum.  Proof of Theorem 3.3. Assume all roots of s on T are also roots of q. Let f = q f ∈ qH p . Factor s = s+ s0 s− as before. Then q = s0 qˆ for some qˆ ∈ P. From Theorem 3.1 we know that s− s+ H p + Pdeg(s− s+ )−1 = H p . Hence qˆf = s− s+ f + r with f ∈ H p and r ∈ P with deg(r) < deg(s− s+ ). Thus f = q f = s0 qˆf = sf + s0 r ∈ sH p + Pdeg(s)−1 , where we used deg(s0 r) = deg(s0 ) + deg(r) < deg(s0 ) + deg(s− s+ ) = deg(s). Hence qH p ⊂ sH p + Pdeg(s)−1 . Conversely, assume α ∈ T such that s(α) = 0 and q(α) = 0. By Lemma 3.5 there exists a f ∈ W+ ⊂ H p which is not in (z − α)H p + C, and hence not in sH p + Pdeg(s)−1 , by Lemma 3.4. Now set f = q f ∈ qH p . We have q(z) = (z − α)q1 (z) + c1 for a q1 ∈ P and c1 = q(α) = 0. Assume f ∈ (z − α)H p + C, say f (z) = (z − α)g(z) + c for a g ∈ H p and c ∈ C. Then ((z − α)q1 (z) + c1 )f(z) = q(z)f(z) = f (z) = (z − α)g(z) + c. Hence f(z) = (z − α)(g(z) − q1 (z)f(z))/c1 + c/c1 , z ∈ D, which shows f ∈ (z−α)H p +C, in contradiction with our assumption. Hence f ∈ (z−α)H p +C. This implies, once more by Lemma 3.4, that there exists a f ∈ qH p which is not in sH p + Pdeg(s)−1 .  The following lemma will be useful in the sequel. Lemma 3.6. Let q, s+ ∈ P, q, s+ ≡ 0 be co-prime with s+ having roots only p p outside T. Then s−1 + (qH + Pdeg(q)−1 ) = qH + Pdeg(q)−1 .

Toeplitz-like operator with symbol having poles on the unit circle I 249 p Proof. Set R := s−1 + (qH + Pdeg(q)−1 ). Since s+ has only roots outside T, we −1 −1 p ∞ have s+ ∈ H and s+ H = H p . Thus −1 p p R = s−1 + (qH + Pdeg(q)−1 ) = qH + s+ Pdeg(q)−1 .

This implies qH p ⊂ R. Next we show Pdeg(q)−1 ⊂ R. Let r ∈ Pdeg(q)−1 . Since P ⊂ qH p + Pdeg(q)−1 , we have rs+ ∈ qH p + Pdeg(q)−1 and thus r = p s−1 + (rs+ ) ∈ R. Hence qH + Pdeg(q)−1 ⊂ R. It remains to prove R ⊂ qH p + Pdeg(q)−1 . Let g = s−1 + (qh + r) ∈ R with h ∈ H p and r ∈ Pdeg(q)−1 . Since q and s+ have no common roots, there exist polynomials a, b ∈ P with qa + s+ b ≡ 1 and deg(a) < deg(s+ ), deg(b) < deg(q). Since rb ∈ P ⊂ qH p + Pdeg(q)−1 , we have −1 −1 p s−1 + r = s+ r(qa + s+ b) = qs+ ra + rb ∈ qH + Pdeg(q)−1 . p p Also qs−1 + h ∈ qH , so we have g ∈ qH + Pdeg(q)−1 . This shows that R ⊂ p qH + Pdeg(q)−1 and completes the proof. 

Remark 3.7. For what other Banach spaces X of analytic functions on D do the above results hold? Note that the following properties of X = H p are used: (1) (2) (3) (4) (5)

P ⊂ W+ ⊂ X, and P is dense in X; W+ X  ⊂ X; ∞ if g = n=0 gn z n ∈ X then gn → 0; if g ∈ X and α ∈ T, then g(z/α) ∈ X as well; if s ∈ P has no roots on T, then sX is closed in X.

To see item 3 for X = H p : note that by H¨ older’s inequality H p ⊂ H 1 , and for p = 1 this follows from the Riemann–Lebesgue Lemma ([12, Theorem I.2.8]), actually a sharper statement can be made in that case by a theorem of Hardy,  see [12, Theorem III 3.16], which states that if f ∈ H 1 , then |fn |n−1 < ∞. Other than X = H p , 1 < p < ∞, the spaces of analytic functions Ap on D with Taylor coefficients p-summable, cf., [13] and reference ([1-5]) given there, also have these properties. For a function f ∈ Ap the norm f Ap is defined as the lp -norm of the sequence (f)k of Taylor coefficients of f . Properties (1), (3) and (4) above are straightforward, property (2) is the fact that a function in the Wiener algebra is an lp multiplier (see, e.g., [13]). It remains to prove property (5). Let s be a polynomial with no roots on T, and let (fn ) be a sequence of functions in Ap such that the sequence (sfn ) converges to g in Ap . We have to show the existence of an f ∈ Ap such that g = sf . Note that fn and g are analytic functions, and convergence of (sfn ) to g in Ap means / that sf lp → 0. Consider the Toeplitz operator Ts : lp → lp . Then n − g . / lp → 0. Since s has no roots on T the Toeplitz sfn = Ts fn . So Ts f. n −g operator Ts is Fredholm and has closed range, and since s is a polynomial Ts is injective. Thus there is a unique f ∈ lp such that Ts f = g. Now define (at ∞ . = g, so at least least formally) the function f (z) = k=0 (f)k z k . Then sf

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formally s(z)f (z) = g(z). It remains to show that f is analytic on D. To see this, consider z = r with 0 < r < 1. Then by H¨older’s inequality "∞ #1/q  1/q ∞   1 k kq    p p |(f )k |r ≤ f l r = f l , 1 − rq k=0

k=0

∞ showing that the series f (z) = k=0 (f)k z k is absolutely convergent on D. Since f (z) = g(z) s(z) is the quotient of an analytic function and a polynomial it can only have finitely many poles on D, and since the series for f (z) converges for every z ∈ D it follows that f is analytic in D.

4. Fredholm properties of Tω for ω ∈ Rat(T) In this section we prove Theorem 1.2. We start with the formula for Ker(Tω ). Lemma 4.1. Let ω ∈ Rat(T), say ω = s/q with s, q ∈ P co-prime. Write s = s− s0 s+ with the roots of s− , s0 , s+ inside, on, or outside T, respectively. Then  < rˆ Ker(Tω ) = | deg(ˆ r) < deg(q) − (deg(s− ) + deg(s0 )) . (4.1) s+ Proof. If g = rˆ/s+ where deg(ˆ r) < deg(q) − (deg(s− ) + deg(s0 )), then sg = s− s0 rˆ which is a polynomial with deg(s− s0 rˆ) < deg(q). Thus ωg = s− s0 rˆ/q ∈ Rat0 (T) which implies that g ∈ Dom(Tω ) and Tω g = 0. Hence g ∈ ker(Tω ). This proves the inclusion ⊃ in the identity (4.1). Conversely suppose g ∈ ker Tω . Then Tω g = 0, i.e., by Lemma 2.3 we have ωg = rˆ/q or equivalently sg = rˆ for some rˆ ∈ Pdeg(q)−1 . Hence s− s0 (s+ g) = sg = rˆ. Thus g = r/s+ with r := s+ g ∈ H p . Note that s− s0 r = rˆ, so that r = rˆ/(s− s0 ). Since r ∈ H p and s− s0 only has roots in D, the identity r = rˆ/(s− s0 ) can only hold in case s− s0 divides rˆ, i.e., rˆ = s− s0 r1 for some r1 ∈ P. Then r = r1 ∈ P and we have deg( r) = deg(s+ g) = deg(ˆ r) − deg(s− s0 ) < deg(q) − (deg(s− ) + deg(s0 )). Hence g is included in the right-hand side of (4.1), and we have also proved the inclusion ⊂. Thus (4.1) holds.  We immediately obtain the following corollaries. Corollary 4.2. Let ω ∈ Rat(T). Then dim(Ker(Tω )) deg(q) and Q = {0} other+  with + ˙ Q, ˙ indicating a direct sum. In particular, wise. Then Pdeg(s)−1 = P  Pdeg(s)−1 = P if and only if deg(s) ≤ deg(q).

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 = {0}. Hence it is trivial that P +Q  Proof. For deg(s) ≤ deg(q) we have Q is a direct sum. Also, in this case Pdeg(s)−1 ⊂ Pdeg(q)−1 and consequently sH p + Pdeg(q)−1 = sH p + Pdeg(s)−1 , and this subspace of H p contains all polynomials. In particular, for any r ∈ Pdeg(s)−1 we have qr ∈ sPdeg(q)−1 +  Hence P  = Pdeg(s)−1 . Pdeg(q)−1 , which shows r ∈ P.  i.e., deg(r) < deg(s) − deg(q). Next, assume deg(s) > deg(q). Let r ∈ Q, In that case deg(rq) < deg(s) so that if we write rq as rq = r1 s + r2 then r1 ≡ 0 and r2 = rq with deg(rq) ≥ deg(q). Thus rq is not in sPdeg(q)−1 +  Hence P ∩Q  = {0}. It remains Pdeg(q)−1 and, consequently, r is not in P. +Q  = Pdeg(s)−1 . Let r ∈ Pdeg(s)−1 . Then we can write rq to show that P as rq = r1 s + r2 with deg(r1 ) < deg(q) and deg(r2 ) < deg(s). Next write r2 as r2 = r1 q + r2 with deg( r2 ) < deg(q). Since deg(r2 ) < deg(s), we have  Moreover, we have deg( r1 ) < deg(s) − deg(q). Thus r1 ∈ Q. rq = r1 s + r2 = r1 s + r1 q + r2 = (r1 s + r2 ) + r1 q, hence (r − r1 )q = r1 s + r2 .  and we can write r = (r − r1 ) + r1 ∈ P  + Q.  Thus r − r1 ∈ P,  We now show that if s has no roots on T, then the reverse inclusions in (4.2) also hold. Theorem 4.7. Let ω ∈ Rat(T), say ω = s/q with s, q ∈ P co-prime. Then Tω  has closed range if and only if s has no roots on T, or equivalently, sH p + P is closed in H p . In case s has no roots on T, we have  Dom(Tω ) = qH p + Pdeg(q)−1 and Ran(Tω ) = sH p + P. (4.4) Proof. The proof is divided into three parts.  Part 1. In the first part we show that s has no roots on T if and only if sH p +P  = Pdeg(s)−1 , and the is closed in H p . Note that for deg(s) ≤ deg(q) we have P  as in Lemma claim coincides with Theorem 3.1. For deg(s) > deg(q), define Q p  is finite dimensional, Q  is closed in 4.6, viewed as a subspace of H . Since Q  is closed, then so is sH p + P  +Q  = sH p +Pdeg(s)−1 . By H p . Hence, if sH p + P Theorem 3.1, the latter is equivalent to s having no roots on T. Conversely, if s has no roots on T, then sH p is closed, by Theorem 8 of [17] (see also  is finite dimensional, and thus the proof of Theorem 3.1). Now using that P p p  closed in H , it follows that sH + P is closed.  being closed implies (4.4). In particular, Part 2. Now we show that sH p + P this shows that s having no roots on T implies that Tω has closed range. Note that it suffices to show Dom(Tω ) ⊂ qH p + Pdeg(q)−1 , since the equalities in  is closed. Then also (4.4) then follow directly from (4.2). Assume sH p + P p sH + Pdeg(s)−1 is closed, as observed in the first part of the proof, and hence sH p + Pdeg(s)−1 = H p . This also implies s has no roots on T. Write s = s− s+ with s− , s+ ∈ P, with s− and s+ having roots inside and outside T only, respectively. Let g ∈ Dom(Tω ). Then sg = qh + r for h ∈ H p and r ∈ Pdeg(q)−1 . Note that sH p = s− H p , since s+ H p = H p . By Theorem 3.1 we have H p = sH p +Pdeg(s− )−1 . Since h ∈ H p = sH p +Pdeg(s− )−1 , we can

Toeplitz-like operator with symbol having poles on the unit circle I 253 write h = sh + r with h ∈ H p and r ∈ Pdeg(s− )−1 . Note that deg(qr + r) < deg(s− q). We can thus write qr + r = r1 s− + r2 with deg(r1 ) < deg(q) and deg(r2 ) < deg(s− ). Then sg = qh + r = qsh + qr + r = qsh + r1 s− + r2 = s(qh + r1 s−1 + ) + r2 . Hence r2 = s(g −qh −r1 s−1 + ). Since deg(r2 ) < deg(s− ), we can evaluate both sides (as well as the derivatives on both sides) at the roots of s− , to arrive at −1  r2 ≡ 0. Hence s(g − qh − r1 s−1 + ) ≡ 0. Dividing by s, we find g = qh + r1 s+ . Since q and s+ are co-prime and r1 ∈ Pdeg(q)−1 , by Lemma 3.6 we have −1 p  p r1 s−1 + ∈ qH + Pdeg(q)−1 . Thus g = qh + r1 s+ ∈ qH + Pdeg(q)−1 . Part 3. In the last part we show that if s has roots on T, then Tω does not have closed range. Hence assume s has roots on T. Also assume Ran(Tω ) is closed.  ⊂ Ran(Tω ) and Ran(Tω ) is closed, also sH p + P  ⊂ Ran(Tω ). Since sH p + P

 is finite dimensional, and hence closed, sH p + P +Q  is closed and Since Q we have +Q  = sH p + P +Q  = sH p + Pdeg(s)−1 = H p . sH p + P Therefore, we have +Q  ⊂ Ran(Tω ) + Q  ⊂ H p. H p = sH p + P  = H p. It follows that Ran(Tω ) + Q p Let h ∈ H such that qh ∈ sH p + Pdeg(s)−1 , which exists by Theorem  Since h ∈ Ran(Tω ), 3.3. Write h = h + r with h ∈ Ran(Tω ) and r ∈ Q. p there exist g ∈ H and r ∈ Pdeg(q)−1 such that sg = qh + r = q(h − r ) + r = qh − qr + r.  Write r as r = sr1 + r2 with r1 , r2 ∈ P, deg(r2 ) < deg(s). Note that r ∈ Q,  so that deg(qr ) < deg(s). Thus qh = sg+qr −r = sg+qr −sr1 −r2 = s(g−r1 )+(qr −r2 ) ∈ sH p +Pdeg(s)−1 , in contradiction with qh ∈ sH p + Pdeg(s)−1 . Hence Ran(Tω ) is not closed.



 and thus, by When s has no roots on T we have Ran(Tω ) = sH p + P p  Lemma 4.6, Ran(Tω ) + Q = H . However, this need not be a direct sum in case s has roots outside T. In the next lemma we obtain a different formula for Ran(Tω ), for which we can determine a complement in H p . Lemma 4.8. Let ω ∈ Rat(T), say ω = s/q with s, q ∈ P co-prime. Assume s has no roots on T. Write s = s− s+ with the roots of s− and s+ inside and outside T, respectively. Define − = {r ∈ P | rq = r1 s− + r2 for r1 , r2 ∈ Pdeg(q)−1 } P − = {0} if − = Pdeg(s )−deg(q)−1 if deg(s− ) > deg(q) and Q and define Q − deg(s− ) ≤ deg(q). Then − ˙P Ran(Tω ) = s− H p +

and

− = H p . ˙Q Ran(Tω )+

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In particular, codim Ran(Tω ) = max{0, deg(s− ) − deg(q)}. − , that is, sH p + P = Proof. It suffices to prove that Ran(Tω ) = s− H p + P p  s− H + P− , by Theorem 4.7. Indeed, the direct sum claims follow since ˙ deg(s− ) is a direct sum decomposition of H p , by Proposition H p = s− H p +P − + − is a direct sum decomposition of Pdeg(s ) , by ˙Q 3.2, and Pdeg(s− ) = P − applying Lemma 4.6 with s replaced by s− .  ⊂ s− H p + P − . Let f = sh + r with We first show that sH p + P p  h ∈ H and r ∈ P, say rq = sr1 + r2 . Then rq = s− (s+ r1 ) + r2 . Now write s+ r1 = q r1 + r2 with deg( r2 ) < deg(q). Since r2 and r2 have degree less than deg(q) and q(r − s− r1 ) = s− (s+ r1 ) + r2 − qs− r1 = s− r2 + r2 , − . Therefore it follows that r − s− r1 ∈ P − . f = sh + r = s− (s+ h + r1 ) + (r − s− r1 ) ∈ s− H p + P  ⊂ s− H p + P − . Thus sH p + P To complete the proof we prove the reverse implication. Let f = s− h + − with h ∈ H p and r ∈ P − , say rq = s− r1 + r2 with r1 , r2 ∈ r ∈ s− H p + P Pdeg(q)−1 . Set p p g = s−1 1 ) ∈ s−1 + (qh + r + (qH + Pdeg(q)−1 ) = qH + Pdeg(q)−1 ,

with the last identity following from Lemma 3.6. Then g ∈ Dom(Tω ) and  We show that Tω g = f resulting in f ∈ sH p + P,  as desired. Tω g ∈ sH p + P. We have sg = s− (qh + r1 ) = s− qh + s− r1 = s− qh + rq − r2 = q(s− h + r) − r2 ∈ qH p + Pdeg(q)−1 . This proves Tω g = s− h + r = f , which completes our proof.



Before proving Theorem 1.2 we first give a few direct corollaries. Corollary 4.9. Let ω ∈ Rat(T) have no zeroes on T. Then codim Ran(Tω )   <  1, to show that (6.1) holds, we have to prove the following:    N −m N −m−1  i + m i + m − 1 N −m−i N z = (z−1) z N −m−i−1 + . (6.2) m−1 m m i=0

i=0

To see this, we shall make use of the so-called hockey-stick formula, which    N −m  i+m−1 N implies that = . Thus, the right-hand side of (6.2) is m−1 m i=0 equal to  N −m−1 N −m   i + m i+m−1 N −m−i−1 (z − 1) z + . (6.3) m m−1 i=0

Note that the remainder of is equal to

N −m  i=0

i=0

N −m  i=0

 i + m − 1 N −m−i z upon dividing by z − 1 k−1

 i+m−1 , so the remainder term in (6.2) is correct. k−1

To finish the proof it remains to compare the coefficients of z k on the left and right-hand sides of (6.2) with k ≥ 1. A straightforward rewriting of the right-hand side shows that the equality (6.2) follows from the basic property of binomial coefficients. 

Toeplitz-like operator with symbol having poles on the unit circle I 261 Example 6.2. Let ω(z) = (z − 1)−m , i.e., s ≡ 1 and q(z) = (z − 1)m . From Proposition 2.1 we know qH p + Pm−1 ⊂ Dom(Tω ) which contains all the polynomials. Put   i+m−1 a−i = , i = 0, 1, 2, . . . m−1 and

 b−j =

0, am−j ,

j < m, j ≥ m.

From Lemma 6.1 above, for N > m we can write  N −m  m−1   i + m − 1 N −m−i  N m N z = (z − 1) z + (z − 1)j + 1 m−1 j i=0 j=0 N −m m−1  N  = q(z) a−i z N −m−i + (z − 1)j + 1 j i=0 j=0 N −m m−1  N  j = q(z) a−(N −m−j) z + (z − 1)j + 1. j j=0

j=0

Put r(z) =

m−1  j=0

N j

 (z − 1)j + 1.

Then r is a polynomial with deg(r) < m = deg q and since s ≡ 1, s(z)z N = q(z)

N −m

a−(N −m−j) z j + r

j=0

and so from Lemma 2.3, for N > m we have Tω z N =

N −m

a−(N −m−j) z j =

N −m

j=0

j=0

b−(N −j) z j =

N 

b−(N −j) z j

j=0

since b−j = 0 for j < m. From Lemma 4.1 we have Ker(Tω ) = span{z j , j < m} and so the matrix representation of Tω will be an upper triangular Toeplitz matrix with the first m columns zero. Proposition 6.3. Let ω ∈ Rat0 (T), say ω = s/q with s, q ∈ P co-prime, q having all its roots in T and deg(s) = n < m = deg(q). Then, for N ≥ m − n, Tω z N =

N −m+n+1 

a−j+1 z N −m+n+1−j

j=1

where a−j = O(j M −1 ) with M the maximum of the orders of the poles of ω on T. Thus the matrix representation [Tω ] of Tω with respect to the standard

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G.J. Groenewald, S. ter Horst, J. Jaftha and A.C.M. Ran

p basis {z n }∞ n=0 of H is given by ⎛ 0 · · · 0 a0 ⎜ 0···0 0 ⎜ [Tω ] = ⎜ 0 · · · 0 0 ⎝ .. .

a−1 a0 0

a−2 a−1 a0 .. .

a−3 a−2 a−1

··· ··· ···

⎞ ⎟ ⎟ ⎟ ⎠

F GH I m−n

Proof. Since ω ∈ Rat0 (T), by Corollary 4.4 we have Pm−n−1 ⊂ Ker(Tω ). Jt Suppose q(z) = j=1 (z − αj )mj with αj ∈ T the poles of ω with multit plicities mj . By partial fractions decomposition we can write ω = j=1 ωj , where sj (z) , for sj ∈ Pmj −1 , j = 1 · · · t. ωj (z) = (z − αj )mj t 2 Note that [Tω ] = j=1 [Tωj ] and if sj (z) = c0 + c1 z + c2 z + · · · +  m −1 j cmj −1 z mj −1 then [Tωj ] = i=0 ci [Tzi /(z−αj )mj ]. From this it follows that it suffices to prove the result for ω(z) = z n /(z − α)m . To this end, assume ω = s/q where s(z) = z n and q(z) = (z − α)m for some α ∈ T. By Lemma 6.1 we have  N −m  m−1   j + m − 1 N −m−j  N z N = (z − 1)m z + (z − 1)j + 1. m−1 j j=0

By replacing z with  z N α

=

z α

j=1

we can write

 −m  m−1   m N j j + m − 1  z N −m−j  N  z −1 + − 1 +1. m−1 j α α α j=0 j=1

z

Multiplying with αN results in  N −m  m−1  N  j+m−1 N m j−1 N −m−j z = (z−α) α z + αN −j (z−α)j +αN . m−1 j j=0

j=1

So, for N > m − n, s(z)z N = z N +n = (z − α)m

+

m−1  j=1

Put a−i =



N +n−m   j=0

 j+m−1 αj−1 z N +n−m−j m−1

 N +n αN +n−j (z − α)j + αN +n . j

 i+m−1 , i = 0, 1, 2, . . . , m−1

 b−j =

0 a−(j−(m−n))

if j < m − n, if j ≥ m − n,

Toeplitz-like operator with symbol having poles on the unit circle I 263 and r(z) =

m−1  j=1

 N +n αN +n−j (z − α)j + αN +n . j

Then deg(r) < m = deg(q) and s(z)z

N

= q(z)

N +n−m   j=0

= q(z)

N +n−m 

 j+m−1 αj−1 z N +n−m−j + r(z) m−1

a−j z N +n−m−j + r(z)

j=0

= q(z)

N +n−m 

a−((N +n)−m−j) z j + r(z)

j=0

= q(z)

N +n−m 

b−(N −j) z j + r(z)

j=0

from which it follows that Tω z N =

N +n−m  j=0

b−(N −j) z j =

N 

b−(N −j) z j

j=0

as b−j = 0 for j < m − n. Thus the matrix representation of Tω is given by ⎞ ⎛ 0 · · · 0 a0 a−1 a−2 a−3 · · · ⎜ 0···0 0 a0 a−1 a−2 · · · ⎟ ⎜ ⎟ ⎜ 0···0 0 0 a0 a−1 · · · ⎟ ⎠ . ⎝ .. .. . . F GH I m−n

Since  a−j =

j+m−1 m−1

 =

we have a−j = O(j m−1 ).

(j + m − 1)(j + m − 2) · · · j (j + m − 1)m−1 ≤ , (m − 1)! (m − 1)! 

Proof of Theorem 1.3. Let ω = s/q ∈ Rat with s, q ∈ P co-prime and q having a root on T. By Lemma 2.4, ω can be written uniquely as ω = ω0 + ω1 with ω0 ∈ Rat0 (T) and ω1 ∈ Rat with no poles on T. In particular, ω1 ∈ L∞ (T). Then [Tω ] = [Tω0 ]+[Tω1 ] and [Tω0 ] is as in Proposition 6.3. Moreover, [Tω1 ] has the form as in Theorem 1.3 and the Fourier coefficients of ω1 are square summable. This completes the proof. 

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7. Examples In the final section we discuss three examples. Example 7.1. Let ω(z) = (z − 1)−1 ∈ Rat0 (T). Then ω = s/q with s ≡ 1 and q(z) = z − 1. By Theorem 1.2 we have Dom(Tω ) = (z − 1)H p + C,

Ker(Tω ) = P0 = C,

Ran(Tω ) = H p ,

and so Tω is Fredholm. These facts can also be shown explicitly. By Proposition 4.5 it suffices to establish that Dom(Tω ) ⊂ (z −1)H p and so consequently H p ⊂ Ran(Tω ). To this end let g ∈ Dom(Tω ). Then there are h ∈ H p and c c ∈ C with ωg = (z − 1)−1 g = h + z−1 . Then g = (z − 1)h + c showing p p that g ∈ (z − 1)H + C. For h ∈ H put g = (z − 1)h, then g ∈ H p with ωg = (z − 1)−1 (z − 1)h = h, showing that g ∈ Dom(Tω ) and Tω g = h. That Ker(Tω ) = C is also easily verified directly, as for c ∈ C, ωc = c 0 + z−1 . Thus Tω c = 0 and so C ⊂ Ker(Tω ). The converse follows from Lemma 2.3 as for g ∈ Ker(Tω ), g = c for some c ∈ C. For the matrix representation, note that z n − 1 = (1 + z + z 2 + · · · + z n−1 )(z − 1) or equivalently (z − 1)−1 z n = 1 + z + z 2 + · · · + z n−1 + (z − 1)−1 , and so Tω z n = 1 + z + z 2 + · · · + z n−1 . From this it follows that the matrix representation [Tω ] with respect to the p standard basis {z n }∞ n=0 of H is given by ⎛ ⎞ 0 1 1 1 1 ··· ⎜ 0 0 1 1 1 ··· ⎟ ⎜ ⎟ [Tω ] = ⎜ 0 0 0 1 1 · · · ⎟ . ⎝ ⎠ .. .. . . ··· Let [T2 ] be given by ⎛ ⎜ ⎜ ⎜ [T2 ] = ⎜ ⎜ ⎝

0 0 1 −1 0 1 0 0 .. .

0 0 0 ··· 0 0 0 ··· −1 0 0 · · · 1 −1 0 · · · .. . ···

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

This is the matrix representation of T2 = S + P1 − I where S = Tz is the forward shift operator, P1 the projection onto the first component and I the identity operator on H p . Then T2 is a generalised inverse of Tω and a right-sided inverse of Tω .

Toeplitz-like operator with symbol having poles on the unit circle I 265 z−α ∈ Rat(T) for α ∈ C, α = 1. From Lemma z−1 2.3  < c p p Dom(Tω ) = g ∈ H : ωg = h + ,h ∈ H ,c ∈ C . z−1 For α ∈ T by Theorem 4.7,

Example 7.2. Let ω(z) =

Dom(Tω ) = (z − 1)H p + C,

Ran(Tω ) = (z − α)H p + P = (z − α)H p + C,

since P = {c ∈ C : c(z − 1) = c1 (z − α) + c2 , c1 , c2 ∈ C} = C by Lemma 4.6. This can be shown explicitly, and also holds for α ∈ T. By Proposition 4.5 it suffices to show Ran(Tω ) ⊂ (z − α)H p + C. Note that for h ∈ H p we have h ∈ Ran(Tω ) if and only if there exist g ∈ H p and c ∈ C such that z−α c g =h+ , z−1 z−1 Now use that

z−1 z−α

=1+

α−1 z−α

i.e.,

z−1 c h+ = g ∈ H p. z−α z−α

and h ∈ H p , to arrive at

α−1 c h+ ∈ H p , for some c ∈ C z−α z−α c z−α p ⇐⇒ h + ∈ H = (z − α)H p , for some c ∈ C α−1 α−1 ⇐⇒ h ∈ (z − α)H p + C.

h ∈ Ran(Tω ) ⇐⇒

So Ran(Tω ) = (z − α)H p + C. For the matrix representation with respect to the basis {z n }∞ n=0 , note z−α 1−α that =1+ = 1 + c(z − 1)−1 where c = 1 − α. Then the matrix z−1 z−1 p representation with respect to the standard basis {z n }∞ n=0 of H is given by ⎞ ⎛ 1 c c c c ··· ⎜ 0 1 c c c ··· ⎟ ⎟ ⎜ [Tω ] = ⎜ 0 0 1 c c · · · ⎟ . ⎠ ⎝ .. .. . . ··· From Theorem 1.1, Tω is Fredholm if and only if α ∈ T, which also follows from the fact that Ran(Tω ) = (z − α)H p + C. From Theorem 5.4 Tω is invertible for α ∈ D and by Lemma 4.1, Ker(Tω ) = {c/(z − α) : c ∈ C} in case α ∈ D. For α ∈ D, let T + be the operator on H p with the matrix representation p with respect to the standard basis {z n }∞ n=0 of H be given by ⎞ ⎛ 1 −c −cα −cα2 −cα3 · · · ⎜ 0 1 −c −cα −cα2 · · · ⎟ ⎜ ⎟ [T + ] = ⎜ 0 0 1 −c −cα · · · ⎟ ⎝ ⎠ .. .. . . ··· Then T + is the bounded inverse for Tω .

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G.J. Groenewald, S. ter Horst, J. Jaftha and A.C.M. Ran

For α ∈ D, Tω is surjective and the operator T  with the matrix representation with respect to the standard basis of H p given by ⎛ ⎞ 1 0 0 0 0 ··· ⎜ cα−1 1 0 0 0 ··· ⎟ ⎜ ⎟ [T  ] = α−1 ⎜ c(α)−2 cα−1 1 0 0 · · · ⎟ ⎝ ⎠ .. .. . . ··· is a right-sided inverse for Tω . z+1 Example 7.3. Let ω(z) = (z−1) 2 ∈ Rat0 (T). From Lemma 2.3  < r p Dom(Tω ) = g ∈ H p : ωg = h + , h ∈ H , deg(r) ≤ 1 . (z − 1)2

From Proposition 4.5 we see that (z − 1)2 H p + P1 ⊂ Dom(Tω ) and (z + 1)H p + C ⊂ Ran(Tω ). In this case ω has a zero on T, so Tω is not Fredholm. Nonetheless, we will show that (z+1)H p +C = Ran(Tω ) and (z−1)2 H p +P1 = Dom(Tω ). To this end, let h ∈ Ran(Tω ). Then there exists an f ∈ H p and a polynomial r with deg(r) ≤ 1 with f= Note that

(z−1)2 z+1

(z − 1)2 r h+ ∈ H p. z+1 z+1

4 = z − 3 + z+1 and

r z+1

c2 = c1 + z+1 for c1 , c2 ∈ C. So we have

4 c2 h + c1 + ∈ H p , for some c1 , c2 ∈ C z+1 z+1 4 c2 ⇐⇒ h+ ∈ H p , for some c2 ∈ C z+1 z+1 c2 z+1 p ⇐⇒ h + ∈ H = (z + 1)H p , for some c2 ∈ C 4 4 ⇐⇒ h ∈ (z + 1)H p + C.

h ∈ Ran(Tω ) ⇐⇒ (z − 3)h +

So Ran(Tω ) = (z + 1)H p + C. From Theorem 1.1 it follows that Tω is not Fredholm, which can also be seen directly from the fact that Ran(Tω ) = (z + 1)H p + C which is not closed. To show Dom(Tω ) = (z − 1)2 H p + P1 , let f ∈ Dom(Tω ) then there are h ∈ H p and az + b = r ∈ P1 with ωf = h +

az + b , (z − 1)2

or equivalently,

f=

(z − 1)2 az + b h+ . z+1 z+1

Since h ∈ Ran(Tω ) = (z + 1)H p + C there are g ∈ H p and c ∈ C with 2 4 az+b b−a h = (z + 1)g + c. As (z−1) z+1 = z − 3 + z+1 and z+1 = a + z+1 , we find that (z − 1)2 az + b ((z + 1)g + c) + z+1 z+1 4c b−a = (z − 1)2 g + (z − 3)c + +a+ . z+1 z+1

f=

Toeplitz-like operator with symbol having poles on the unit circle I 267 This implies that 4c + b − a = f − (z − 1)2 g − (cz − 3c + a) ∈ H p , z+1 which, by Lemma 2.2, can only happen if 4c+b−a = 0. z+1 2 As 4c+b−a = f − (z − 1) g − (cz − 3c + a) ∈ H p from Lemma 2.2 we z+1 get 4c+b−a = 0. Thus f = (z − 1)2 g + (cz − 3c + a) ∈ (z − 1)2 H p + P1 and z+1 so Dom(Tω ) = (z − 1)2 H p + P1 . From Lemma 4.1 it follows that Ker(Tω ) = C and for the matrix representation with respect to {z n }∞ n=0 note k−1 z+1 k  (2k + 3)z − (2k + 1) z = (2n + 1)z k−n−1 + 2 (z − 1) (z − 1)2 n=0

and so the matrix representation ⎛ 0 ⎜ 0 ⎜ [Tω ] = ⎜ 0 ⎝

[Tω ] given by 1 0 0 .. .

3 1 0

5 3 1 .. .

⎞ 7 ··· 5 ··· ⎟ ⎟ . 3 ··· ⎟ ⎠ ···

Acknowledgement. The present work is based on research supported in part by the National Research Foundation of South Africa. Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard. We would also like to thank the anonymous referee for the useful suggestions, corrections and comments.

References [1] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz operators, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006. [2] J.B. Conway, A course in Functional analysis, 2nd ed., Springer, 1990. [3] V. Dybin and G.M. Grudsky, Introduction to the theory of Toeplitz operators with infinite index, Oper. Theory Adv. Appl. 137, Birkh¨ auser Verlag, Basel, 2002. [4] S. Goldberg, Unbounded linear operators and applications, Macgraw-Hill, New York, 1966. [5] I Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of linear operators. Vol. I, Oper. Theory Adv. Appl. 49, Birkh¨ auser Verlag, Basel, 1990. [6] I Gohberg, S. Goldberg, and M.A. Kaashoek, Basic classes of linear operators, Birkh¨ auser Verlag, Basel, 2002. [7] G.J. Groenewald, S. ter Horst, J. Jaftha, and A.C.M. Ran, A Toeplitz-like operator with rational symbol having poles on the unit circle II: the spectrum, Oper. Theory Adv. Appl., accepted. [8] P. Hartman, On unbounded Toeplitz Matrices, Am. J. Math 85 (1963), 59–78.

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[9] P. Hartman and A. Wintner, On the spectra of Toeplitz’s Matrices, Am. J. Math 72 (1950), 359–366. [10] H. Helson, Large analytic functions, in: Linear Operators and Function spaces (Timi¸soara, 1988), Oper. Theory Adv. Appl. 43, pp. 209–216, Birkh¨ auser Verlag, Basel, 1990. [11] J. Janas, Unbounded Toeplitz operators in the Bargmann-Segal space, Studia Mathematica 99 no. 2 (1991), 87–99. [12] Y. Katznelson, An introduction to Harmonic Analysis, Dover, 1968. [13] V.V. Lebedev, On p -multipliers of functions analytic on the disk (Russian). Funktsional. Anal. i Prilozhen. 48 no. 3 (2014), 92–96; translation in: Funct. Anal. Appl. 48 no. 3 (2014), 231–234. [14] J. Rovnyak, On the theory of unbounded Toeplitz operators, Pac. J. Math. 31 (1969), 481–496. [15] D. Sarason, Unbounded Toeplitz operators, Integr. Equ. Oper. Theory 61 no. 2 (2008), 281–298. [16] N.L. Vasilevski, Commutative algebras of Toeplitz operators on the Bergman space, Oper. Theory Adv. Appl. 185, Birkh¨ auser Verlag, Basel, 2008. [17] D. Vukoti´c, Analytic Toeplitz operators on the Hardy space H p : a survey, Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 101–113. G.J. Groenewald and S. ter Horst Department of Mathematics, Unit for BMI, North-West University Potchefstroom, 2531 South Africa e-mail: [email protected] [email protected] J. Jaftha Numeracy Center, University of Cape Town Rondebosch 7701, Cape Town South Africa e-mail: [email protected] A.C.M. Ran Department of Mathematics, Faculty of Science, VU Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands and Unit for BMI, North-West University, Potchefstroom, South Africa e-mail: [email protected]

Canonical form for H-symplectic matrices G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran Dedicated to Rien Kaashoek on the occasion of his eightieth birthday

Abstract. In this paper we consider pairs of matrices (A, H), with A and H either both real or both complex, H is invertible and skew-symmetric and A is H-symplectic, that is, AT HA = H. A canonical form for such pairs is derived under the transformations (A, H) → (S −1 AS, S T HS) for invertible matrices S. In the canonical form for the pair, the matrix A is brought in standard (real or complex) Jordan normal form, in contrast to existing canonical forms. Mathematics Subject Classification (2010). Primary 15A21, 15B57, 15A63; Secondary 47B50. Keywords. Indefinite inner product space, canonical forms, H-symplectic matrices.

1. Introduction In this paper we shall consider pairs of matrices (A, H), where A and H are either real or complex matrices and A is H-symplectic. Recall, when H = −H T is invertible, a matrix A is called H-symplectic when AT HA = H. Obviously, when S is an invertible matrix, then S −1 AS is S T HS-symplectic. Under these transformations one might ask: what is the canonical form for the pair (A, H)? Such canonical forms already exist in the literature, see for instance [14, 15], and [4, 9, 16, 17, 18] for several slightly different versions. The canonical forms available in the literature keep H in as simple a form as possible, and simultaneously bring A into a form from which the Jordan canonical form of A may be read off more or less easily, with blocks that are at best of the form Jn (λ) ⊕ Jn ( λ1 )−T , where the superscript −T indicates the transpose of the inverse. (As usual, Jn (λ) denotes the n × n upper triangular Jordan block with eigenvalue λ.) In some cases blocks in the canonical form are much more complicated. It is our goal to present here a canonical form where A is completely in (real) Jordan form. © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_11

269

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G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran

In our previous paper [8] we considered matrices A which were unitary in an indefinite inner product given by a symmetric (or Hermitian) matrix H. Canonical forms for unitary matrices in indefinite inner product spaces have been the subject of many papers, we mention here the books [6, 7] where the canonical forms were deduced from corresponding canonical forms for selfadjoint matrices in an indefinite inner product space, and the papers [10, 19, 20, 21, 22, 23], see also [1]. General theory for operators and matrices in indefinite inner product spaces may be found in the books [2, 3, 6, 7, 11]. We shall make use of results from [14, 15] where unitary and symplectic matrices are studied from the point of view of normal matrices in an indefinite inner product space, and where also canonical forms have been given. Closest to our development in [8] is the paper [13], although a complete canonical form is not deduced there. Our point of view is that we wish to bring the matrix A in (real) Jordan canonical form, and see what this implies for the matrix H representing the bilinear or sesquilinear form. The start of our considerations was the simple form for expansive matrices in an indefinite inner product, developed in [12] and [5]. We consider both the complex case, as well as the real case, where all matrices involved are assumed to be real. In fact, there are three cases to be considered: 1. A and H are complex matrices, with H = −H ∗ invertible and A∗ HA = H, considered as matrices acting inthe space Cn equipped with the n standard sesquilinear form x, y = i=1 xi yi , 2. A and H are complex matrices, with H = −H T invertible and AT HA = n H, considered as matrices acting nin the space C equipped with the standard bilinear form x, y = i=1 xi yi , 3. A and H are real matrices,with H = −H T invertible and AT HA = H, considered as matrices acting in the space Rn equipped with the standard bilinear form. The first case is easy: put H1 = iH, then H1 = H1∗ is invertible and A H1 A = H1 . Hence, A is H1 -unitary, and a canonical form can be deduced from canonical forms for unitary matrices in indefinite inner product spaces, such as given in, e.g., our recent paper [8]. The resulting theorem is presented in the final section of this paper. The focus of this paper will be on the two remaining cases above. In the third case the matrix A will be called H-symplectic; this is the classical case. The second case is less well-studied. In that case, for lack of a better term, we shall call the matrix A also H-symplectic, it will be clear from the context whether we work in Cn or Rn . It should be stressed that in both cases the space is equipped with the standard bilinear form (so, in particular, in the complex case not the standard sesquilinear form). A canonical form for the second case seems to be less well-known, and probably appeared for the first time in [14]. It is our aim to derive a canonical form with A in standard (real or complex) Jordan canonical form (see [24]). This would be analogous to the ∗

Canonical form for H-symplectic matrices

271

canonical form we recently derived for H-unitary matrices in [8]. Of course, starting from any canonical form where A is not exactly in Jordan canonical form, we can transform the pair (A, H) via an appropriate basis transformation S to the pair (S −1 AS, S T HS) with S −1 AS in Jordan canonical form. This will have the desired effect, although it may not directly produce the same form for S T HS that is achieved in our main results. Indeed, this is caused by the fact that there are many invertible matrices S such that, when A is in Jordan canonical form, also S −1 AS is in Jordan canonical form. We shall make use of the freedom this provides in our proofs. The authors thank the anonymous referee for pointing out that the main results (Theorem 2.6 and Theorem 2.11) of this paper may be derived in this manner from results in paper [16] (albeit with a considerable amount of work for some of the cases). However, we have chosen to take a more direct approach here, and develop the desired canonical form from scratch. To further motivate our choice to keep A in Jordan canonical form, consider the problem of finding a function f (A) of the matrix A, which is greatly facilitated by having A in Jordan normal form. We can once again use the results on the indecomposable blocks ([14, 15]) to limit the number of cases we have to consider. In particular, Theorem 8.5 in [14] gives a canonical form, but also tells us that the indecomposable blocks in the complex case are of three types and is given in the following proposition, where U in [14] is replaced by our A, and Q in [14] is our S. Proposition 1.1. Let H = −H T be invertible and let A be H-symplectic. Then there is an invertible matrix S such that S −1 AS = ⊕kj=1 Aj ,

S T HS = ⊕kj=1 Hj ,

(1.1)

−HjT

where in each pair (Aj , Hj ) the matrix Hj = is invertible, and Aj is Hj -symplectic, and each pair is of one of the following indecomposable forms: (i) (complex eigenvalues) 1 1 1 Aj = Jnj (λ) ⊕ Jnj ( ) with Re λ > Re or Im λ > Im if λ λ λ 1 Re λ = Re , (1.2) λ

0 H12 Hj = ; T −H12 0 (ii) ±1, even partial multiplicity, Aj = Jnj (±1), with nj even, Hj = −HjT ; (iii) ±1, odd partial multiplicities,



0 Aj = Jnj (±1) ⊕ Jnj (±1) with nj odd, Hj = T −H12

(1.3)

H12 . 0

(1.4)

The matrices Hj , and in particular the form of the matrices H12 in (1.2) and (1.4) may be further reduced to a canonical form as is described in the main results of this paper.

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In [15], Theorem 5.5, the canonical form and the indecomposable blocks for the real case are described, in that case the first class of blocks for the complex case has to be subdivided into three classes: when λ is real, when λ is unimodular but not ±1, and when λ is non-real and non-unimodular (in the latter case actually there is a quadruple of eigenvalues involved). In particular, note that odd sized blocks with eigenvalue one or minus one come in pairs. This was proved in e.g. [14], see in particular Proposition 3.4 there and its proof, and also in [1], Proposition 3.1. As a consequence of this, all one needs to do to arrive at a canonical form for the pair (A, H) is to derive canonical forms for each of these indecomposable blocks.

2. Main Results In this section we will present the main results of this article. In the first subsection is the main result for the complex case and in the second subsection the main result for the real case. Each subsection makes use of a number of definitions which will be presented first. Most of these definitions have their origin in the canonical form for H-unitary matrices described in [8]. 2.1. Complex case We start by giving the definitions needed for the main theorem in the complex case. Definition 2.1. For odd n > 1 the defined as follows:

n+1 2

×

n−1 2 −i+1

= (−1)

p n+1 j = (−1)j · 2

n−1   n+1 2 , 2 matrix Pn = pij i=1,j=1 is

n−1 , 2 n+1 for i = 1, . . . , − 1, 2 n−1 for j = 1, . . . , , 2

when i + j ≤

pi j = 0 pi

n−1 2

n−1 2 −i+1

1 2

and all other entries are defined by pi

j+1

= −(pi j + pi+1 j ).

The numbers   pij have the following explicit formula, with the underj standing that = 0 whenever k < 0 or j < k: k     (−1)j j+1 j−1 pij = − . n+1 n+1 2 2 −i 2 −i−2 Indeed, it can be easily checked that these numbers satisfy the recursion and the initial values given in Definition 2.1.

Canonical form for H-symplectic matrices To get a feeling for how such ⎡ 0 ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ P11 = ⎢ ⎢ 0 ⎢ ⎢ ⎢ −1 ⎣ − 12

273

a matrix looks like, we give P11 below: ⎤ 0 0 0 −1 0 0 1 − 92 ⎥ ⎥ ⎥ 7 16 ⎥ 0 −1 2 − 2 ⎥ ⎥. ⎥ 1 − 52 92 − 14 2 ⎥ ⎥ 3 − 42 52 − 62 ⎥ 2 ⎦ 1 1 1 1 −2 2 −2 2

Also, P9 is the submatrix of P11 formed by deleting the last column and first row. Observe that the recursion for the entries of Pn actually holds for all its entries, provided the first column and last row are given, or the last column and first row. Also note that the recursion is the same as the one for Pascal’s triangle, modulo a minus sign. Take note that the entries are not the numbers in the Pascal triangle because the starting values are different: if we consider the entries in the first column and last row of Pn as the starting values, then the nonzero starting numbers are 1, 12 rather than 1, 1 as would be the case for the Pascal triangle. n+1 Definition 2.2. We also introduce for odd n the n+1 2 × 2 matrix Zn which n+1 n+1 has zeros everywhere, except in the ( 2 , 2 )-entry, where it has a 1. For instance, Z5 is given by ⎡ ⎤ 0 0 0 Z 5 = ⎣ 0 0 0⎦ . 0 0 1

We shall also make use of the matrices Zn ⊗ I2 , which is the (n + 1) × (n + 1) matrix which has zeros everywhere except in the two

by two lower right block 0 1 where it has I2 , and Zn ⊗ K1 , where K1 = , i.e., the (n + 1) × (n + 1) 1 0 matrix which has zeros everywhere except in the two by two lower right block where it has K1 . Definition 2.3. Next we introduce for even n the as follows:

n 2 −i+1

  n2 , n2 matrix Qn = qij i=1, j=1

n , 2 n for i = 1, . . . , , 2 n for j = 1, . . . , , 2 when i + j ≤

qi j = 0 qi

n n 2×2

n

= (−1) 2 −i

q n2 j = (−1)j−1 and all other entries are defined by qi

j+1

= −(qi j + qi+1 j ).

274

G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran Again, we give an example: Q10 is ⎡ 0 0 ⎢0 0 ⎢ Q10 = ⎢ ⎢0 0 ⎣0 −1 1 −1

given by 0 0 1 2 1

0 −1 −3 −3 −1

⎤ 1 4⎥ ⎥ 6⎥ ⎥. 4⎦ 1

Also, Q8 is formed from this by deleting the first row and last column. Note that the numbers involved, apart from a minus sign, are exactly the numbers from Pascal’s triangle, so in this case we can even give a precise formula: when i + j ≥ n2 + 1 we have   j−1 qij = (−1)j−1 n . 2 −i For λ ∈ C \ {−1, 0, 1} we define the following: n−1 Definition 2.4. Let n > 1 be an odd integer, then the n+1 2 × 2 matrix Pn (λ) is defined as follows: n−1 $ % n+1 2 , 2 n+1 (2.1) Pn (λ) = pij λ 2 +j−i i=1, j=1

where pij are the entries of the matrix Pn introduced above. For example, P5 (λ) is the 3 × 2 matrix given by ⎡ ⎤ 0 λ4 ⎢ 2 3 3⎥ ⎥ P5 (λ) = ⎢ 2λ ⎦ . ⎣ −λ − 12 λ

1 2 2λ

Definition 2.5. Let n > 1 be an even integer, then the is defined as follows:   n2 , n2 n Qn (λ) = qij λ 2 +j−i−1 i=1, j=1

n 2

×

n 2

matrix Qn (λ) (2.2)

where qij are the entries of the matrix Qn introduced earlier. With these definitions in place we state now the main theorem for the complex case. The reader should realize that in Pn , Qn , Zn the subscript n does not indicate that these are n × n matrices, but that the dimensions of these matrices depend on n as indicated in the previous definitions. Theorem 2.6. Let A be H-symplectic, with both A and H complex. Then the pair (A, H) can be decomposed as follows. There is an invertible matrix S such that p p   S −1 AS = Al , S T HS = Hl , l=1

l=1

where each pair (Al , Hl ) satisfies one of the following conditions for some n depending on l; (i) σ(Al ) = {1} and the pair (Al , Hl ) has one of the following two forms:

Canonical form for H-symplectic matrices ⎛

275

⎡

⎤⎞ 0 0 Z n Pn ⎜ ⎢ 0 ⎟ 0 PnT 0⎥ ⎢ ⎥⎟ with n odd. Case 1: ⎜ J (1) ⊕ J (1), n n T ⎝ ⎣−Zn −Pn 0 0 ⎦⎠ −PnT 0 0 0 

 0 Qn with n even. Case 2: Jn (1), −QTn 0 (ii) σ(Al )⎛= {−1} and the pair ⎡ (Al , Hl ) has one of the following two forms: ⎤⎞ 0 0 Zn Pn (−1) ⎜ ⎢ ⎟ 0 0 PnT (−1) 0 ⎥ ⎢ ⎥⎟ with Case 1: ⎜ ⎝Jn (−1)⊕Jn (−1), ⎣ −ZnT −Pn (−1) 0 0 ⎦⎠ −PnT (−1) 0 0 0 n odd. 

 0 Qn (−1) Case 2: Jn (−1), with n even. −QTn (−1) 0 (iii)  σ(Al ) = {λ, λ1 } with λ ∈ C\{−1,

0, 1} and the pair (Al , Hl ) is of the form 0 H 12 Jn (λ) ⊕ Jn ( λ1 ), , where H12 is of one of the following T −H12 0 two forms, depending on whether n is odd

or even: Zn Pn (λ) Case 1: n is odd: H12 = . Pn ( λ1 )T 0

0 Qn (λ) Case 2: n is even: H12 = . − λ12 Qn ( λ1 )T 0 2.2. Real case First we present a number of definitions needed for the main theorem of the real case. Take note that analogues of some of the definitions presented earlier in the complex case are

also needed in the real case. α β Let γ = with β = 0 and α2 + β 2 = 1. As usual Jn (γ) denotes −β α the matrix ⎡ ⎤ γ I2 ⎢ ⎥ .. .. ⎢ ⎥ . . ⎥ Jn (γ) = ⎢ ⎢ ⎥ .. ⎣ . I2 ⎦ γ of size 2n × 2n. We also define the following: n−1 ˜ Definition 2.7. For n odd the n+1 2 × 2 block matrix Pn (γ) with two by two matrix blocks is defined as: n−1 $ % n+1 2 , 2 n+1 P˜n (γ) = pij H0 (γ T ) 2 +j−i , i=1, j=1

where p ij are the entries of the matrix Pn as in Definition 2.1, and H0 = 0 1 . −1 0

276

G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran As an example,

⎡

0

⎢ T 2 P˜5 (γ) = ⎢ ⎣−H0 (γ ) − 12 H0 γ T

H0 (γ T )4

⎤ ⎥

3 T 3⎥ 2 H0 (γ ) ⎦ 1 T 2 2 H0 (γ )

˜ n (γ) with two by two Definition 2.8. For even n the n2 × n2 block matrix Q matrix blocks is defined as:  n n ˜ n (γ) = qij (γ T ) n2 +j−i 2 , 2 Q , i=1, j=1 where qij are the entries of the matrix Qn as in Definition 2.3.

α β Also, for γ = , with α2 + β 2 = 1, we define −β α n−1 Definition 2.9. Let n > 1 be an odd integer, then the n+1 2 × 2 block matrix Pn (γ) with two by two matrix blocks is defined as follows: n−1 $ % n+1 2 , 2 n+1 Pn (γ) = pij K1 γ 2 +j−i (2.3) i=1, j=1

where pij are the entries of the matrix Pn introduced earlier, and K1 = 0 1 . 1 0 For example, P5 (γ) is the 3 × 2 matrix given by ⎡ ⎤ 0 K1 γ 4 ⎢ ⎥ 3 2 3⎥ P5 (γ) = ⎢ 2 K1 γ ⎦ . ⎣ −K1 γ − 12 K1 γ 12 K1 γ 2 Definition 2.10. Let n > 1 be an even integer, then the Qn (γ) is defined as follows:   n2 , n2 n Qn (γ) = qij K1 γ 2 +j−i−1 i=1, j=1

n 2

×

n 2

block matrix (2.4)

where qij are the entries of the matrix Qn introduced earlier. With these definitions in place, we state the main theorem in the real case. Theorem 2.11. Let A be H-symplectic, with both A and H real. Then the pair (A, H) can be decomposed as follows. There is an invertible real matrix S such that p p   S −1 AS = Al , S T HS = Hl , l=1

l=1

where each pair (Al , Hl ) satisfies one of the following conditions for some n depending on l; (i) σ(Al ) = {1} and the pair (Al , Hl ) has one of the following two forms:

Canonical form for H-symplectic matrices ⎛

277

⎡

⎤⎞ 0 0 Z n Pn ⎜ ⎢ 0 ⎟ 0 PnT 0⎥ ⎢ ⎥⎟ with n odd. Case 1: ⎜ J (1) ⊕ J (1), n n T ⎝ ⎣−Zn −Pn 0 0 ⎦⎠ −PnT 0 0 0

  0 Qn with n even, and ε = ±1. Case 2: Jn (1), ε −QTn 0 (ii) σ(Al )⎛= {−1} and the pair ⎡ (Al , Hl ) has one of the following two forms: ⎤⎞ 0 0 Zn Pn (−1) ⎜ ⎢ ⎟ 0 0 PnT (−1) 0 ⎥ ⎢ ⎥⎟ with J Case 1: ⎜ (−1)⊕J (−1), n n T ⎝ ⎣ −Zn −Pn (−1) 0 0 ⎦⎠ −PnT (−1) 0 0 0 n odd. 

 0 Qn (−1) Case 2: Jn (−1), ε with n even, and ε = ±1. −QTn (−1) 0 (iii)  σ(Al ) = {λ, λ1 } with λ ∈ R\{−1,

0, 1} and the pair (Al , Hl ) is of the form 0 H12 1 Jn (λ) ⊕ Jn ( λ ), , where H12 is of one of the following T −H12 0 two forms, depending on whether n is odd

or even: Zn Pn (λ) Case 1: n is odd: H12 = . Pn ( 1 ) T 0 λ

0 Qn (λ) Case 2: n is even: H12 = . − λ12 Qn ( λ1 )T 0 (iv) σ(Al ) = {α ± iβ} with α2 + β 2 = 1 and β = 0, and

the pair (Al , Hl ) has α β one of the following two forms with γ = : −β α 

 Z ⊗ H0 P˜n (γ) Case 1: Jn (γ), ε n˜ with n odd, and ε = ±1. −Pn (γ)T 0 

 ˜ n (γ) 0 Q Case 2: Jn (γ), ε with n even, and ε = ±1. ˜ n (γ)T −Q 0 −1 2 2 (v) σ(Al ) = {α ± iβ, (α ± iβ)  1 and β =  0, and the pair  } with α + β = 0 H12 −1 (Al , Hl ) is of the form Jn (γ) ⊕ Jn (γ ), , where H12 T −H12 0 is of one of the following two forms, depending on whether n is odd or even:

Zn ⊗ K1 Pn (γ) Case 1: n is odd: H12 = . Pn (γ −1 )T 0

0 Qn (γ) Case 2: n is even: H12 = . −(In ⊗ −γ −2 )Qn (γ −1 )T 0

Note that the columns of the matrix S in the theorem form a special real Jordan basis for A. The theorem should be compared with the canonical form obtained in [15], in particular with Theorem 5.5 there.

278

G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran

The proofs of Theorems 2.6 and 2.11 will be given in the sections to follow. In subsection 3.1 part (i) Case 1 and 2 is proved with λ = 1 for the complex case. Part (ii) follows similarly. Part (iii) needs no proof as this is equivalent to what has been done in the unitary case. The real case is proved in Section 4. Parts (i) and (ii) are the same for the real and complex case, as is part (iii). Part (v) is the same as in the unitary case, so needs no extra proof. Only part (iv) needs a new proof here, and this is presented in Section 4 in detail. The main theorem for the complex H-symplectic case is given in the final section of this paper without proof, as the proof of this theorem once again follows directly from the unitary case, see [8].

3. The complex case We prove the main theorem, Theorem 2.6, of the complex case in the subsections to follow. 3.1. Eigenvalue one We begin by proving Case 1 (with n odd) of Part (i) of Theorem 2.6. Proof. We may assume, based on Proposition 1.1, that in this case

0 H (0) A = Jn (1) ⊕ Jn (1), H= −(H (0) )T 0 for an invertible n × n matrix H (0) . By Section 2 in [5], (see also [12]), we (0) have that Hi j = 0 for i + j ≤ n. Moreover, from AT HA = H we have that Jn (1)T H (0) Jn (1) = H (0) , and so (0)

(0)

(0)

Hi j + Hi j+1 + Hi+1 j = 0,

for i > 1 and j > 1.

(3.1)

˜ where Sˆ and S˜ are n × n upper triangular Toeplitz Consider S = Sˆ ⊕ S, matrices, in particular, the upper triangular Toeplitz matrix Sˆ with first row h1 , · · · , hn will be denoted by toep(h1 , · · · , hn ). Then S −1 AS = A, and

0 SˆT H (0) S˜ S T HS = , 0 −S˜T (H (0) )T Sˆ and note that S is chosen such that (3.1) holds, with H replaced by H  =  S T HS and Hij = 0 for i + j ≤ 0. Indeed, we take S˜ = I and will show that Sˆ can be chosen so that SˆT H (0) has the form given in Part (i) Case 1 of both Theorems 2.6 and 2.11, that is,

˜ = SˆT H (0) = ZnT Pn . H Pn 0

Canonical form for H-symplectic matrices

279

In order to achieve this, it suffices, in view of (3.1) to show that Sˆ can be chosen such that ˜ n+1 n+1 = 1, H ˜ n+1 n+1 = − 1 , H 2 2 2 2 +1 2 ˜ n+1 n+1 = 0, for j = 1, . . . , n − 1 , H 2 +j 2 +j 2 n−1 ˜ n+1 n+1 H = 0, for j = 1, . . . , − 1. 2 +j 2 +j+1 2 ˜ i j = (SˆT H (0) )i j : Let Sˆ = toep(h1 , · · · , hn ), and compute H ⎧ ⎨ 0 if i + j ≤ n, ˜i j = H (3.2) (0) ⎩h H (0) + h H (0) + · · · + h if i + j > n. 1 ij 2 i−1 j i+j−n Hn+1−j j Put S1 = h1 In , and H (1) = S1T H (0) . For i + j = n + 1 with i = j= we have n+1 2 ,

(1)

H n+1 2

(0)

n+1 2

= h1 H n+1 2

n+1 2

and hence H n+1 2

n+1 2

= 1. Formula (3.1) with H

and

.

Because of the invertibility of H (0) we can take h1 so that (1)

n+1 2

(0)

(0)

1 h1

= H n+1 2

replaced by H

(1)

n+1 2

,

, ensures

that all anti-diagonal entries of H (1) with i + j = n + 1 alternates between +1 and −1. Now let S2 = toep(1, h2 , 0, · · · , 0), and let H (2) = S2T H (1) . Then for i + j > n we have from (3.2) that (2)

(1)

(1)

Hi j = Hi j + h2 Hi−1 j .

(3.3)

In particular, for i + j = n + 1 we have (2) n+1−i

Hi (2)

(1) n+1−i

= Hi

(1)

+ h2 Hi−1

n+1−i

(1) n+1−i ,

= Hi

(1)

and so Hi j = Hi j for i + j ≤ n + 1. Furthermore, for i + j = n + 2 we have from (3.3) that (2) n+2−i

Hi (1)

We have Hi−1

n+2−i

(2)

H n+1 2

= H n+1

(1)

− 12 .

(1)

+ h2 Hi−1

= ±1. In particular, for i = (1)

n+1 2 +1

since H n+1 −1 2

(1) n+2−i

= Hi

2

n+1 2

(1)

n+1 2 +1

+ h2 H n+1 −1 2

n+2−i .

(j =

n+1 2

+ 1), we have

(1)

n+1 2 +1

= H n+1 2

n+1 2 +1

− h2 ,

(2)

n+1 2 +1

= −1. Now we can choose h2 such that H n+1 2

n+1 2 +1

=

A repeated application of (3.1) determines all entries for which i + j = n + 2. Next, put S3 = toep(1, 0, h3 , 0, · · · , 0) and H (3) = S3T H (2) . By (3.2) we have for i + j > n that (3)

(2)

(2)

Hi j = Hi j + h3 Hi−2 j .

280

G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran (3)

(2)

(2)

If i + j ≤ n + 2 this gives Hi j = Hi j , since i + j − 2 ≤ n, so that Hi−2 j = 0. For i + j = n + 3 we obtain from the identity above that (3) n+3−i

Hi For i = j =

n+1 2

(2) n+3−i

= Hi

(2)

+ h3 Hi−2

n+3−i .

+ 1, we have

(3)

(2)

(2)

H n+1 +1 n+1 +1 = H n+1 +1 n+1 +1 + h3 H n+1 −1 n+1 +1 . 2

2

2

2

2

(2)

2

(2)

Since H n+1 −1 n+1 +1 = −1 we can take h3 = H n+1 +1 n+1 +1 to obtain 2

(3)

2

2

2

H n+1 +1 n+1 +1 = 0. Similar as before, this can be used to determine all entries 2

2

(3)

Hi j for which i + j = n + 3 by using (3.1). Now put S4 = toep(1, 0, 0, h4 , 0, . . . , 0) and H (4) = S4T H (3) . From (3.2) we have for i + j > n that (4)

(3)

(3)

Hi j = Hi j + h4 Hi−3 j . (4)

(3)

As before Hi j = Hi j for all i + j ≤ n + 3. For i + j = n + 4 we have from (3.2) that (4) n+4−i

Hi If we take i =

n+1 2

(4)

H n+1 +1 2

+ 1, then j = (3)

n+1 2 +2

= H n+1 +1

(3)

since H n+1 −2 2

2

(3) (3) n+4−i + h4 Hi−3 n+4−i . n+1 2 + 2 and we have

= Hi

(3)

n+1 2 +2

+ h4 H n+1 −2 2

(3)

n+1 2 +2

= H n+1 +1 2

n+1 2 +2

+ h4 ,

(4)

n+1 2 +2

= 1. So, we can choose h4 such that H n+1 +1 2

(4)

n+1 2 +2

= 0.

By repeated application of updates of (3.1), all entries of Hi j for which i + j = n + 4 can now be computed. Now we can continue by induction to finish the proof, where Sˆ = S n+1 · · · S2 S1 .  2

Proof of Theorem 2.6 Part (i) Case 2 with n even. We may assume that A = Jn (1) with respect to the Jordan basis x1 , . . . , xn . Denote H = [Hij ]ni,j=1 (so Hij = Hxj , xi ). Analogously to what has been done in [5] and [12], we already know the following: Hij = 0

when

i + j ≤ n,

and Hij + Hi j+1 + Hi+1 j = 0

for i > 1 and j > 1.

(3.4)

Furthermore, we have from the fact that H is skew-symmetric that all entries on the main diagonal are zero and we have that Hij = −Hji . Let us denote for convenience H n2 n2 +1 := c, where c is complex and c = 0 because of the invertibility of H. By repeated application of (3.4) we have along the main anti-diagonal of H the entries alternating between c and −c, n i.e., Hi n+1−i = (−1) 2 −i · c. This determines all entries Hij for i + j < n + 2. Since H n2 n2 = 0 it follows from (3.4) that H n2 n2 +2 = −c. Continuing in this way gives that for i = 2, . . . , n2 we have Hi n+2−i = c · qi n2 −i+2 , where qij is

Canonical form for H-symplectic matrices

281

defined as in Definition 2.3. This then defines all entries of Hij for i+j < n+3. It is important to note that S T HS preserves the previous properties for H. If we show that there is an invertible S such that S −1 AS = A and the right lower corner block of S T HS is zero, then by repeated application of (3.4) the bottom row of the upper right corner block of S T HS has only entries alternating between −c and c. From this point, again by repeated application of (3.4), one proves that the upper right corner block of S T HS is given by c · Qn . Finally take S = √1c I to finish the proof. It remains to find such an S. We do this by changing the Jordan basis step by step. First we define a new Jordan basis as follows: let z1 = x 1 ,

z2 = x2 ,

z3 = x3 + h1 x1 ,

z4 = x4 + h1 x2 , . . . .

So, in general we have (i)

zj = x j

for

j = 1, 2

and

(i)

zj = xj + h1 xj−2

for

j > 2,

with h1 ∈ C. The superscript is because of the iterative process in our proof. It can easily be verified that this is indeed a Jordan chain. Using this new Jordan basis, we construct a new form for H. Note that the required S must satisfy S −1 AS = A, so nothing is lost in the Jordan form of A. In the first iteration step for finding an appropriate S, we show how to obtain a zero for the entry H n2 +1 n2 +2 . Put S1 = toep(1 h1 · · · 0), then for H (1) = S1T HS1 we have (S1T HS1 )ij = h21 Hi−1 j−1 + h1 Hi−1 j + h1 Hi j−1 + Hij . Hence, for i = (1)

H n +1 2

n 2

n 2 +2

+ 1, j = = h21 H n2 = =

n 2

+ 2 we have that

n 2 +1

+ h1 H n2

n 2 +2

+ h1 H n2 +1

n 2 +1

+ H n2 +1

n 2 +2

ch21 + (−c)h1 + H n2 +1 n2 +2 c(h21 − h1 ) + H n2 +1 n2 +2 . (1)

There always is a solution in order to obtain H n +1 2

n 2 +2

= 0, since h1 ∈ C. (1)

Now that this entry is zero, we can use (3.4) to compute all entries in Hij (1)

for i + j = n + 3 in terms of c. In particular we have H n (1) Using this equation again and the fact that H n +2 n +2 2 2 (1) (1) symmetric) gives us H n +1 n +3 = 0 and H n n +4 = −c. 2 2 2 2

n 2 2 +3

= c from (3.4).

= 0 (since H is skew

In the next step of the algorithm we aim to obtain a zero for the ( n2 + 2, n2 + 3)-th entry of H (1) . For this let S2 = toep(1 0 h2 0 · · · 0), then the (i, j)-th entry of H (2) = S2T H (1) S2 is given by (2)

(1)

(1)

(1)

(1)

Hij = (S2T H (1) S2 )ij = h22 Hi−2 j−2 + h2 Hi−2 j + h2 Hi j−2 + Hij .

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G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran

For i =

n 2 (2)

+ 2 and j =

H n +2 2

n 2 +3

n 2

+ 3 we have

(1)

= h22 H n

n 2 2 +1

(1)

+ h2 H n

n 2 2 +3

= ch22 + ch2 + h2 (0) + =

c(h22

+ h2 ) +

(1) H n +2 2

(1)

+ h2 H n +2 2

(1) H n +2 2 n 2 +3

n 2 +1

(1)

+ H n +2 2

n 2 +3

. (2)

This again is solvable with h2 ∈ C and thus we have H n +2 2 (3.4) a number of times we have: (2)

H n +1 2

n 2 +3

(2)

= H n +1 2

n 2 +4

=0

n 2 +3

and

(2)

Hn

n 2 2 +5

n 2 +3

= 0. Using

= c.

(2)

Knowing these entries, we can determine all entries in Hij for i + j = n + 4. Continuing in this way by induction, we obtain a sequence of matrices (k) Sk , k = 1, . . . n2 in which each iteration simplifies the form of Hij leaving all previously simplified entries unchanged. Setting S = S n2 S n2 −1 . . . S2 S1 we get the full lower right corner block of S T HS zero. The upper right corner block will then be the matrix c · Qn . Finally we can scale by setting S = √1 I. |c|

Since H = −H T , this determines the whole matrix H.



3.2. Eigenvalue −1 Here we only note that the difference with λ = 1 lies in what we called

the 0 H12 magic wand formula. If A = Jn (−1) ⊕ Jn (−1) and H = , then T −H12 0 T T we see that A HA = H implies that Jn (−1) H12 Jn (−1) = H12 must hold. If we let the (i, j)-th entry of H12 be denoted as Hi j , then the last expression gives us our magic wand formula, i.e., Hi j − Hi j+1 − Hi+1 j = 0. The proof is analogous to the case when λ = 1 for both n odd and n even, with the exception that Pn and Qn now depend on λ = −1. 3.3. Eigenvalues not ±1 We consider the case where the indecomposable block is of the form 1 A = Jn (λ) ⊕ Jn ( ). λ From the results of Section 3 in [14] we may assume that the corresponding form for the skew-symmetric matrix H is given by

0 H12 H= , T −H12 0 for some invertible n × n matrix H12 , which does not have any additional structure. Writing out AT HA = H, we see that this results in 1 Jn (λ)T H12 Jn ( ) = H12 . λ

Canonical form for H-symplectic matrices

283

Now compare Section 4 in [8]. The formula above is the same as formula (38) there. As a consequence, the canonical form for H12 is given by Theorem 4.1 in [8] and its proof. This leads to statement (iii) in Theorem 2.6 for this case.

4. The real case In this section we consider the real case. The case where the eigenvalues are ±1 is the same as in the complex case, the case of a real pair of eigenvalues λ, λ1 will also be the same as in the complex case. We need to consider only the non-real eigenvalues.

α β So, let γ = , with β = 0, and set −β α ⎡ ⎤ γ I2 ⎢ ⎥ .. .. ⎢ ⎥ . . ⎢ ⎥. Jn (γ) = ⎢ ⎥ . .. I ⎦ ⎣ 2 γ 4.1. Complex non-unimodular eigenvalues Proof of Theorem 2.11 Part (v). First we consider non-real, non-unimodular eigenvalues, so α2 + β 2 = 1. By Corollary 3.5 in [14] we may restrict ourselves to the case where

0 H12 A = Jn (γ) ⊕ Jn (γ −1 ), H= T −H12 0 for some invertible 2n × 2n matrix H12 . Then AT HA = H is equivalent to Jn (γ)T H12 Jn (γ −1 ) = H12 . This, however, is exactly the same as equation (51) in [8]. That means that the canonical form for this case can be deduced as in Theorem 5.1 in [8].  4.2. Complex unimodular eigenvalues In this subsection we first prove three preliminary lemmas which are needed for the proof of Case (iv) of Theorem 2.11. Assume that H = −H T is invertible, and Jn (γ)T HJn (γ) = H, where γ is as above, but with α2 + β 2 = 1, so that γ T = γ −1 . We can rely on quite a number of results of Section 2 in [8], compare for instance Lemma 2.6 there.  We n consider A = Jn (γ) with β = 0 and α2 + β 2 = 1. We denote H = Hi,j i,j=1 , where each Hi,j is a 2 × 2 T matrix, and Hj,i = −Hi,j . We keep the convention that comma separated subindices indicate that we are working with 2 × 2 blocks. This convention was used by the authors in earlier papers [5, 8, 12] as well.

284

G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran Comparing the corresponding (i, j)-th blocks of AT HA and H, yields (Jn (γ)T HJn (γ))i,j ⎧ T γ H1,1 γ, i = 1, j ⎪ ⎪ ⎪ ⎨γ T H T i = 1, j 1,j−1 + γ H1,j γ, = T ⎪ H γ + γ H γ, i > 1, j i−1,1 i,1 ⎪ ⎪ ⎩ T T Hi−1,j−1 + γ Hi,j−1 + Hi−1,j γ + γ Hi,j γ, i > 1, j

= 1, > 1, = 1, > 1,

= Hi,j .

(4.1)

[8]

the definition of E, that

is, E = {aI 2 + bH

0 | a, b ∈ R} 1 1 0 0 1 and also K0 = and K1 = . 0 0 −1 1 0

h h12 For a 2 × 2 matrix H = 11 we have h21 h22 2



−β αβ −αβ −β 2 γ T Hγ = H + (h11 − h22 ) + (h + h ) (4.2) 12 21 αβ β 2 −β 2 αβ

Recall from 0 with H0 = −1

= H + β(h11 − h22 )(−βK0 + αK1 ) + β(h12 + h21 )(−αK0 − βK1 ). (4.3) Observe, if F ∈ E, then always γ T F γ = F , as E is a commutative set of matrices and γ T γ = I2 . Lemma 4.1. For all i, j = 1, . . . , n we have Hi,j ∈ E. T Proof. Note that H1,1 ∈ E as H1,1 = −H1,1 . Now let i = 1, j > 1, then by (4.1) γ T H1,j γ − H1,j = −γ T H1,j−1 . Now suppose that we have already shown that H1,j−1 ∈ E, then also γ T H1,j−1 ∈ E, so γ T H1,j γ − H1,j ∈ E. From (4.2) it follows that

−β(h11 − h22 ) − α(h12 + h21 ) = 0, α(h11 − h22 ) − β(h12 + h21 ) = 0, or equivalently



−β α

−α −β



h11 − h22 = 0. h12 + h21

As α2 + β 2 = 1 it follows that h11 − h22 = 0 and h12 + h21 = 0, so that H1,j ∈ E. By induction we see that H1,j ∈ E for all j = 1, . . . , n, and likewise that also Hi,1 ∈ E for all i = 1, . . . , n. Using (4.1) for i > 1, j > 1 we have γ T Hi,j γ − Hi,j = −(Hi−1,j−1 + γ T Hi,j−1 + Hi−1,j γ). Assuming that Hi,j−1 , Hi−1,j and Hi−1,j−1 are all in E, we obtain that γ T Hi,j γ − Hi,j ∈ E, and as we have seen above this implies that Hi,j ∈ E. By induction we have proved the lemma.  The lemma allows us to state the following important result.

Canonical form for H-symplectic matrices

285

Lemma 4.2. For i > 1 and j > 1 we have Hi−1,j−1 + γ T Hi,j−1 + Hi−1,j γ = 0.

(4.4)

Proof. Since by the previous lemma Hi,j ∈ E for all i, j, we have that γ T Hi,j γ − Hi,j = 0 for all i, j. Then (4.1) becomes (4.4).  Next, we show that H has a (block) triangular form. Lemma 4.3. For i + j ≤ n we have Hi,j = 0. Proof. Since H1,j ∈ E we have from (4.1) that γ T H1,j−1 = 0, and by invertibility of γ it follows that H1,j−1 = 0 for j = 2, . . . , n. Likewise Hi−1,1 = 0 for i = 2, . . . , n. Now consider i > 1 and j > 1, and suppose that we have already shown that Hi−1,j−1 = 0. By (4.4) we have 0 = γ T Hi,j−1 + Hi−1,j γ, which implies Hi,j−1 = −Hi−1,j γ 2 ,

(4.5)

using again the commutativity of E. Since we know that H1,j = 0 for j = 1, . . . , n − 1, repeated application of (4.5) gives Hi,j = 0 for i + j ≤ n.  From here on we have to distinguish between n even and n odd. 4.2.1. The case when n is odd. Proof of Theorem 2.11, Part (iv), Case 1 with n odd. If n is odd, then H n+1 , n+1 is skew symmetric, so H n+1 , n+1 = cH0 for some real number c. 2 2 2 2 Then by (4.5) we have H n−1 , n+3 = −cH0 (γ T )2 , and H n−3 , n+5 = cH0 (γ T )4 , 2 2 2 2 and so on. Up to this point we have derived results that hold for any Jordan basis. The next step is to construct a special Jordan basis such that H is in the canonical form. Equivalently, we find a matrix S such that S −1 Jn (γ)S = Jn (γ), while S T HS is in the canonical form. This will be done step by step. Ultimately we shall show that it is possible to take S such that Hi,j = 0 for n+1 n+3 i, j > n+1 2 . In addition, the matrix S is such that the block entry H 2 , 2 is equal to 12 cγ T H0 . Note that the latter is in accordance with (4.4). We follow a similar line of reasoning as in the proof in Section 2.3 in [8]. For an invertible block upper triangular Toeplitz matrix S with block entries in E we have that S −1 Jn (γ)S = Jn (γ). We shall use such matrices to construct a canonical form via S T HS. First, let S1 = toep(h1 , 0, · · · , 0), with 0 = h1 ∈ E. Let H (1) = S1T HS1 . Then (S1T HS1 )i,j = hT1 Hi,j h1 = hT1 h1 Hi,j = dHi,j for some positive real n+1 number d. So we see that we can scale the entry in the ( n+1 2 , 2 )-position to εH0 with ε = ±1. After that we can extract ε from every entry in H (1) , and put it in front of the matrix. This way we may assume that H n+1 , n+1 = H0 . 2 2 From now on we shall assume that this is the case. n+3 Next, consider the entry at position ( n+1 2 , 2 ). By (4.4) we have (1)

(1)

H0 + γ T H n+3 , n+1 + H n+1 , n+3 γ = 0. 2

2

2

2

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G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran

 T (1) (1) Since H n+3 , n+1 = − H n+1 , n+3 , this amounts to 2

2

2

2



T

(1)

H0 − γ T H n+1 , n+3 2

(1)

+ H n+1 , n+3 γ = 0.

2

2

(1)

One easily checks that since H n+1 2

n+3 2

2

is in E, that for some real number d

1 (1) H n+1 , n+3 = − γ T H0 + dγ T . (4.6) 2 2 2 Let S2 = toep(I2 , h2 , 0, · · · , 0) with 0 = h2 ∈ E and put H (2) = S2T H (1) S2 . Then (2)

(1)

(1)

(1)

(1)

Hi,j = Hi,j + Hi,j−1 h2 + hT2 Hi−1,j + hT2 Hi−1,j−1 h2 . (2)

(1)

For i + j ≤ n + 1 we have Hi,j = Hi,j . Indeed, in this case by Lemma 4.3, (1)

(1)

(1)

Hi,j−1 = Hi−1,j = Hi−1,j−1 = 0. For i = 4.2 and 4.3 that (2)

(1)

n+1 2 ,j

=

n+3 2

we have from Lemma

(1)

(1)

H n+1 , n+3 = H n+1 , n+3 + H n+1 , n+1 h2 + hT2 H n−1 , n+3 2

2

2

2

2

2

2

2

(1)

= H n+1 , n+3 + H0 h2 − hT2 H0 (γ T )2 2

2

(1)

= H n+1 , n+3 + γ T (H0 (γh2 − hT2 γ T )), 2

2

where in the last equality we used the commutativity of E and γ T γ = I2 . Now observe that H0 (γh2 − hT2 γ T ) is a scalar multiple of the identity matrix depending on h2 ∈ E. This implies that by choosing h2 appropriately we can obtain using (4.6) 1 (2) H n+1 , n+3 = − γ T H0 . 2 2 2 In the next steps we shall show that we can choose the Jordan basis so that the entries Hi,j with both i > n+1 and j > n+1 are all zero. For 2 2 this it suffices, by (4.4), to show that S can be chosen so that Hi,i = 0 and Hi,i+1 = 0 for i > n+1 2 . As a first step, take S3 = toep(I2 , 0, h3 , 0 · · · , 0) with h3 ∈ E, and let (3) (2) H (3) = S3T H (2) S3 . Then Hi,j = Hi,j for i + j ≤ n + 2, as one easily checks, and (3)

(2)

(2)

(2)

(2)

H n+3 , n+3 = H n+3 , n+3 + hT3 H n−1 , n+3 + H n+3 , n−1 + hT3 H n−1 , n−1 h3 2

2

2

2

2

2

2

2

2

2

(2)

= H n+3 , n+3 − hT3 H0 (γ T )2 − γ 2 H0 h3 . 2

(2)

2

(3)

Now H n+3 , n+3 and H n+3 , n+3 are both skew symmetric, hence a real multiple 2

2

2

2

of H0 . Since hT3 (γ T )2 + γ 2 h3 is a multiple of the identity it is clear that we (3) can choose h3 such that H n+3 , n+3 = 0. 2

2

(3)

(3)

By (4.4), and the fact that H n+5 , n+3 = −(H n+3 , n+5 )T we obtain that 2

(3)

2

2

(3)

2

−γ (H n+3 , n+5 ) + H n+3 , n+5 γ = 0. T

2

T

2

2

2

Canonical form for H-symplectic matrices

287

(3)

In turn, this implies that H n+3 , n+5 is a real multiple of γ T . 2

2

As a second step, take S4 = toep(I2 , 0, 0, h4 , 0 · · · , 0) with h4 ∈ E, and (4) (3) let H (4) = S4T H (3) S4 . Then Hi,j = Hi,j for i + j ≤ n + 3, as one easily checks, and (4)

(3)

(3)

(3)

(3)

H n+3 , n+5 = H n+3 , n+5 + hT4 H n−3 , n+5 + H n+3 , n−1 h4 + hT4 H n−3 , n−1 h4 2

2

2

2

(3)

= H n+3 , n+5 + 2

2

2

2

2

(3) hT4 H n−3 , n+5 2

(3)

2

2

2

2

(3)

+ H n+3 , n−1 h4 . 2

2

(3)

Since H n−3 , n+5 = H0 (γ T )4 and H n+3 , n−1 = −H0 γ 2 (as noted in the begin2

2

2

2

ning of the proof) the latter equation becomes (4)

(3)

H n+3 , n+5 = H n+3 , n+5 + hT4 H0 (γ T )4 − H0 γ 2 h4 2

2

2

2

(3)

= H n+3 , n+5 + γ T H0 (hT4 (γ T )3 − γ 3 h4 ). 2

2

Now H0 (hT4 (γ T )3 − γ 3 h4 ) is a real multiple of I2 and as we already know (3) that H n+3 , n+5 is a real multiple of γ T , it follows that we can take h4 so that 2

(4)

2

H n+3 , n+5 = 0. 2

2

We can now continue in this manner by an induction argument (compare [8], Section 2) and by setting S = S n+1 · · · S2 S1 .  2

4.2.2. The case when n is even. Proof of Theorem 2.11, Part (iv), Case 2 with n even. If n is even, then by (4.5) H n2 +1, n2 = −H n2 , n2 +1 γ 2 . Since also H n2 +1, n2 = −H Tn , n +1 we have 2 2 H Tn , n +1 = H n2 , n2 +1 γ 2 . A straightforward computation then gives H n2 , n2 +1 = 2 2 cγ T for some real number c. Then H n2 −1, n2 +2 = −c(γ T )3 , H n2 −2, n2 +3 = c(γ T )5 , and so on. Up to this stage we have derived a ”form” that holds for a general Jordan basis. The next step is to construct a special Jordan basis such that the ”form” for H is transformed into the canonical form given in Theorem 2.11. Equivalently, we find step by step a matrix S such that S −1 Jn (γ)S = Jn (γ), while S T HS is in the canonical form. We shall show that it is possible to take S such that Hi,j = 0 for i, j > n2 . First we take S1 = toep(h1 , 0, · · · , 0), with 0 = h1 ∈ E. Then H (1) = has block entries equal to hT1 h1 times the corresponding entry in H. T Since h1 h1 is a positive multiple of the identity this means we can arrange (1) for H n , n +1 = εγ T , with ε = ±1. Taking out ε from all the entries, we may S1T HS1 2

2

(1)

assume that H n , n +1 = γ T , and this will be assumed from now on. 2

2

288

G.J. Groenewald, D.B. Janse van Rensburg and A.C.M. Ran

Take S2 = toep(I2 , h2 , 0 · · · , 0) with h2 ∈ E, and put H (2) = S2T H (1) S2 . Then (2)

(1)

(1)

(1)

2

2

(1)

H n +1, n +1 = H n +1, n +1 + hT2 H n , n +1 + H n +1, n h2 + hT2 H n , n h2 2

2

2

2

2

2

2

2

(1)

= H n +1, n +1 + hT2 γ T − γh2 . 2

2

(1)

hT2 γ T −γh2

Since is a multiple of H0 , and since H n +1, n +1 is a skew-symmetric 2 2 2 × 2 matrix, and hence also is a multiple of H0 , it is possible to choose h2 (2) so that H n +1, n +1 = 0. 2

2

Next, take S3 = toep(I2 , 0, h3 , 0, · · · , 0) and put H (3) = S3T H (2) S3 . Then (3)

(2)

(2)

(2)

(2)

H n +1, n +2 = H n +1, n +2 + hT3 H n −1, n +2 + H n +1, n h3 + hT3 H n , n h3 2

2

2

=

2

2

(2) H n +1, n +2 2 2

−

2

hT3 (γ T )3

2

2

2

2

− γh3

(2)

= H n +1, n +2 − γ T (hT3 (γ T )2 + γ 2 h3 ). 2

2

Now by (4.4) we have (2)

(2)

0 = H n +1, n +2 γ + γ T H n +2, n +1 , 2

and since

(2) H n +2, n +1 2 2

=

2

2

(2) −(H n +1, n +2 )T 2 2

2

this amounts to

(2)

(2)

0 = H n +1, n +2 γ − γ T (H n +1, n +2 )T . 2

This implies that

(2) H n +1, n +2 2 2

2

= dγ

2

T

2

for some real number d. Then it follows (3)

that it is possible to choose h3 such that H n +1, n +2 = 0. 2 2 We can now continue and finish the argument by induction showing that Hi,j = 0 for i, j > n2 . This in turn implies by repeated application of (4.4) that the entries in the right upper corner block of H have the form as described in the theorem by setting S = S1 S2 . . . S n2 . 

5. The complex H-symplectic case To complete the paper, let us consider finally the case where A and H are complex matrices, and, with H = −H ∗ invertible and A∗ HA = H. As we observed in the introduction, the matrix A is then iH-unitary, and iH is Hermitian. Thus we can apply the result of [8], Theorem 6.1, to arrive at the following theorem. Theorem 5.1. Let H be a complex skew-Hermitian invertible matrix, and let A be complex H-symplectic, so A∗ HA = H. Then the pair (A, H) can be decomposed as p p   Al , S ∗ HS = Hl , S −1 AS = l=1

l=1

where the pairs (Al , Hl ) have one of the following forms with n depending on l

Canonical form for H-symplectic matrices ¯ −1 } with |λ| = (i) σ(Al ) = {λ, λ  1, and ¯ −1 ), Al = Jn (λ) ⊕ Jn (λ

Hl =

0 ∗ −H12

289

H12 , 0

where H12 has one of the following two forms depending on whether n is odd or even:

¯ Zn Pn (λ) Case 1: H12 = −i when n is odd, ¯ −1 )T Pn ( λ 0

¯ 0 Qn (λ) Case 2: H12 = −i when n is even. ¯ −2 Qn (λ ¯ −1 )T −λ 0 (ii) σ(Al ) = {λ} with |λ| = 1, and the pair (Al , Hl ) has one of the following two forms:

¯ Zn Pn (λ) Case 1: (Jn (λ), −iε ) with ε = ±1 and n is odd, Pn (λ)T 0

¯ n (λ) ¯ 0 λQ Case 2: (Jn (λ), ε ) with ε = ±1 and n is even. −λQn (λ)T 0

References [1] Y.-H. Au-Yeung, C.K. Li, L. Rodman, H-unitary and Lorentz matrices, SIAM J. Matrix Anal. Appl. 25 (2004), 1140–1162. [2] T.Ya. Azizov, I.S. Iohvidov, Linear operators in spaces with an indefinite metric, John Wiley and Sons, Chicester, 1989 [Russian original 1979]. [3] J. Bogn´ ar, Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 78, Springer-Verlag, New York–Heidelberg, 1974. [4] A. Ferrante, B.C. Levy, Canonical form of symplectic matrix pencils, Linear Algebra and its Applications 274 (1998), 259–300. [5] J.H. Fourie, G. Groenewald, D.B. Janse van Rensburg, A.C.M. Ran, Simple forms and invariant subspaces of H-expansive matrices, Linear Algebra and its Applications 470 (2015), 300–340. [6] I. Gohberg, P. Lancaster, L. Rodman, Matrices and Indefinite Scalar Products, Oper. Theory: Adv. and Appl., vol. 8, Birkh¨ auser Verlag, Basel, 1983. [7] I. Gohberg, P. Lancaster, L. Rodman, Indefinite Linear Algebra and Applications, Birkh¨ auser Verlag, Basel, 2005. [8] G.J. Groenewald, D.B. Janse van Rensburg, A.C.M. Ran, A canonical form for H-unitary matrices, Operators and Matrices 10 no. 4 (2016), 739–783. [9] J. Gutt, Normal forms for symplectic matrices, Port. Math. 71 (2014), 109– 139. [10] B. Huppert, Isometrien von Vektorr¨ aumen 1, Arch. Math. 35 (1980), 164–176. [11] I.S. Iohvidov, M.G. Krein, H. Langer, Introduction to the spectral theory of operators in spaces with an indefinite metric, Mathematical Research, vol. 9, Akademie-Verlag, Berlin, 1982.

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[12] D.B. Janse van Rensburg, Structured matrices in indefinite inner product spaces: simple forms, invariant subspaces and rank-one perturbations, Ph.D. thesis, North-West University, Potchefstroom, 2012, http://www.nwu.ac.za/content/mam-personnel. [13] I. Krupnik, P. Lancaster, H-selfadjoint and H-unitary matrix pencils, SIAM J. Matrix Anal. Appl. 19 (1998), 307–324. [14] Chr. Mehl, On classification of normal matrices in indefinite inner product spaces, Electronic Journal of Linear Algebra 15 (2006), 50–83. [15] Chr. Mehl, Essential decomposition of polynomially normal matrices in real indefinite inner product spaces, Electronic Journal of Linear Algebra 15 (2006), 84–106. [16] Chr. Mehl, V. Mehrmann, A.C.M Ran and L. Rodman, Eigenvalue perturbation theory of structured matrices under generic structured rank one perturbations: Symplectic, orthogonal, and unitary matrices, BIT Numer. Math. 54 (2014), 219–255. [17] A.J. Laub, K. Meyer, Canonical forms for symplectic and Hamiltonian matrices, Celestial Mechanics 9 (1974), 213–238. [18] W.-W. Lin, V. Mehrmann, H. Xu, Canonical forms for Hamiltonian and symplectic matrices and pencils, Linear Algebra and its Applications 302–303 (1999), 469–533. [19] V. Mehrman, H. Xu, Structured Jordan canonical forms for structured matrices that are Hermitian, skew Hermitian or unitary with respect to indefinite inner products, Electronic Journal of Linear Algebra 5 (1999), 67–103. [20] J. Milnor, On Isometries of Inner Product Spaces, Inventiones Math. 8 (1969), 83–97. [21] L. Rodman, Similarity vs unitary similarity: Complex and real indefinite inner products, Linear Algebra and its Applications 416 (2006), 945–1009. [22] V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (1988), 481–501 (Translation from Russian). [23] V.V. Sergeichuk, Canonical matrices of isometric operators on indefinite inner product spaces, Linear Algebra and its Applications 428 (2008), 154–192. [24] L. Smith, Linear Algebra, Springer-Verlag, New York, 1984. G.J. Groenewald and D.B. Janse van Rensburg Department of Mathematics, Unit for BMI, North-West University Potchefstroom, 2531, South Africa e-mail: [email protected] [email protected] A.C.M. Ran Department of Mathematics, FEW, VU university Amsterdam De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands and Unit for BMI, North-West University Potchefstroom, South Africa e-mail: [email protected]

A note on the Fredholm theory of singular integral operators with Cauchy and Mellin kernels Peter Junghanns and Robert Kaiser Abstract. A result on the necessity and sufficiency of conditions for the Fredholmness of singular integral operators in weighted Lp -spaces on the interval (0, 1) of the real axis R is proved together with a respective Fredholm index formula. These operators are composed by the Cauchy singular integral operator, Mellin operators and operators of multiplication by piecewise continuous functions. Mathematics Subject Classification (2010). Primary 45E05; Secondary 45E10. Keywords. Singular integral operators, Cauchy kernel, Mellin kernel, Fredholm theory.

1. Introduction This paper is devoted to the Fredholm theory of integral operators, which are the sum of a Cauchy singular integral operator and Mellin operators. More precisely, these operators are of the form     1 x u(y) dy b(x) 1 u(y) dy + c+ (x) k+ a(x)u(x) + πi 0 y − x y y 0 (1.1)    1 1 − x u(y) dy + c− (x) k− 1−y 1−y 0 and are considerd in weighted Lp -spaces, where a, b, c± : [−1, 1] −→ C are piecewise continuous functions and k± : (0, ∞) −→ C are continuous functions. We are interested in necessary and sufficient conditions for the Fredholmness of the operator as well as an index formula based on the winding number of a certain closed curve in C associated with the operator (1.1). To the knowledge of the authors, proofs of such results are only presented in special cases in the literature. Of course, the case c± ≡ 0 is well known due to © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_12

291

292

P. Junghanns and R. Kaiser

the famous book of Gohberg and Krupnik [4, 5]. Moreover, in [2] J. Elschner and in [9] J.E. Lewis and C. Parenti considered the case of C∞ -functions a(x) and b(x) for p = 2 and arbitrary 1 < p < ∞ , respectively, and R. Duduchava [1] presented results for a, b ∈ C and special Mellin kernel functions, namely k± (t) =

tw , (1 + t)k+1

0 ≤ Re w ≤ k ∈ {0, 1, 2, . . .}

and linear combinations of them. In [8], a general result for (1.1) is formulated without proof. The aim of the present paper is to close this gap. In Section 2, we collect a series of properties of the Mellin transformation, which are used to prove boundedness and Fredholm properties of the operators under consideration in Sections 3.1 and 3.2, respectively. The proof of the main result (Theorem 3.27) is based on the above mentioned results and ideas presented in [4, 5, 2, 1] as well as the application of the local principle of Gohberg and Krupnik. For a Banach space X , by L(X) we denote the Banach algebra of all linear and bounded operators A : X −→ X , equipped with the norm AL(X) = sup {AxX : x ∈ X, xX ≤ 1} , and by K(X) the (two-sided and closed) ideal in L(X) of all compact operators. The subset of L(X) of all Fredholm operators is denoted by Φ(X) . Remember that an operator A ∈ L(X) is called Fredholm if and only if the image A(X) is closed in X and if the null space N(A) = {x ∈ X : Ax = 0} of A and the null space of the adjoint operator A∗ : X∗ −→ X∗ are finite dimensional. For A ∈ Φ(X) , the number ind (A) := dim N(A) − dim N(A∗ ) is called the (Fredholm) index of A .

2. Properties of the Mellin transformation For z ∈ C and a measurable function f : (0, ∞) −→ C, for which tz−1 f (t) is integrable on each compact subinterval of (0, ∞), the Mellin transform f(z) is defined as  R  f (z) = lim tz−1 f (t) dt , (2.1) R→∞

R−1

if this limit exists. Under weaker conditions on f we have to modify this definition. To this end, we start with some well known properties of the Fourier transform which is well defined for u ∈ L1 (R) by  ∞ (Fu) (η) := e−iηt u(t) dt , η ∈ R . −∞

At first we state the theorem of Riemann-Lebesgue, which is fundamental in the theory of Fourier integrals. Lemma 2.1. For u ∈ L1 (R) , we have Fu ∈ C(R) and limη→±∞ (Fu) (η) = 0 .

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293

Proof. By using Lebesgue’s dominated convergence theorem we easily get the continuity of Fu. The limit relation lim (Fu) (η) = 0 is an immediate η→±∞



consequence of [10, Section 1.8, Theorem 1].

Lemma 2.2. If k ∈ N and u, v ∈ L (R) , where v(t) = t u(t) , then Fu is k times differentiable and (Fu)(k) (η) = (−i)k (Fv)(η) , η ∈ R . k

1

Proof. We have (Fu)(η + h) − (Fu)(η) = h with g(h, t) =



∞

g(h, t)e−iηt tu(t) dt

(2.2)

−∞

e−iht − 1 . Obviously, ht sup {|g(h, t)| : (h, t) ∈ [0, 1] × R} < ∞ .

Consequently, we can apply Lebesgue’s dominated convergence theorem to (2.2) and get  ∞  ∞  −iηt (Fu) (η) = lim g(h, t)e tu(t) dt = −i e−iηt tu(t) dt . h→0

−∞

−∞

Now, the proof can be finished by induction. Lemma 2.3. Let u ∈ L1 (R) and Fu ∈ L1 (R). Then there holds  ∞ 1 u(t) = eiηt (Fu)(η) dη 2π −∞



(2.3)

for almost every t ∈ R. Moreover, if u is additionally continuous on R, then (2.3) holds for all t ∈ R. Proof. In view of [10, Section 1.24, Theorem 22], the relation  t  λ itη 1 e −1 u(s) ds = lim (Fu)(η) dη λ→∞ 2π iη 0 −λ  ∞ itη e −1 F u∈L1 (R) 1 = (Fu)(η) dη 2π −∞ iη is true for all t ∈ R. Since u belongs to L1 (R), the left-hand side of this equation is differentiable for almost every t ∈ R. Moreover, due to the fact that Fu ∈ L1 (R), we can again make use of Lebesgue’s dominated convergence theorem and get   ∞ 1 d ∞ eitη − 1 1 d eitη − 1 u(t) = (Fu)(η) dη = (Fu)(η) dη 2π dt −∞ iη 2π −∞ dt iη  ∞ 1 = eiηt (Fu)(η) dη 2π −∞ for almost every t ∈ R. If u is additionally continuous on R, then (2.3) holds for all t ∈ R, since both the left-hand side and the right-hand side of relation (2.3) are continuous functions in t . 

294

P. Junghanns and R. Kaiser For 0 < R < ∞ and u ∈ L1 (−R, R) , set  R (FR u) (η) = e−iηt u(t) dt , −R

η ∈ R.

Lemma 2.4. Let u ∈ L1 (R) ∩ L2 (R). Then Fu ∈ L2 (R) and FR u converges for R −→ ∞ in L2 (R) to Fu . Proof. By [10, Section 3.1, Theorem 48], we have that FR u converges in the  , belonging to L2 (R) . Morenorm of the space L2 (R) to a function, say Fu over, the second method of the proof of [10, Theorem 48] yields that the equation (see [10, Section 3.3])  ∞ e−itη − 1 d  f (t) dt (Fu)(η) = dη −∞ −iη holds for almost all η ∈ R . Since f belongs to L1 (R) we can again change  = Fu almost everywhere. Hence integration and differentiation and get Fu the lemma is proved.  Lemma 2.5 ([10], Theorem 74). Let u ∈ L2 (R). Then FR u converges for R −→ ∞ in L2 (R). If we denote this limit by Fu, then the so defined operator F : L2 (R) −→ L2 (R) is linear and bounded. Moreover, for v = Fu , we have − u = F − v, where F − v = limR→∞ FR v in L2 (R) and  R  −  1 FR v (t) = eiηt v(η) dη , t ∈ R . 2π −R According to Lemma 2.4, we will call the operator F : L2 (R) −→ L2 (R), which is defined by Lemma 2.5, Fourier transformation and the function Fu 1 Fourier transform of u ∈ L2 (R). Note that (F − v)(t) = 2π (Fv)(−t) holds, − 1 such that F : L (R) −→ C(R) is also well defined Lemma 2.6 ([10], Section 3.1, Theorems 48, 49). The above defined Fourier transformation F : L2 (R) −→ L2 (R) is an isomorphism, where uL2 (R) = 1 2 2π FuL2 (R) for all u ∈ L (R) . Let ξ ∈ R, p ∈ [1, ∞) , and R+ = (0, ∞). By Lpξ := Lpξ (R+ ) we denote the weighted Lp -space defined by the norm  ∞  p1 f ξ,p,∼ := |f (t)|p tξ dt . (2.4) 0

Moreover, for a function f : R+ −→ C , we define (Tξ f ) (t) = eξ t f (et ), t ∈ R . The operator Tξ : Lpξp−1 −→ Lp (R) is an isometric isomorphism, where the inverse is given by (Tξ−1 f )(t) = t−ξ f (ln t) . For f ∈ L1ξ−1 and η ∈ R , we have  ∞ tξ+iη−1 f (t) dt = 2π(F − Tξ f )(η) = (FTξ f ) (−η) (2.5) f(ξ + iη) = 0

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295

and, in view of Lemma 2.1, f(ξ + i·) ∈ C(R) . Let Γξ := {z ∈ C : Re z = ξ} . By Lp (Γξ ) we denote the Lp -space given by the norm  ∞  p1 p gΓξ ,p = |g(ξ + iη)| dη . −∞

For a measurable function g : Γξ −→ C, for which g is integrable   on− each interval of the form [ξ −iR, ξ +iR], R > 0, and for t ∈ R , we define Mξ g (t) by  ξ+iR  R 1 1 −z (M− g)(t) = lim t g(z) dz = lim t−(ξ+iη) g(ξ + iη) dη , ξ R→∞ 2πi ξ−iR R→∞ 2π −R (2.6) if this limit exists. For g ∈ L1 (Γξ ) and t ∈ R+ , we have  t−ξ R (M− g)(t) = lim g(ξ + iη)e−iη(ln t) dη ξ R→∞ 2π −R    (2.7) = (2π)−1 lim Tξ−1 FR g(ξ + i·) (t) R→∞



  = (2π)−1 Tξ−1 F g(ξ + i·) (t) . : ; Let C∞ (Γξ ) := u ∈ C(Γξ ) : lim|η|→∞ u(ξ + iη) = 0 . With the help of the equations (2.5) and (2.7) in combination with Lemma 2.1 and Lemma 2.3, we can state the following two lemmas. Lemma 2.7. For ξ ∈ R , the Mellin transformation Mξ : L1ξ−1 −→ C∞ (Γξ ) , f → f as well as the operator M− : L1 (Γξ ) −→ C(R+ ) are well defined. ξ

Lemma 2.8. For ξ ∈ R and f ∈ L1ξ−1 with f ∈ L1 (Γξ ) , there holds      −   f (t) = 2π)−1 Tξ−1 F (2π)F − Tξ f (t) = M− ξ Mξ f (t) = Mξ f (t)

(2.8)

for almost every t ∈ R . Moreover, if f is additionally continuous on R+ , then (2.8) holds for all t ∈ R+ . +

In view of Lemma 2.5 together with relation (2.5), we can define the Mellin transformation on the space L22ξ−1 by  x − Mξ f := 2πF − Tξ f = 2π lim FR Tξ f = lim tξ+i ·−1 f (t) dt , x−→∞

R→∞

x−1

where the limit is taken in the space L (R). The operator M− ξ can also be defined on the space L2 (Γξ ), namely by (cf. (2.7)) 2

−1 −1 M− Tξ Fg = (2π)−1 lim Tξ−1 FR g , ξ g := (2π) R→∞

where the limit is taken in the space L22ξ−1 (R+ ) . Applying Lemma 2.6, we can formulate the following lemma.   Lemma 2.9. For ξ ∈ R , the Mellin transformation Mξ : L2ξ −→ L2 Γ 1+ξ is − an isomorphism, where the inverse mapping is given by M−1 ξ = Mξ .

2

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Lemma 2.10. Let n ∈ N , ρ ∈ R , and f ∈ Cn (R+ ) with f () ∈ L2ρ+2

for  = 0, 1, . . . , n .

Then, there hold the relations lim f () (t) t

t−→0

1+ρ 2 +

=0

lim f () (t) t

and

1+ρ 2 +

t−→∞

=0

(2.9)

for  = 0, . . . , n − 1. Moreover, for f (t) = t f () (t) , g = f () , and  = 0, 1, . . . , n , we have f (z) = (−1) z(z + 1) · · · (z +  − 1)f(z)

(2.10)

for a.e. z ∈ Γ 1+ρ , and 2

g (z) = (−1) (z − 1) · · · (z −  + 1)(z − )f(z − )

(2.11)

for a.e. z ∈ Γ 1+ρ + . 2

() Proof. Let  ∈ {0, . . . , n − 1} , a = 1+ρ (t)t+a . Then, g ∈ 2 , and g(t) = f 2  2 L−1 and g ∈ L1 . Because of the relation  x [g(x)]2 = 2 g(t)g  (t) dt + [g(1)]2 1

and the estimate



∞ 0

|g(t)g  (t)| dt ≤ g−1,2,∼ g  1,2,∼ ,

the value [g(x)]2 converges if x goes to zero or to infinity. These limits vanish, since g 2 belongs to the space L1−1 . We immediately get limt−→0 g(t) = 0 as well as limt−→∞ g(t) = 0 , and (2.9) is proved. Now, let z ∈ Γa , ψ (z) = z +  − 1 , and χm (z) =

1,

Im z ∈ [−m, m],

0,

otherwise,

for m ∈ N. For x > 0 and  ∈ {1, . . . , n} , we have  x f (t)tz−1 dt = f (−1) (x)xz+−1 − f (−1) (x−1 )x−z−+1 x−1

 − (z +  − 1)

(2.12)

x

f

(−1)

(t)t

−1 z−1

t

dt .

x−1

Let’s set I1 (x, z) := f (−1) (x)xz+−1 − f (−1) (x−1 )x−z−+1 and

 I2 (x, z) := −(z +  − 1)

x

f (−1) (t)t−1 tz−1 dt x−1

and let x tend to ∞ . In view of Lemma 2.9, the left-hand side of (2.12) converges in L2 (Γa ) to f ∈ L2 (Γa ). Moreover, χm I2 (x, ·) converges in L2 (Γa )

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297

to −χm ψ f−1 . Using (2.9), we see, that the function I1 (x, z) converges uniformly w.r.t. z ∈ Γa to zero . Thus, by (2.12), χm f = −χm ψ f−1 . Since m can be chosen arbitrary, we arrive at f = −ψ f−1 in the L2a -sense. By induction we get f = (−1) ψ ψ−1 · · · ψ1 f in L2a , which proves (2.10). The formula (2.11) follows then directly from the relation g = f (· − ) .  Lemma 2.11. Let α, β ∈ R with α < β. If f ∈ L22ξ−1 for every ξ ∈ (α, β),   then f ∈ L1α+β . Moreover, by Lemma ∩ L2α+β−1±2μ for all μ ∈ 0, β−α 2 2

−1±μ

2.9, f ∈ L2 (Γξ ) for all ξ ∈ (α, β) . If, additionally, f is holomorphic in the strip Γα,β := {z ∈ C : α < Re z < β} with   @ ?   (2.13) sup (1 + |z|)1+δ f (z) : α < Re z < β < ∞ for some δ > 0, then there hold the relation

  f(· + μ) − f(· − μ) ∈ L1 Γ α+β 2

and the representation f (t) =

tμ 2πi(1 − t2μ )

 Re z=

α+β 2

? @ t−z f(z − μ) − f(z + μ) dz

(2.14)

  for almost every t ∈ R+ and for all μ ∈ 0, β−α . Moreover, if the function 2 f is additionally continuous on R+ , then (2.14) holds for all t ∈ R+ , from which we can conclude lim tα+ε f (t) = 0

t−→0

and

lim tβ−ε f (t) = 0

(2.15)

t−→∞

for every ε > 0.

  Proof. Let δ > 0 be sufficiently small and μ ∈ 0, β−α . We have 2  ∞ α+β−1 1 |f (t)|t− 2 + 2 ±μ dt 0



1

=

t

δ− 12

|f (t)t

α+β−1 ±μ−δ 2



∞

| dt +

0

! ! α+β−1 ! ! ≤ const !f (t)t 2 ±μ−δ !

1

t−δ− 2 |f (t)t

α+β−1 ±μ+δ 2

1

0,2,∼

! ! α+β−1 ! ! + !f (t)t 2 ±μ+δ !

?

| dt

< 0,2,∼

@

= const f α+β−1±2μ−2δ,2,∼ + f α+β−1±2μ+2δ,2,∼ < ∞, and, consequently, f ∈ L1α+β 2

−1±μ

. The relation

  f(· + μ) − f(· − μ) ∈ L1 Γ α+β 2

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P. Junghanns and R. Kaiser

follows by the proofof [7, Corollary 2.7]. Thus, g = f(· + μ) − f(· − μ) belongs to L1 Γ α+β , where g ∈ L1α+β and g(t) = f (t)t−μ − f (t)tμ , i.e., 2

2

−1

tμ g(t) f (t) = . Now, Lemma 2.8 applied to g leads to (2.14). The limits 1 − t2μ (2.15) follow from the estimate α+β ! ! μ− 2   ! (· + μ)! f (t) ≤ t f (· − μ) − f , ! ! |1 − t2μ | Γ α+β ,1 2   where μ ∈ 0, β−α . 2



Corollary 2.12. Let α, β ∈ R with α < β. Moreover, let f ∈ Cn (R+ ) and f () ∈ L22ξ+2−1 for all  ∈ {0, 1, · · · , n} , for all ξ ∈ (α, β) , and for some n ∈ N. We suppose that the Mellin transform f is holomorphic in the strip Γα,β with     ? @     sup (1 + |z|)n+δ f(z) + (1 + |z|)n+1+δ f (z) : z ∈ Γα,β < ∞ (2.16) for some δ > 0. Then, for every  ∈ {0, 1, . . . , n} and every ε > 0 , the relations lim tα++ε f () (t) = 0

t−→0

lim tβ+−ε f () (t) = 0

and

t−→∞

(2.17)

are fulfilled. Proof. Let n ∈ N. The limits (2.17) for  ∈ {0, 1, . . . , n − 1} follow directly from Lemma 2.10 by setting ρ = 2(α + ε) − 1 and ρ = 2(β − ε) − 1 . Now, we consider the case  = n . For this, let y(t) = tn f (n) (t). Again using Lemma 2.10, we get y(z) = (−1)n z(z + 1) · · · (z + n − 1)f(z) ,

z ∈ Γα,β ,

where y is holomorphic in Γα,β and fulfils (2.13) for some δ > 0. Thus, with the help of Lemma 2.11, we get the assertion for  = n.  For α, β ∈ R we define fβ (t) := e−βt f (t) and the set : ; LEα = LEα (R+ ) := f : R+ −→ C : fβ ∈ L10 (R+ ) ∀ β > α . If f ∈ LEα , then the Laplace transform  ∞ (Lf )(z) = f (t)e−zt dt 0

is well defined for all z ∈ C with Re z > α. Lemma 2.13 ([11], Chapter II, Section 2, Theorem 5a). For α ∈ R and f ∈ LEα , the Laplace transform Lf is holomorphic in the half plane {z ∈ C : Re z > α} and there holds  ∞ f (t)tn e−zt dt , Re z > α , n ∈ N . (Lf )(n) (z) = (−1)n 0

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299

Lemma 2.14. Let f ∈ L1ξ−1 for every ξ ∈ (α, β) and some α, β ∈ R with α < β. Then, the Mellin transform f is holomorphic in the strip Γα,β and there holds  ∞ f(k) (z) = f (t)(ln t)k tz−1 dt , z ∈ Γα,β , k ∈ N . 0

Proof. Clearly, defining (Z± f )(x) = f (e±x ) , we have Z− f ∈ LEα and Z+ f ∈ LE−β . Taking into account Lemma 2.13 and the relation  ∞  ∞  ∞ −x −zx −x −zx  f (z) = f (e )e dx = f (e )e dx + f (ex )ezx dx −∞

0

0

= (LZ− f )(z) + (LZ+ f )(−z) , we get that the Mellin transform f is holomorphic in the strip Γα,β , where  ∞  ∞ (n) n −x n −zx  f (z) = (−1) f (e )x e dx + f (ex )xn ezx dx 0



0



1

∞

f (t)(ln t)n tz−1 dt +

= 

0 ∞

=

f (t)(ln t)n tz−1 dt

1

f (t)(ln t)n tz−1 dt ,

z ∈ Γα,β , n ∈ N ,

0



and the lemma is proved.

Let I ⊂ R be an interval. For f : I −→ C, we denote by supp(f ) := {x ∈ I : f (x) = 0} the support of the function f . Moreover, we set : ; + ∞ + + C∞ 0 (R ) := f ∈ C (R ) : supp(f ) ⊂ R , supp(f ) compact and

∞ C∞ 0 [0, 1] := {f ∈ C [0, 1] : supp(f ) ⊂ (0, 1)} .

+  Lemma 2.15. Let f ∈ C∞ 0 (R ). Then, the Mellin transform f is an entire function and the equation (2.8) holds for all t ∈ R+ and all ξ ∈ R. Moreover, f(z) suffices  @ ?   sup (1 + |z|)m f(k) (z) : α < Re z < β < ∞ (2.18)

for all α, β ∈ R with α < β and for all k = 0, 1, 2, . . . and m ∈ R. ()

Proof. Define gk (t) = f (t)(ln t)k and gk (t) = t gk (t) . According to Lemma 2.14, the Mellin transform f is holomorphic on C and f(k) (z) = gk (z) , where + gk also belongs to C∞ 0 (R ) . Let α, β ∈ R with α < β and  ∈ N. Since, for k = 0, 1, 2, . . . , the function gk has compact support, we get | gk (z)| ≤ C0

and

| gk (z)| ≤ C1 ,

z ∈ Γα,β ,

(2.19)

where C0 , C1 are constants not depending on z . Since  ∈ N is arbitrary, relation (2.18) follows directly from (2.19) and gk (z) = (−1) z(z + 1) · · · (z +  − 1) gk (z)

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P. Junghanns and R. Kaiser

(see (2.10)). For all ξ ∈ R , we have f ∈ L1ξ−1 due to the compact support of f and we have f ∈ L1 (Γξ ) due to (2.18). Consequently, Lemma 2.9 is applicable and (2.8) holds for all t ∈ R+ .  Let ξ ∈ R and let b : Γξ −→ C be a measurable function. By bξ (∂) we + denote the so-called Mellin operator, which, for f ∈ C∞ 0 (R ) , is given by    1 (bξ (∂)f )(t) := M−1 b M f (t) = t−z b(z)f(z) dz , t ∈ R+ . ξ ξ 2πi Re z=ξ Lemma 2.16. Let b ∈ L2 (Γξ ), ξ ∈ (α, β) be a holomorphic function on Γα,β , which satifies  ; : sup (1 + |z|)−m b(z) : α < Re z < β < ∞, −∞ < α < β < ∞ (2.20) + ∞ + for some m ∈ R. The mapping bξ (∂) : C∞ 0 (R ) −→ C (R ) is well defined, where ? @  (k)   sup tk+ξ bξ (∂)f (t) : t ∈ R+ , k ∈ N0 < ∞ (2.21) + holds for all functions f ∈ C∞ 0 (R ) and all ξ ∈ (α, β). Moreover, the operator bξ (∂) is independent of ξ ∈ (α, β) and we have the representation  ∞   t  f (τ )   −1 + bξ (∂)f (t) = Mξ b dτ , f ∈ C∞ 0 (R ) , τ τ 0

for all t ∈ R+ . + Proof. Let f ∈ C∞ 0 (R ). For every N ∈ N0 , we have, due to Lemma 2.15 and (2.20),  ? @   sup (1 + |z|)N b(z)f(z) : z ∈ Γα,β < ∞ .

Hence, by taking into account Lemma 2.2, the mapping + ∞ + bξ (∂) : C∞ 0 (R ) −→ C (R ),

f → bξ (∂)f

(2.22)

is well defined (cf. also (2.23)). By applying Cauchy’s theorem to  t−z b(z)f(z) dz Γ

with Γ = [ξ1 − iR, ξ2 − iR] ∪ [ξ2 − iR, ξ2 + iR] ∪ [ξ2 + iR, ξ1 + iR] ∪ [ξ1 + iR, ξ1 − iR] , α < ξ1 < ξ2 < β , R > 0 and letting R tend to ∞, we see that the map in (2.22) is independent of ξ ∈ (α, β). Note, that    t−ξ ∞ −iη ln t bξ (∂)f (t) = e b(ξ + iη)f(ξ + iη) dη 2π −∞

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301

and, due to Lemma 2.2, 

 ξ t−ξ−1 bξ (∂)f (t) = − 2π



∞ −∞

e−iη ln t b(ξ + iη)f(ξ + iη) dη

 i t−ξ ∞ −iη ln t 1 − e η b(ξ + iη)f(ξ + iη) dη 2π −∞ t  t−ξ−1 ∞ −iη ln t =− e (ξ + iη)b(ξ + iη)f(ξ + iη) dη. 2π −∞ By induction, we get, for k ∈ N ,  (k) tk+ξ bξ (∂)f (t)  (−1)k ∞ −iη ln t = e (ξ + iη)(ξ + 1 + iη) · · · 2π −∞

(2.23)

· · · (ξ + k − 1 + iη)b(ξ + iη)f(ξ + iη) dη . Consequently, by (2.20) and Lemma 2.15,   (k)   k+ξ  bξ (∂)f (t) ≤ C1 t

∞ −∞

(1 + |ξ + iη|)k+m |f(ξ + iη)| dη ≤ C2 .

This proves, with the help of (2.19) and Lemma 2.15, that (2.21) holds. For ξ ∈ (α, β) and R > 0 , we set b(ξ + is),

bR (ξ + is) :=

s ∈ [−R, R] , s ∈ R \ [−R, R] .

0,

It follows that 

∞ 0





M−1 ξ b

∞

=



t τ

M−1 ξ b

f (τ ) τ −1 dτ

t

0

τ

−



M−1 ξ bR



∞

+ 0

=: A1R (t) + A2R (t).



  t 

τ

M−1 ξ bR

f (τ )

t τ

dτ τ

f (τ )

dτ τ

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P. Junghanns and R. Kaiser

Using Fubini’s theorem, we get #  R  −(ξ+is)  ∞" t 1 dτ 2 b(ξ + is) ds f (τ ) AR (t) = 2π τ τ 0 −R 1 = 2π =

1 2π



R

t

−(ξ+is)

−R



R −R



∞

b(ξ + is)

τ ξ+is−1 f (τ ) dτ ds

0

t−(ξ+is) b(ξ + is)f(ξ + is) ds .

Since bf ∈ L1 (Γξ ), we have, due to (2.7), lim A2R (t) = (M−1 ξ bMξ f )(t)

R−→∞

for every t ∈ R+ . For some 0 < ε < M < ∞ , we have supp f ⊂ [ε, M ] . Consequently,  εt   εt 2     1 2   2ξ−1 −1 AR (t) ≤ b (σ) − M b (σ) σ dσ σ −2ξ−1 dσ  M−1  R ξ ξ t M

! ! ! ! ≤ C !M−1 (b − b ) R ! ξ

t M

2ξ−1,2,∼

with a constant C = C(t) depending on t ∈ R+ . Since lim b − bR Γξ ,2 = 0 R→∞

and since M−1 : L2 (Γξ ) −→ L22ξ−1 is continuous by Lemma 2.9, we can ξ conclude limn−→∞ A1n (t) = 0 for every t ∈ R+ . 

3. Singular integral operators with Cauchy and Mellin kernels 3.1. Boundedness Let κ : R −→ C be a measurable function. The integral operator Wκ defined by  ∞ (Wκ f )(s) = κ(s − t)f (t) dt, s ∈ R+ 0

is called Wiener–Hopf integral operator. Recall the notion of Lpξ (cf. (2.4)), which means that Lp0 = Lp (R+ ) . Lemma 3.1. Let p ∈ [1, ∞) and κ ∈ L1 (R). Then, Wκ : Lp0 −→ Lp0 is a linear and bounded operator. Proof. The well known Young’s convolution inequality delivers Wκ f 0,p,∼ ≤ κLp (R) f 0,p,∼

∀ f ∈ Lp0 .

Thus, Wκ : Lp0 −→ Lp0 is a well defined linear and bounded operator.



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303

 p := L  p (0, 1) the weighted Let p ∈ [1, ∞) , ρ, σ ∈ R. We denote by L ρ,σ ρ,σ Lp -space equipped with the norm 

1/p

1

f ρ,σ,p :=

|f (x)| υ  p

ρ,σ

(x) dx

,

υ ρ,σ (x) = xρ (1 − x)σ .

0

For a measurable function k : R+ −→ C we define the Mellin operator Bk by  1   x f (y) (Bk f )(x) = k dy, x ∈ (0, 1) . y y 0 Moreover, by S and SR+ we denote the Cauchy singular integral operators  1 1 f (y) (Sf )(x) = dy, x ∈ (0, 1) πi 0 y − x and 1 (SR+ f )(s) = πi



∞ 0

f (t) dt, t−s

s ∈ R+ ,

where the integrals are considered as Cauchy principal value integrals. The following lemma is well known. See, for example, [4, Sections 1.2, 1.5] or [1, Theorem 1.16, Remark 1.17]. Lemma 3.2. The operator S resp. SR+ is a linear and bounded operator in  p resp. Lp , if and only if the conditions L ρ,σ ρ 1 < p < ∞,

−1 < ρ, σ < p − 1

resp.

1 < p < ∞,

−1 < ρ < p − 1

are fulfilled. + Lemma 3.3. For f ∈ C∞ 0 (R ), we have   (SR+ f )(t) = M−1 ξ (−i cot π·)Mξ f (t)

1 =− 2π



t−z cot(πz) f(z) dz

(3.1)

Re z=ξ

for all ξ ∈ (0, 1) and all t ∈ R+ . Proof. Let ξ ∈ (0, 1) and ρ = 2ξ − 1 . From [6, p. 48] or [1, (4.10),(4.11)] we infer the equation SR+ f = Tξ−1 F −1 coth π(iξ + ·)FTξ f = Tξ−1 F coth π(iξ − ·)F −1 Tξ f for all f ∈ L2ρ . Thus, since coth π(iξ − η) = −i cot π(ξ + iη) , SR+ f = M−1 ξ (−i cot π·)Mξ f

(3.2)

+ for all f ∈ L2ρ . In case of f ∈ C∞ 0 (R ), the left- and the right-hand side of (3.2) are continuous functions and so (3.1) holds pointwise on R+ . 

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+ Corollary 3.4. Let f ∈ C∞ 0 (R ) and γ ∈ R. For ξ ∈ (−γ, 1 − γ), we have   (ψγ SR+ ψ−γ f ) (t) = M−1 (−i cot π(γ + ·))M f (t) ξ ξ

=−

1 2π



t−z cot π(γ + z) f(z) dz

(3.3)

Re z=ξ

for all t ∈ R , where ψγ (t) = t . +

γ

Proof. In view of Lemma 3.3 we have the equation  1 (ψγ SR+ ψ−γ f ) (t) = − t−(z−γ) cot(πz) f(z − γ) dz 2π Re z=α  1 =− t−z cot π(γ + z) f(z) dz 2π Re z=α−γ for arbitrary α ∈ (0, 1). By setting ξ := α − γ we get the assertion.



Corollary 3.5. Let α, β ∈ R with α < β and k ∈ L22ξ−1 for all ξ ∈ (α, β). The Mellin transform  k(z) is supposed to fulfil the condition  @ ?   (3.4) sup (1 + |z|)−m  k(z) : α < Re z < β < ∞ for some m ∈ R . Using Lemma 2.16, in the same way as in the proof of Corollary 3.4 one can show that the relation  ∞  γ     t t f (τ )  k dτ = M−1 k(γ + ·)M f (t) ξ ξ τ τ τ 0  1 = t−z  k(γ + z)f(z) dz 2πi Re z=ξ + holds true for t ∈ R+ , f ∈ C∞ 0 (R ) , and ξ ∈ (α − γ, β − γ) .

For a given function f : (0, 1) −→ C let P+ f : R+ −→ C be defined by (P+ f )(t) =

f (t), 0,

t ∈ (0, 1) , t > 1.

(3.5)

Corollary 3.6. Let α, β ∈ R with α < β and k ∈ L22ξ−1 for all ξ ∈ (α, β). The Mellin transform  k(z) is supposed to fulfil condition (3.4) for an appropriate m ∈ R. For a, b, c ∈ C , let A := aI + bS + cBk . Moreover, we assume that (α, β) ∩ (0, 1) = ∅ holds if b = 0 and c = 0. Then, for γ ∈ R , ψγ (t) = tγ , and f ∈ C∞ 0 (0, 1) , we have (P+ ψγ Aψ−γ f )(t)      = χ(0,1) M−1 a − bi cot(π(γ + ·)) + c k(γ + ·) M P f (t) , ξ + ξ

(3.6)

Fredholm theory for singular integral operators for all t ∈ R+ and for all ⎧ (−γ, 1 − γ), ⎪ ⎨ (−γ, 1 − γ) ∩ (α − γ, β − γ), ξ∈ ⎪ ⎩ (α − γ, β − γ), where χ(0,1) (t) =

1,

t ∈ (0, 1) ,

0,

t > 1.

305

c = 0, b = 0, c = 0 , b = 0,

Proof. Equation (3.6) is an immediate consequence of Corollaries 3.4 and 3.5.  Lemma 3.7. Let p ∈ [1, ∞). If k ∈ L11+ρ −1 holds for some ρ ∈ R, then the p

 p −→ L  p is bounded. integral operator Bk : L ρ,0 ρ,0 Proof. At first we notice, that the mapping  p −→ Lp , Z 1+ρ : L ρ,0 0 p

f → e−

1+ρ p ·

f (e−· )

is an isometrical isomorphism and that, for t ∈ R+ ,  1  −t    (1+ρ)t 1+ρ f (− ln y) e − p Z 1+ρ Bk Z −1 f (t) = e k dy y− p 1+ρ p y y p 0  ∞ 1+ρ = k(es−t )e p (s−t) f (s) ds . 0

Hence, Bk can be transformed into a Wiener–Hopf integral operator with 1+ρ kernel function κ(t) = k(e−t )e− p t , t ∈ R. Taking into account  ∞  ∞ (1+ρ)s 1+ρ κL1 (R) = e− p |k(e−s )| ds = t p −1 |k(t)| dt = k 1+ρ −1,1,∼ −∞

p

0



we can apply Lemma 3.1 to finish the proof.

Lemma 3.8. Let p ∈ (1, ∞) and ρ, σ ∈ R. Moreover, let A be an integral operator with the measurable kernel function k : (0, 1) × (0, 1) −→ C , i.e.,  1 k(x, y)u(y) dy , 0 < x < 1 . (Au)(x) = 0

 p −→ L  p is compact, if the condition Then A : L ρ,σ ρ,σ ⎡ ⎤p−1 p   p−1  σp  1  1   pρ   x 1 − x   ⎣ k(x, y) dy ⎦ dx < ∞   1−y 0 0  y

(3.7)

is fulfilled. Proof. First of all, we have to remark that the assertion is well known for the values ρ = σ = 0 (cf. [3, Prop. 2, Section 4.5.2]). Since  p −→ L p , ωI : L ρ,σ 0,0

f → ωf

306

P. Junghanns and R. Kaiser ρ

σ

with ω(x) = x− p (1 − x)− p is an isometrical isomorphism, the compactness of A is equivalent to the compactness of the integral operator  p −→ L p . ω −1 AωI : L 0,0

0,0

But, since the kernel function of this integral operator is equal to   pρ  σ x 1−x p k(x, y) y 1−y 

the latter is compact, if (3.7) is fulfilled.

Lemma 3.9. Let p ∈ (1, ∞) and k : R+ −→ C be a measurable function, which is bounded on (0, 1 + ε) for some ε > 0 and which fulfills  ∞  ∞$ p % p−1 1+ρ 2+ρ t p −1 |k(t)| dt < ∞ and t p −1 |k(t)| dt < ∞ (3.8) 0

1

 p −→ L  p is for some ρ ∈ (−1, p − 1). Then, the integral operator Bk : L ρ,σ ρ,σ bounded for all σ ∈ (−1, p − 1). Proof. Let ε > 0 be sufficiently small, such that (1 − ε)−1 ≤ 1 + ε is fulfilled. Moreover, let 0, x ∈ (1 − ε, 1) , ψ(x) = 1, x ∈ (0, 1 − ε) . Now, we decompose the operator B = Bk in the following way: B = ψBψI + ψB(1 − ψ)I + (1 − ψ)BψI + (1 − ψ)B(1 − ψ)I .  p if and only if the operator χψBψχ−1 I The operator ψBψI is bounded in L ρ,σ  p −→ L  p is an  p , where χ(x) = (1 − x) σp and χI : L is bounded in L ρ,σ ρ,0 ρ,0 −1 isometrical isomorphism. Since χψ and ψχ are bounded functions on (0, 1)  p (cf. Lemma 3.7), we get the and B is a bounded operator in the space L ρ,0 p . boundedness of χψBψχ−1 I and hence the boundedness of ψBψI in L ρ,σ Now we turn to the operators ψB(1 − ψ)I ,

(1 − ψ)BψI ,

and

(1 − ψ)B(1 − ψ)I .

 p by checkWe are going to show, that these operators are compact in L ρ,σ ing the respective condition (3.7). Regarding to ρ, σ ∈ (−1, p − 1) and the boundedness of k on (0, 1 + ε), we can estimate ⎡ ⎤p−1 p  p−1  σp    1  1   pρ   x x 1 − x   −1 ⎣ k dy ⎦ dx y [1 − ψ(x)] [1 − ψ(y)]   1−y y 0 0  y 

1



1

= 1−ε

1−ε



1

p−1 p   p−1    −ρ  σ x − −1 y p (1 − y) p k dy xρ (1 − x)σ dx y   y



1

≤ const 1−ε

1−ε

σ

(1 − y)− p−1 dy

p−1 (1 − x)σ dx < ∞

Fredholm theory for singular integral operators

307

(using 1 − ε < y −1 x < (1 − ε)−1 for 1 − ε < x, y < 1) and ⎡ ⎤p−1 p  p−1  σp    1  1   pρ   x 1 − x x   ⎣ k dy ⎦ dx y −1 ψ(x)[1 − ψ(y)]   1−y y 0 0  y 

1−ε



1

≤ const 0



1−ε 1−ε



p−1 p    p−1   ρ (1 − y)− p k x  dy xρ dx  y 

1

≤ const

(1 − y) 0

ρ − p−1

(3.9)

p−1 xρ dx < ∞

dy

1−ε

(due to 0 < y −1 x < 1 for 0 < x < 1 − ε and 1 − ε < y < 1). Moreover, with the help of (3.8) we get ⎡ ⎤p−1 p  p−1  σp    1  1   pρ   x 1 − x x   ⎣ k dy ⎦ dx y −1 [1 − ψ(x)]ψ(y)   1−y y 0 0  y 



1

1−ε

≤ const 1−ε



∞

≤ const

0

p−1 p    p−1  − ρ −1  x y p k  dy (1 − x)σ dx  y 

p−1 p  2+ρ  p−1  p −1  k(t) dt < ∞, t

1



and the proof is complete. Lemma 3.10. Let p ∈ [1, ∞) and γ ∈ R. Moreover, let f ∈ some δ > 0. Then, we have f ∈ L11+γ −1 .

Lpγ−δ

∩

Lpγ+δ

for

p

Proof. In case of p = 1 the assertion is obviously fulfilled. Let p ∈ (1, ∞) . Due to  ∞  1  ∞ 1+γ γ−δ γ+δ 1+δ 1−δ −1 −1 p p p t |f (t)| dt = t |f (t)|t dt + t p −1 |f (t)|t p dt 0

0

1



1

≤

   p 1+δ t p −1 p−1 dt

p−1 p



|f (t)| t

p γ−δ

0

 p1

1

dt

0





∞

+

t 1

1−δ p −1



p p−1

 p−1  p

∞

dt

|f (t)|p tγ+δ dt

 p1

1

and due to the relations     p 1−δ p 1+δ −1 > −1 and −1 < −1 , p p−1 p p−1 ? @ we get f  1+γ −1,1,∼ ≤ const f γ−δ,p,∼ + f γ+δ,p,∼ < ∞ . Hence, the p

function f belongs to L11+γ −1 . p



308

P. Junghanns and R. Kaiser

Proposition 3.11. Let p ∈ (2, ∞), ρ ∈ (−1, p − 1) and k ∈ Lpρ−δ ∩ Lpρ+δ for some δ > 0. Moreover, the function k is supposed to be bounded on (0, 1 + ε)  p −→ L  p is bounded for for some ε > 0. Then, the integral operator Bk : L ρ,σ ρ,σ all σ ∈ (−1, p − 1). Proof. In view of Lemma 3.9 and Lemma 3.10 it remains to show that  ∞$ p % p−1 2+ρ t p −1 |k(t)| dt < ∞ 1

is satisfied. But, this follows from  ∞$ p % p−1 2+ρ t p −1 |k(t)| dt 1



∞

≤

t

ρ+δ

|k(t)| dt p

1  p−1 

∞

t

1

δ −1− p−2

 p−2 p−1 dt

< ∞,

1



and the proposition is proved. Proposition 3.12. Let ρ ∈ (−1, 1) and k Moreover, the function k is supposed to be point 1. Then, the integral operator Bk : σ ∈ (−1, 1).

∈ L2ρ−δ ∩ L2ρ+δ for some δ > 0. bounded in a neighbourhood of the  2 −→ L  2 is bounded for all L ρ,σ ρ,σ

Proof. In view of Lemma 3.10 we have  ∞ 1+ρ t 2 −1 |k(t)| dt < ∞ and



0

∞

tρ |k(t)|2 dt < ∞ .

0

Thus, the conditions in (3.8) are fulfilled and, considering the proof of Lemma 3.9, it remains to prove that an estimate like (3.9) can be shown in case p = 2 and under the present assumptions. But this is really the case, since 2 σ    1  1   ρ2   1−x 2 x  x  −1 k y ψ(x)(1 − ψ(y)) dy dx    y 1 − y y 0 0 

1−ε



1

≤ const 0



1−ε



1

1−ε

= const 1−ε



0

  2   k x  (1 − y)−σ dy xρ dx  y    2   k x  xρ dx (1 − y)−σ dy  y 

1 2

≤ const

|k(t)| tρ dt 0

and k ∈ L2ρ .



Proposition 3.13. Let p ∈ (1, ∞) , ρ ∈ (−1, p−1) , and k ∈ C(R+ ). Moreover, we assume that there are real numbers α, β with α < β such that 1+ρ p ∈ (α, β) and such that (3.10) lim k(t)tα = 0 and lim k(t)tβ = 0 . t→0

t→∞

Fredholm theory for singular integral operators

309

 p −→ L  p is a bounded one for all Then, the integral operator Bk : L ρ,σ ρ,σ σ ∈ (−1, p − 1). Proof. Refering to the proof of Lemma 3.9, we have to verify (3.8) and (3.9). The inequality ρ − αp > −1 delivers p−1 p   p−1  1−ε  1    ρ x − (1 − y) p k  dy xρ dx   y 0 1−ε 



1−ε

≤ const

1

xρ−αp dx 0



ρ

(1 − y)− p−1 dy

p−1

1−ε 1−ε

xρ−αp dx < ∞ ,

= const 0

and (3.9) is endorsed. Using 1 + ρ − p − αp > −1 p we get  ∞ t

1+ρ p −1



1

|k(t)| dt =

t

0

1+ρ p −1

0

and

1 + ρ − p − βp < −1 , p 



1

t

1+ρ−p−αp p

0

∞

$ t

t

1+ρ p −1

|k(t)| dt

1

≤ const Similarly one can show 

∞

|k(t)| dt +

2+ρ p −1



∞

dt +

t

1+ρ−p−βp p

< dt < ∞ .

1

|k(t)|

p % p−1

dt < ∞ ,

0



such that (3.8) is also checked.

Let α, β ∈ R with α < β and (α, β) ∩ (0, 1) = ∅ . We say that f belongs to the class Σ−1 α,β if f ∈ L22α−1 ∩ L22β−1 ∩ C(R+ ) and

  @ ?   sup (1 + |z|)1+k f(k) (z) : α < Re z < β < ∞,

k = 0, 1, 2, . . .

(3.11)

hold true. Proposition 3.14. Let p ∈ (1, ∞) and k ∈ Σ−1 α,β . Then, the operator Bk :  p is bounded for all σ ∈ (−1, p − 1) and ρ ∈ (pα − 1, pβ − 1) ∩  p −→ L L ρ,σ ρ,σ (−1, p − 1). Proof. By Lemma 2.11 we get lim tα+ε k(t) = lim tβ−ε k(t) = 0 , where ε > 0 t→0

can be chosen sufficiently small, such that 3.13 delivers the assertion.

t→∞ 1+ρ p ∈ (α

+ ε, β − ε) . Proposition 

310

P. Junghanns and R. Kaiser

 2 −→ L  2 by (Rf )(x) = f (1 − x) . We define the operator R : L ρ,σ σ,ρ 2 2   Since R : Lρ,σ −→ Lσ,ρ is an isometric isomorphism, we can formulate the following corollary. Corollary 3.15. Let p ∈ (1, ∞) and k ∈ Σ−1 α,β . Then, the operator RBk R : p p   Lρ,σ −→ Lρ,σ is bounded for all real numbers ρ and σ in (−1, p − 1) and (pα − 1, pβ − 1) ∩ (−1, p − 1), respectively. 3.2. Fredholmness We will call a function a : [0, 1] → C piecewise continuous if it is continuous at 0 and 1, if the one-sided limits a(x ± 0) exist for all x ∈ (0, 1) and at least one of them coincides with a(x) . The set of all piecewise continuous functions, which have a finite number of jumps will be denoted by PC[0, 1]. Lemma 3.16. Let p ∈ (1, ∞), c ∈ PC[0, 1] with c(0) = 0 and k ∈ Σ−1 α,β . The operators υ 0,−γ c Bk υ 0,γ I

and

υ 0,−γ Bk c υ 0,γ I

 p for every are compact in L ρ,σ ρ ∈ (pα − 1, pβ − 1) ∩ (−1, p − 1)

σ ∈ (−1 + pγ, p(γ + 1) − 1) .

and

Proof. For ε ∈ (0, 1), we set cε (x) :=

c(x),

x ∈ [ε, 1] ,

0,

x ∈ [0, ε) .

Moreover, we choose an ε1 > 0 such that (2.15), we get ⎡



1 0

⎣



1 0

1+ρ p

< β − ε1 is fulfilled. Using

⎤p−1 p   ρ   p−1    x p 1 − x  σp −γ  x   cε (x)k dy ⎦ dx y −1    y  1−y y 

1



ε

≤ const ε

0

⎡



1

≤ const

⎣



ε

p−1 p    p−1  − ρ −1  y p k x  dy (1 − x)σ−pγ dx + const  y 

ε

y

−ρ−p+(β0 −ε1 )p p−1

0

⎤p−1    p  x β−ε1  x  p−1   k dy ⎦    y y  · (1 − x)σ−pγ dx + const



1



ε

≤ const

y ε

0

−ρ−p+(β−ε1 )p p−1

p−1 dy

(1 − x)σ−pγ dx + const < ∞ .

Fredholm theory for singular integral operators

311

 p . Together Lemma 3.8 delivers the compactness of υ 0,−γ cε B(k) υ 0,γ I in L ρ,σ with ! ! ! 0,−γ ! !(c − cε )  υ 0,−γ Bk υ 0,γ I !  p  ≤ const c − cε  !υ Bk υ 0,γ !  p  ∞

 ρ,σ L L

 ρ,σ L L

≤ const c − cε ∞ Bk    p

L Lρ,σ−pγ

≤ const c − cε ∞ −→ 0,



ε −→ +0 ,

p . we immediately get the compactness of the operator υ 0,−γ c Bk υ 0,γ I in L ρ,σ The second operator can be dealt with analogously.  Lemma 3.17. Let p ∈ (1, ∞), f ∈ PC[0, 1] and k ∈ Σ−1 α,β . The operator p  f Bk − Bk f I is compact in Lρ,σ for every ρ ∈ (pα − 1, pβ − 1) ∩ (−1, p − 1)

and

σ ∈ (−1, p − 1) .

Proof. Since f Bk − Bk f I = (f − f (0))Bk − Bk (f − f (0))I holds, it remains to apply Lemma 3.16.



Lemma 3.18. Let p ∈ (1, ∞), c ∈ PC[0, 1] with c(0) = 0 and k ∈ Σ−1 α,β .  p and let ρ be in Moreover, let N be a linear and bounded operator in L ρ,σ (pα − 1, pβ − 1) ∩ (−1, p − 1) and σ in (−1, p − 1). If the operator f N − N f I  p for every function f ∈ C[0, 1], then the operators c N Bk is compact in L ρ,σ p . and Bk N c I are also compact in L ρ,σ Proof. We choose a smooth function χ : [0, 1] −→ [0, 1] such that χ(0) = 1 and cχ ∈ C[0, 1] hold. With the help of Lemma 3.16 and of c N Bk = c N (χ + 1 − χ)Bk = cχN Bk + c(N χ − χN )Bk + cN (1 − χ)Bk = N cχBk + (cχN − N cχ)Bk + c(N χ − χN )Bk + c N (1 − χ)Bk  p follows. The compactness of the operator the compactness of cN Bk in L ρ,σ Bk N c I can be shown analogously.  Lemma 3.19 ([4], Section 1.4, Theorem 4.3). Let p ∈ (1, ∞) and f ∈ C[0, 1].  p for all ρ, σ ∈ (−1, p − 1). The operator f S − Sf I is compact in the space L ρ,σ For 0 < δ < π and two points z1 , z2 ∈ C , let (z1 , z2 ; δ) be the circular arc connecting the points z1 and z2 , where the line segment [z1 , z2 ] is on the left of (z1 , z2 ; δ) and where from every point z ∈ (z1 , z2 ; δ) \ {z1 , z2 } this line segment is seen under the angle δ . In case π < δ < 2π , by definition the segment [z1 , z2 ] lies on the right of the circular arc, where from every point z ∈ (z1 , z2 ; δ) \ {z1 , z2 } this line segment is seen under the angle

312

P. Junghanns and R. Kaiser

2π − δ . Finally, we set (z1 , z2 ; π) = [z1 , z2 ] . If we define, for 0 ≤ μ ≤ 1 and 0 < δ < 2π , the function ⎧ e2i(π−δ)μ − 1 ⎨ sin[(π − δ)μ] i(π−δ)(μ−1) e = 2i(π−δ) , 0 < δ = π , fδ (μ) = (3.12) e −1 ⎩ sin(π − δ) μ, δ = π, then we get

: ; (z1 , z2 ; δ) = z1 + (z2 − z1 )fδ (μ) : 0 ≤ μ ≤ 1 (3.13) : ; since fδ (μ) : 0 ≤ μ ≤ 1 = (0, 1; δ) (cf. [5, Section 9.1]). It is easily seen that, for 0 < ξ < 1 , also (0, 1, 2πξ) = {vξ (t) : −∞ ≤ t ≤ ∞} holds, where

1 − i cot(πξ − iπt) 1 = . (3.14) −2π(t+iξ) 2 1−e If Γ = {γ(t) : t1 ≤ t ≤ t2 } ⊂ C is a closed curve not containing the zero point, then the winding number of Γ around 0 is denoted by wind(Γ) , i.e.,  t=t2 wind(Γ) = arg γ(t) t=t , where argz , z ∈ Γ , is a continuous branch of the 1 argument on Γ . In order to prepare the proof of the following Lemma 3.24 we recall some definitions from [9]. (Note that in [2] one can find analogous investigations in the special case p = 2 .) For α ∈ R , by (α] we denote the greatest integer d smaller than α . In what follows, ∂ stands for the differential operator −t . dt Moreover, for −∞ < α < β < ∞ , by Jα,β we refer to the class of all functions f ∈ C∞ (R+ ) for which there are polynomials pα (t) and pβ (t) of degree less than or equal to max {0, (−α]} and max {0, (β]} , respectively, such that   ; : sup tα+δ ∂ k [f (t) − pα (t)] : 0 < t < 1 < ∞ vξ (t) =

and

  : ; sup tβ−δ ∂ k [f (t) − pβ (t)] : 1 < t < ∞ < ∞

for all nonnegative integers k and all δ > 0 (cf. [9, Definition 1.1]). For −∞ ≤ α < β ≤ ∞ and m ∈ R , let Om α,β denote the class of all holomorphic functions f : Γα,β −→ C satisfying   ? @   sup (1 + |z|)k−m f (k) (z) : α < Re z < β  < ∞ for all [α , β  ] ⊂ (α, β) and all nonnegative integers k (cf. [9, Definition 1.3]). If −∞ < α < β < ∞ and −∞ < γ < δ < ∞ , then by Jα,β (Om γ,δ ) we refer to the class of all functions a : R+ × Γγ,δ −→ C for which a(·, z) belongs to + Jα,β for every z ∈ Γγ,δ and for which a(t, ·) is in Om γ,δ for every t ∈ R . For −∞ < γ < δ < ∞ and m ∈ R , let (cf. [9, Definition 1.4]) K m Σ Jγ−γ  ,δ−δ (Om γ,δ = γ  ,δ  ) . [γ  ,δ  ]⊂(γ,δ)

Let 1 < p < ∞ . A function a(t, z) belongs to the symbol class Σ1/p if there are 0 , γ, δ ∈ [0, 1] and functions a± ∈ Jγ−δ,δ−γ satisfying γ < p−1 < δ , a ∈ Σ γ,δ

Fredholm theory for singular integral operators

313

 −1 , where and aθ ∈ Σ γ,δ aθ (t, z) = a(t, z) − a+ (t)θ(z) − a− (t)[1 − θ(z)] and θ(z) =

1 (3.15) 1 − e2πiz

 1/p and f ∈ C∞ (R+ ) , define a(., ∂)f by (cf. [9, Definition 3.1]). For a ∈ Σ 0 (cf. [9, Definition 3.2])  1 [a(., ∂)f ](t) = t−z a(t, z)f(z) dz , t > 0 . 2πi Re z=p−1 The principal symbol σ(a(., ∂)) of the operator a(., ∂) is then defined as the restriction of the function a(t, z) to the boundary of the “rectangle” : ; R1/p = (t, p−1 + iξ) ∈ C : 0 ≤ t ≤ ∞, −∞ ≤ ξ ≤ ∞ . Remember the definition of the operator P+ in (3.5). Moreover, for a function f : [0, 1] −→ C , let (Rf )(x) = f (1 − x) (cf. the end of Section 3.1). By a cut-off function we mean a function ϕ ∈ C∞ 0 [0, 1) which is identically zero in a neighbourhood of 1 . We recall [9, Definition 4.1].  p ) is called a pseudodifferential opDefinition 3.20. An operator A ∈ L(L 0,0 erator of class OΣ1/p (0, 1) if and only if the following three conditions are fulfilled:  p −→ (a) If ψ1 , ψ2 ∈ C∞ [0, 1] and supp ψ1 ∩supp ψ2 = ∅ , then ψ2 Aψ1 I : L 0,0  p is a compact operator. L 0,0 (b) If ϕ1 , ϕ2 ∈ C∞ 0 [0, 1) are cut-off functions, then there are a function 0  p −→ L  p such that, a0ϕ1 ϕ2 ∈ Σ1/p and a compact operator Kϕ :L 0,0 0,0 1 ϕ2 ∞ for all f ∈ C0 (0, 1) , 0 (ϕ2 Aϕ1 f )(t) = [a0ϕ1 ϕ2 (., ∂)P+ f ](t) + (Kϕ f )(t) , 1 ϕ2

0 < t < 1.

(c) The operator AR = RAR satisfies (a) and (b) with AR , a1ϕ1 ϕ2 (., ∂) , 1 0 and Kϕ instead of A , a0ϕ1 ϕ2 (., ∂) , and Kϕ , respectively. 1 ϕ2 1 ϕ2 Lemma 3.21 ([9], pp. 518–520). If A ∈ OΣ1/p (0, 1) , then there are functions a0 (t, z) and a1 (t, z) defined on : ; Λ1/p = (t, z) ∈ ∂R1/p : 0 ≤ t < 1 such that, for ϕ1 , ϕ2 ∈ C∞ 0 [0, 1) and x ∈ {0, 1} , [σ(axϕ1 ϕ2 (., ∂))](t, z) = ϕ2 (t)ϕ1 (t)ax (t, z) ,

(t, z) ∈ Λ1/p .

Moreover, for 0 < t < 1 , a0 (t, p−1 ± i∞) = a1 (1 − t, p−1 ∓ i∞) .

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According to Lemma 3.21 we associate to an operator A ∈ OΣ1/p (0, 1) the (continuous) curve : ; Γ0A = a0 (t, p−1 + i∞) = a1 (1 − t, p−1 − i∞) : 0 < t < 1 ; : ∪ a1 (0, p−1 + iξ) : −∞ ≤ ξ ≤ +∞ (3.16) ; : ∪ a0 (1 − t, p−1 − i∞) = a1 (t, p−1 + i∞) : 0 < t < 1 ; : ∪ a0 (0, p−1 + iξ) : −∞ ≤ ξ ≤ ∞ . Lemma 3.22 ([9],Theorems 4.1, 4.2). An operator A ∈ OΣ1/p [0, 1) is Fred p if and only if 0 ∈ Γ0 . In the case of Fredholmness, the Fredholm holm in L 0,0 A index of A is equal to the winding number of Γ0A w.r.t. the zero point. Lemma 3.23. Let Θ(z) = −i cot(πz) . For 0 < α < β < 1 , m ∈ R , and k = 1, 2, . . . , we have sup {|Θ(z)| : z ∈ Γα,β } < ∞ ,   ? @   sup (1 + |z|)k+m Θ(k) (z) : z ∈ Γα,β < ∞ , and, for α − β < μ < β − α , and θ(z) from (3.15),   ? @   sup (1 + |z|)k+m θ(k) (z) − θ(k) (μ + z) : z ∈ Γαμ ,βμ < ∞ ,

(3.17)

(3.18)

where αμ = max {α, α − μ} and βμ = min {β, β − μ} . Proof. For z = x + iy , 0 < x < 1 , and x ∈ R , the formula Θ(x + iy) =

e2πix + e2πy e2πix − e2πy

shows the boundedness of Θ(z) in Γα,β . Of course, it is sufficient to consider F (z) = cot(z) in Γα0 ,β0 with 0 < α0 < β0 < π instead of Θ(z) . Now, F  (z) = [sin(z)]−2 and F  (z) = −2 F  (z)F (z) . Consequently, for k > 2 , k−2  k − 2 F (k) (z) = −2 F (k−1−j) (z)F (j) (z) . j j=0 Since there are constants c0 , c1 such that |F  (z)| ≤ c1 e−2|Im z| and |F (z)| ≤ c0 for z ∈ Γα,β , (3.17) follows by induction. Furthermore, in view of   e2πiz 1 − e2πiμ   , αμ < Re z < βμ , θ(z) − θ(μ + z) = (1 − e2πiz ) 1 − e2πi(μ+z) we get the existence of a further constant c2 such that |θ(z) − θ(μ + z)| ≤ c2 e−2|Im z| for z ∈ Γαμ ,βμ . That (3.18) is also true for k > 0 , is a consequence of (3.17) and θ(z) = 12 [1 − Θ(z)] . 

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We set A := aI + bS + c+ Bk+ + c− RBk− R , ∞

(3.19)

where a, b, c± ∈ L (0, 1) and k± ∈ C(R ) . +

Lemma 3.24. Let p ∈ (1, ∞), a, b, c± ∈ C∞ [0, 1] and k± ∈ Σ0α± ,β± . Moreover, we assume σ± ∈ (pα± − 1, pβ± − 1) ∩ (−1, p − 1) . (3.20) p Then, the operator A defined by (3.19) is Fredholm in the space L σ+ ,σ− if and only if the curve  <    A : = a+ 0, 1 + σ+ + it : t ∈ R Γ p ? @ ∪ a(x) − b(x) : x ∈ [0, 1] (3.21)    < 1 + σ− − ∪ a 1, + it : t ∈ R p ? @ ∪ a(1 − x) + b(1 − x) : x ∈ [0, 1] does not run through the zero point, where a± (t, z) = a(t) ∓ b(t)i cot(πz) + c± (t) k± (z) . p p In this case, the Fredholm index of A : L σ+ ,σ− −→ Lσ+ ,σ− is equal to  A ) , where the orientation of Γ  A is given by the inherent parametrizawind(Γ tion in (3.21). σ+

σ−

Proof. Setting ω(x) = x− p (1 − x)− p we note that the operator A is Fred−1 p  holm in the space L AωI is Fredσ+ ,σ− if and only if the operator A := ω p  , where the Fredholm indices coincide. holm in the space L 0,0

First, we check that A satisfies condition (a) of Definition 3.20: If ψ1 , ψ2 ∈ C∞ [0, 1] have disjoint support, then ψ2 Sψ1 I = (ψ2 S − Sψ2 I)ψ1 I p p and by Lemma 3.19 we get the compactness of ψ2 Sψ1 I : L σ+ ,σ− −→ Lσ+ ,σ− . Since at least one of the functions ψ1 and ψ2 is zero in a neighbourhood p p of 0 (resp. 1) the compactness of ψ2 Bk+ ψ1 I : L σ+ ,σ− −→ Lσ+ ,σ− (resp. p p ψ2 B k ψ1 I : L −→ L ) is a consequence of Lemma 3.16. Hence, −

σ+ ,σ−

σ+ ,σ−

p .  1 I = ω −1 ψ2 Aψ1 ωI is compact in L ψ2 Aψ 0,0 Now, we turn to condition (b) of Definition 3.20: For this, write ω(x) = σ+ σ− χ(x)ζ(x) with χ(x) = x− p and ζ(x) = (1−x)− p and let ϕ1 , ϕ2 ∈ C∞ 0 [0, 1) be cut-off functions. Due to ϕ1 ζ ∈ C[0, 1] and ϕ1 ζ (1) = 0 , we conclude the  σ ,0 −→ L  σ ,0 by Lemma 3.19, compactness of the operator Sϕ1 ζI −ϕ1 ζS : L + +  σ ,0 −→ L  σ ,0 by Lemma 3.17, and of the operator Bk+ ϕ1 ζI − ϕ1 ζBk+ : L + +  σ ,0 −→ L  σ ,0 by Lemma 3.16. Consequently, of the operator Bk− Rϕ1 ζR : L + +

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taking into the account ϕ2 ζ −1 ∈ C[0, 1] , the operator   0  1 I − ϕ2 ϕ1 aI + bχ−1 SχI + c+ χ−1 Bk χI Kϕ = ϕ2 Aϕ + 1 ϕ2  = ϕ2 ζ −1 χ−1 b (Sϕ1 ζI − ϕ1 ζS)    + c+ Bk+ ϕ1 ζI − ϕ1 ζBk+ + c− RBk− Rϕ1 ζI χI  p . By Corollary 3.6, for f ∈ C∞ (0, 1) , we get turns out to be compact in L 0 0,0 the formula      0   1 f (t) = a0ϕ ϕ (., ∂)P+ f (t) + Kϕ f (t) , 0 < t < 1 , ϕ2 Aϕ 1 2 1 ϕ2 where a0ϕ1 ϕ2 (t, z) = [P+ ϕ2 ϕ1 a0 (·, z)] (t) and 

   σ+ σ+  a0 (t, z) = a(t) − ib(t) cot π +z + c+ (t) k+ +z . p p Since ϕ2 ϕ1 is identically 1 in a neighbourhood of 0 , the function a0ϕ1 ϕ2 (·, z) belongs to J−ε,ε for every ε > 0 and every z ∈ Γ0,1 ∩ Γα,β . Lemma 3.23 and k+ ∈ Σ0α+ ,β+ imply a0ϕ1 ϕ2 (t, ·) ∈ O0γ+ ,δ+ for every t ≥ 0 , where γ+ = max {α+ , 0} −

σ+ , p

δ+ = min {β+ , 1} −

σ+ p

0 and, due to (3.20), γ+ < p1 < δ+ . Hence a0ϕ1 ϕ2 ∈ Σ γ+ ,δ+ . Moreover, again using Lemma 3.23 and setting a± = P+ ϕ2 ϕ1 [a ∓ b] , we get that the function a0θ defined by a0θ (t, z) := a0ϕ1 ϕ2 (t, z) − a+ (t)θ(z) − a− (t)[1 − θ(z)] 

   σ+ σ+ = 2 [P+ ϕ2 ϕ1 b] (t) θ(z) − θ +z + [P+ ϕ2 ϕ1 c+ ] (t) k +z p p 0  −1 belongs to Σ max{γ+ ,0},min{δ+ ,1} . All this together gives aϕ1 ϕ2 ∈ Σ1/p . To check condition (c) in Definition 3.20, we remark that RSR = −S , such that analogously to the previous considerations we have, for every function f ∈ C∞ 0 (0, 1) ,      1   1 f (t) = a1ϕ ϕ (., ∂)P+ f (t) + Kϕ ϕ2 RARϕ f (t) , 0 < t < 1 , 1 2 1 ϕ2 1  p −→ L  p and a function a1 with a compact operator Kϕ :L ϕ1 ϕ2 ∈ Σ1/p 0,0 0,0 1 ϕ2 satisfying the equation

a1ϕ1 ϕ2 (t, z) = [P+ ϕ2 ϕ1 a1 (·, z)] (t) with



   σ− σ− +z + c− (1 − t)  +z . k− a1 (t, z) = a(1 − t) + ib(1 − t) cot π p p

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317

Summarizing, we get A ∈ OΣ1/p (0, 1) , where the associated curve Γ0A , defined in (3.16), is equal to Γ0A = {a(t) − b(t) : 0 < t < 1} 

 ? 1 + σ− ∪ a(1) + ib(1) cot π + iξ p   @ 1 + σ− + c(1)  k− + iξ : −∞ ≤ ξ ≤ +∞ p ∪ {a(1 − t) + b(1 − t) : 0 < t < 1}  

? 1 + σ+ ∪ a(0) − ib(0) cot π + iξ p   @ 1 + σ+  + c(0) k+ + iξ : −∞ ≤ ξ ≤ +∞ p  A given by (3.21). Note that, for 0 < μ < 1 resp. and coincides with Γ α± < μ < β± , the relations −i cot [π(μ ± i∞)] = ∓1 and  k+ (μ ± i∞) =  k− (μ ± i∞) = 0 hold.  Remark 3.25. The condition, which is necessary and sufficient for the Fredholmness of the operator A , formulated in Lemma 3.24 is equivalent to the following: For all x ∈ [0, 1] , we have a(x) − b(x) = 0 , and the curve 9 c+ (0) k+ (ξ+ − it) ΓA := 1 + [c(0) − 1]vξ+ (t) + : −∞ ≤ t < ∞ a(0) − b(0) ? @ ∪ c(x) : 0 ≤ x ≤ 1 ∪

9 c− (1) k− (ξ− − it) c(1) + [1 − c(1)]vξ− (t) + : −∞ < t ≤ ∞ a(1) − b(1)

a(x) + b(x) 1 + σ± , ξ± = , and a(x) − b(x) p where vξ (t) is given in (3.14). If this is the case, then the Fredholm index of p p A:L σ+ ,σ− −→ Lσ+ ,σ− is equal to the negative winding number of the curve ΓA w.r.t. zero, indL pσ ,σ (A) = −wind(ΓA ) .

does not run through zero, where c(x) =

+

−

 A implies a(x) − b(x) = 0 for all x ∈ [−1, 1] . Then, Proof. Trivially, 0 ∈ Γ   p  p A∈Φ L σ+ ,σ− if and only if B ∈ Φ Lσ+ ,σ− , where B=

b c+ c− a I+ S+ B k+ + RBk− R , a−b a−b a−b a−b

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 B , defined by (3.21), is equal and the Fredholm indices coincide. The curve Γ to : ; B = Γ  + ∪ {1} ∪ Γ  − ∪ c(1 − x) : 0 ≤ x ≤ 1 Γ with + = Γ

= and − = Γ

=

9   a(0) − b(0) i cot π(ξ+ + it) c+ (0) k+ (ξ+ + it) + : −∞ < t < ∞ a(0) − b(0) a(0) − b(0) c+ (0) k+ (ξ+ + it) 1 + [c(0) − 1]vξ+ (−t) + : −∞ < t < ∞ a(0) − b(0)

9

9   a(1) + b(1) i cot π(ξ− + it) c− (1) k− (ξ+ + it) + : −∞ < t < ∞ a(1) − b(1) a(1) − b(1) c− (1) k− (ξ− + it) c(1) + [1 − c(1)]vξ− (−t) + : −∞ < t < ∞ a(1) − b(1)

9 ,

 B and ΓA coincide, where the winding number of Γ  B is equal to so that Γ minus the winding number of ΓA .  Corollary 3.26. Let p ∈ (1, ∞), a, b ∈ C[0, 1], c± ∈ PC[0, 1] and k± ∈ Σ0α± ,β± . Moreover, we assume σ± ∈ (pα± − 1, pβ± − 1) ∩ (−1, p − 1) . p Then, the operator A defined by (3.19) is Fredholm in the space L σ+ ,σ− , if and only if a(x) = b(x) for all x ∈ [0, 1] and if the curve ΓA given in Remark 3.25 does run through the zero point. In this case, the index of A equals the negative winding number of the curve ΓA w.r.t. zero, where the orientation is given by the inherent parametrization, indL pσ ,σ (A) = −wind(ΓA ) . +

−

Proof. Lemma 3.16 delivers the compactness of the operator   A − aI + bS + c+ (0)Bk+ + c− (1)RBk− R  2 . Hence, we can suppose that c± ∈ C are constants. At first, in the space L ρ,σ we assume the operator A is Fredholm and that a(x) = b(x) is not true for all x ∈ [0, 1] or that the curve ΓA contains the zero point. If there is an x∗ ∈ [0, 1] such that a(x∗ ) = b(x∗ ) or a(x∗ ) = −b(x∗ ) , then, for every ε > 0 , there exist functions aε , bε ∈ C∞ [0, 1] satisfying aε (x∗ ) = a(x∗ ) , bε (x∗ ) = b(x∗ ) , and aε − a∞ , bε − b∞ < ε . If there is not such an x∗ , but 0 ∈ ΓA , then we can choose aε , bε ∈ C∞ [0, 1] with aε (t) = a(t) and bε (t) = b(t) for t = 0 and t = 1 as well as aε − a∞ , bε − b∞ < ε . In both cases, for sufficiently small ε > 0 , the operator aε I + bε S + c+ Bk+ + c− RBk− R is Fredholm in p L σ+ ,σ− in contradiction to Remark 3.25. Hence, a(t) = b(t) for all t ∈ [0, 1] and 0 ∈ ΓA if A is Fredholm.

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319

The converse assertion can be shown analogously to Theorem 3.27,(a) below. The index formula follows also from the fact that the Fredholm index is stable with respect to small perturbations and the fact that continuous functions can be approximated with arbitrary accuracy by infinitely differentiable functions.  Theorem 3.27. Let p ∈ (1, ∞), a, b, c± ∈ PC[0, 1] and k± ∈ Σ0α± ,β± . Moreover, we assume that σ± ∈ (pα± − 1, pβ± − 1) ∩ (−1, p − 1) and that the operator A is defined by (3.19). p (a) The operator A is a Fredholm operator in the space L σ+ ,σ− if and only if a(x ± 0) − b(x ± 0) = 0 for all x ∈ (0, 1) , if a(x) − b(x) = 0 for x ∈ {0, 1} , and if the closed curve 



ΓA := Γ+ ∪ Γ1 ∪ Γ1 ∪ . . . ∪ ΓN ∪ ΓN ∪ ΓN +1 ∪ Γ− , (3.22) does not contain the point 0 . Here N stands for the number of discontia(x) + b(x) nuity points xj , j = 1, . . . , N, of the function c(x) = chosen a(x) − b(x) in such way that x0 := 0 < x1 < · · · < xN < xN +1 := 1. Using these  xj , the curves Γj , j = 1, . . . , N + 1 and Γj , j = 1, . . . , N are given by : ; Γj := c(x) : xj−1 < x < xj and

? @  Γj := c(xj − 0) + [c(xj + 0) − c(xj − 0)]v p1 (t) : −∞ ≤ t ≤ ∞ .

The curves Γ± , connecting the point 1 with one of the endpoints of Γ1 and ΓN +1 , respectively, are given by the formulas 9 c+ (0) k+ (ξ+ − it) Γ+ := 1 + [c(0) − 1]vξ+ (t) + : −∞ ≤ t ≤ ∞ a(0) − b(0) and Γ− :=

9 c− (1) k− (ξ− − it) c(1) + [1 − c(1)]vξ− (t) + : −∞ ≤ t ≤ ∞ a(1) − b(1)

± with ξ± = 1+σ .  p  p  (b) If A ∈ Φ Lσ+ ,σ− , then indL pσ ,σ (A) = −wind(ΓA ) , where the ori+ − entation of ΓA is due to the above given parametrization.

Remark 3.28. In case of c± ≡ 0 , the validity of Theorem 3.27 is well known (cf. [5, Chapter 9, Theorem 4.1]). To prove this theorem we are going to use the local principle of Gohberg and Krupnik. For that we need the following definitions (cf. [4, Section 5.1]). Definition 3.29. Let A be a Banach algebra with identity e.

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• We call a set M ⊂ A localizing class, if the zero element of A does not belong to M and if, for every a1 , a2 ∈ M, there exists an element a ∈ M such that ak a = aak = a for k = 1, 2 holds. • Let X be an arbitrary non-empty set. A family {Mx : x ∈ X} of localizing classes Mx ⊂ A is called covering if, for each choice {ax : x ∈ X} of elements ax ∈ Mx , there is a finite number of elements ax1 , . . . , axn , the sum of which is invertible in A. • Let M ⊂ A be a localizing class. The elements a, b ∈ A are called Mequivalent, if inf (a − b)g = inf g(a − b) = 0

g∈M

g∈M

M

holds. In this case, we write a ∼ b. • Let M ⊂ A be a localizing class. An element a ∈ A is called M-left invertible (M-right invertible), if there exists b ∈ A and g ∈ M such that bag = g (gab = g). It is called M-invertible if it is M-left and M-right invertible. Lemma 3.30 ([4], Chapter 5, Lemma 1.1). Let A be a Banach algebra with identity e and M ⊂ A a localizing class. Moreover, let a, b ∈ A. If a is MM

invertible and a ∼ b, then b is also M-invertible. Now the local principle of Gohberg and Krupnik claims the following. Proposition 3.31 ([4], Chapter 5, Theorem 1.1). Let {Mx : x ∈ X} be a covering system of localizing classes of the Banach algebra A with identity e. If Mx

a, aLx ∈ A, where a ∼ ax for all x ∈ X and if a commutes with all elements of x∈X Mx , then a is invertible in A if and only if ax is Mx -invertible for all x ∈ X. For what follows, we agree some notations: By [L](X) we denote the quotient algebra L(X)/K(X) equipped with the norm : ; [A] = inf A + T  : T ∈ K(X) . By [A] ∈ [L](X) we denote the coset containing the element A ∈ L(X). For x ∈ [0, 1] , let Mx ⊂ C[0, 1] stand for set of all functions f ∈ C[0, 1] , which are identically 1 in a neighbourhood of x . Moreover, we set Mx = ? @ p  [f I] ∈ [L](Lσ+ ,σ− ) : f ∈ Mx , where f I is the multiplication operator in  by the function f . It is easy to see that {Mx : x ∈ [0, 1]} forms a Lp σ+ ,σ−

p covering system of localizing classes in the quotient space [L](L σ+ ,σ− ) .

p p Corollary 3.32. Let A : L σ+ ,σ− −→ Lσ+ ,σ− be the operator defined in (3.19) with a, b, c± ∈ PC[0, 1] , k± ∈ Σ0α± ,β± , and 1 < p < ∞ . p (a) The coset [A] ∈ [L](L σ+ ,σ− ) commutes with all elements from the set L x∈[0,1] Mx .

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321

 with (b) For  a, b,  c± ∈ PC[0, 1], the coset [A] p p A =  aI + bS +  c + B k+ +  c− RBk− R : L σ+ ,σ− −→ Lσ+ ,σ− is Mx -equivalent to the coset [A] if ⎧  a(x ± 0) = a(x ± 0) and b(x ± 0) = b(x ± 0), ⎪ ⎪ ⎨  a(0) = a(0) , b(0) = b(0) , and  c+ (0) = c+ (0), ⎪ ⎪ ⎩  a(1) = a(1) , b(1) = b(1) , and  c− (1) = c− (1), (c) The coset [A] is Mx -invertible if ⎧ a(x ± 0) − b(x ± 0) = 0, ⎪ ⎨ a(0) − b(0) = 0, ⎪ ⎩ a(1) − b(1) = 0,

a(x) + b(x) and ξ± = a(x) − b(x)

1+σ± p

x = 0, x = 1,

x ∈ (0, 1) , x = 0, x = 1,

and if, for −∞ ≤ t ≤ ∞ , ⎧ c(x − 0) + [c(x + 0) − c(x − 0)]v p1 (t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ c+ (0) k+ (ξ+ − it) 1 − [c(0) − 1]vξ+ (t) + = 0, a(0) − b(0) ⎪ ⎪ ⎪ ⎪ ⎪ c (1) k− (ξ+ − it) ⎪ ⎩ c(1) − [1 − c(1)]vξ− (t) + − = 0, a(1) − b(1) where c(x) =

0 < x < 1,

x ∈ (0, 1) , y = 0, y = 1,

.

Proof. (a) This is a consequence of Lemma 3.17 and Lemma 3.19. (b) One has to take into account assertion (a) and the relation [f I] ≤ f ∞ = sup {|f (x ± 0)| : 0 ≤ x ≤ 1} for f ∈ PC[0, 1] .   (c) In case of y ∈ (0, 1) , we can define ay , by ∈ C [0, 1] \ {y} in such a way that ay (y ± 0) = a(y ± 0) , by (y ± 0) = b(y ± 0) , by (0) = by (1) = 0 , and ay (x)±by (x) = 0 for all x ∈ [0, 1]\{y} . Indeed, there are continuous functions γ± : [0, y] −→ C\{0} and δ± : [y, 1] −→ C\{0} satisfying γ± (0) = δ± (1) = 1 , γ± (y) = a(y −0)±b(y −0) , and δ± (y) = a(y +0)±b(y +0) , and we can choose ay (x) = 12 [γ+ (x) + γ− (x)] , by (x) = 12 [γ+ (x) − γ− (x)] , 0 ≤ x < y , as well as ay (x) = 12 [δ+ (x) + δ− (x)] , by (x) = 12 [δ+ (x) − δ− (x)] , y < x ≤ 1 . Due to p Remark 3.28, the operator ay I + by S belongs to Φ(L σ+ ,σ− ) , and by (b), the coset [A] is My -equivalent to [ay I + by S] and consequently My -invertible (cf. Lemma 3.30). In case of y = 0 , we take functions a0 , b0 ∈ C∞ [0, 1] satisfying a0 (0) = a(0) , b0 (0) = b(0) , b0 (1) = 0 , and a(x) ± b(x) = 0 for all x ∈ (0, 1] . In view of Lemma 3.24 together with Remark 3.25, the operator a0 I + b0 S + p c+ (0)Bk+ belongs to Φ(L σ+ ,σ− ) , and [a0 I + b0 S + c+ (0)Bk+ ] is, due to (b), M0 -equivalent to [A] . Again by Lemma 3.30, we get the M0 -invertibilty of [A] . The case y = 1 can be treated analogously. 

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Proof of Theorem 3.27 (a) At first, we show that the conditions, formulated in (a), imply the Fredholmness of the operator A. By Corollary 3.32,(c), the coset A is Mx -invertible for all x ∈ [0, 1] . which, by Proposition 3.31, implies p p the invertibility of [A] and so the Fredholmness of A : L σ+ ,σ− −→ Lσ+ ,σ− . Now we are going to show the necessity of the conditions in (a). Let A p belong to Φ(L σ+ ,σ− ) and assume that the conditions are not fulfilled. In case of a(y0 ) − b(y0 ) = 0 or a(y0 ) + b(y0 ) = 0 for some y0 ∈ (xj−1 , xj ) and j ∈ {1, . . . , N + 1} , we can choose a0 , b0 ∈ C∞ [0, 1] such that a0 (y0 ) = a(y0 ) , b0 (y0 ) = b(y0 ) , b0 (0) = b0 (1) = 0 , and a0 (x) ± b0 (x) = 0 ∀ x ∈ [0, 1] \ {y0 } . In view of Corollary 3.32, the coset [a0 I + b0 S] is Mx -invertible for all x ∈ p [0, 1] \ {y0 } . Since [A] is My0 -invertible (due to A ∈ Φ(L σ+ ,σ− )) and My0 equivalent to [a0 I + b0 S] (cf. Corollary 3.32,(b)), [a0 I + b0 S] is also My0 p invertible. Proposition 3.31 yields a0 I + b0 S ∈ Φ(L σ+ ,σ− ) in contradiction to Lemma 3.24. In case a(xj − 0) − b(xj − 0) = 0 or a(xj + 0) − b(xj + 0) = 0 or    0 ∈ Γj for some j ∈ {1, . . . , N } , we choose a0 , b0 ∈ C [0, 1] \ {xj } such that a0 (xj ± 0) = a(xj ± 0) , b0 (xj ± 0) = b(xj ± 0) , and a0 (x) ± b(x) = 0 for all x ∈ [0, 1] \ {xj } as well as b0 (0) = b0 (1) = 0 . Analogously to the p p previous step, we get the Fredholmness of a0 I + b0 S : L σ+ ,σ− −→ Lσ+ ,σ− in contradiction to Remark 3.28. If a(0) − b(0) = 0 or 0 ∈ Γ+ we take function a0 , b0 ∈ C∞ [0, 1] satisfying a0 (0) = a(0) , b0 (0) = b(0) , b0 (1) = 0 , and a0 (x)±b0 (x) = 0 for all x ∈ (0, 1] . p In the same manner as before, we get a0 I +b0 S ∈ Φ(L σ+ ,σ− ) in contradiction to Lemma 3.24. The case a(1) − b(1) = 0 or 0 ∈ Γ− is treated analogously. (b) In view of Lemma 3.16 it suffices to prove the assertion for the operator A = aI + bSJ + c0 Bk+ + c1 RBk− R

with

c0 = c+ (0) , c1 = c− (1) ,

i.e., w.l.o.g. we can set c± ≡ 1 . Moreover, we can assume that the operators APC = aI + bS and Ax = ax I + bx S with ax = a(x), bx = b(x) for x = 0 and x = 1 are Fredholm. Otherwise, we can slightly modify the values a(0), b(0) and a(1), b(1) in such a way that the index of A remains unchanged, but APC and A0 as well as A1 become Fredholm. In this case APC as well as A0 are one-sided invertible (cf. [5, Section 9.4, Theorem 4.1]). W.l.o.g. we assume, that these operators are invertible from the left side, and denote respective −1 left inverses by A−1 PC and Ax . Hence −1 −1 −1 A−1 PC APC = I, APC APC = I − TPC , and Ax Ax = I, Ax Ax = I − Tx ,

where TPC and Tx are projections onto a complement of the image of APC and Ax , respectively. The images of TPC and of Tx are finite dimensional, p since APC and Ax are Fredholm operators. Thus, TPC , Tx ∈ K(L σ+ ,σ− ) . We set A+ := aI + bS + c0 Bk+ = APC + c0 Bk+ and A− := aI + bS + c1 RBk− R = APC + c1 RBk− R .

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If we write (3.22) in the form ΓA = ΓA = Γ+ ∪ Γ0A ∪ Γ− , where

Γ0A



(3.23)



= Γ1 ∪ Γ1 ∪ . . . ∪ ΓN ∪ ΓN ∪ ΓN +1 , then we have the relations

ΓAPC = Γ0+ ∪Γ0A ∪Γ0− , where

ΓA+ = Γ+ ∪Γ0A ∪Γ0− ,

ΓA− = Γ0+ ∪Γ0A ∪Γ− , (3.24)

: ; Γ0+ = 1 + [c(0) − 1]vξ+ (t) : −∞ ≤ t ≤ ∞

and

: ; Γ0− = c(1) + [1 − c(1)]vξ− (t) : −∞ ≤ t ≤ ∞ . Thus, since by assumption 0 ∈ ΓA ∪ ΓAPC , we have also 0 ∈ ΓA+ ∪ ΓA− , p which means A± ∈ Φ(L σ+ ,σ− ) . Moreover,       wind ΓA+ + wind ΓA− = wind ΓA + wind (ΓAPC ) . (3.25) Now, let χ : [0, 1] −→ [0, 1] be a continuous function, which vanishes in a neighbourhood of the point 0 and is identically one in a neighbourhood of −1 −1 the point 1. Then, the operator A−1 PC (APC χI − χAPC ) APC − APC χTPC = −1 −1 χAPC − APC χI is compact. From Lemma 3.16 we infer K0 := Bk+ A−1 PC RBk− R −1 p = Bk+ A−1 PC χRBk− R + Bk+ APC (1 − χ)RBk− R ∈ K(Lσ+ ,σ− ) .

Consequently,

  −1   A+ A−1 PC A− = APC + c0 Bk+ APC APC + c1 RBk− R   = APC + c0 Bk+ + c1 APC + c0 Bk+ A−1 PC RBk− R = APC + c0 Bk+ + c1 (I − TPC )RBk− R + c0 c1 Bk+ A−1 PC RBk− R = A − K

with the compact operator K = c1 TPC RBk− R−c0 c1 K0 . Taking into account Remark 3.28, this implies  = ind (A+ ) − ind (APC ) + ind (A− ) ind (A) = ind (A+ ) + wind(ΓAPC ) + ind (A− ) . Hence, by (3.25), it remains to prove that

  ind (A± ) = − wind ΓA±

holds. We have A+ = APC + Bk+   A0 + Bk+ + (A0 − APC ) A−1 = APC A−1 0 0 B k+ + T 0 B k+ .

(3.26)

(3.27)

−1 In view of Lemma 3.19, the operators (cf. the above proof for χA−1 PC −APC χI) −1 f A−1 0 − A0 f I

and

−1 f SA−1 0 − SA0 f I

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are compact for every continuous function f , which implies the compactness of the operator −1 −1 (A0 − APC ) A−1 0 Bk+ = (a − a0 )A0 Bk+ + (b − b0 )SA0 Bk+

by taking into account Lemma 3.18. By applying Lemma 3.24 together with Remark 3.25, Remark 3.28, and (3.27) we get   ind (A+ ) = ind (APC ) − ind (A0 ) + ind A0 + Bk+   = − wind (ΓAPC ) + wind (ΓA0 ) − wind ΓA0 +Bk+   = − wind ΓA+ . where, for the last equality, we have used (cf. (3.23) and (3.24)) ΓA+ = Γ+ ∪Γ0A ∪Γ0− , ΓAPC = Γ0+ ∪Γ0A ∪Γ0− , ΓA0 = Γ0+ ∪Γ0− , and ΓA0 +Bk+ = Γ+ ∪Γ0− .   The proof of ind (A− ) = − wind ΓA− can be realized analogously with using A1 instead of A0 . ( ' Having criteria for the Fredholmness on hand, the question arises in which situations such an operator is automatically one-sided invertible (cf., for example, [1, Theorem 8.1]). The description of subclasses of operators of the form (1.1) with this property in weighted Lp -spaces will be the subject of a forthcoming paper.

References [1] R. Duduchava, Integral equations in convolution with discontinuous presymbols, singular integral equations with fixed singularities, and their applications to some problems of mechanics, BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1979. With German, French and Russian summaries, Teubner-Texte zur Mathematik. [Teubner Texts on Mathematics]. [2] J. Elschner, Asymptotics of solutions to pseudodifferential equations of Mellin type, Math. Nachr. 130 (1987), 267–305. [3] S. Feny¨ o and H.W. Stolle, Theorie und Praxis der linearen Integralgleichungen. 1, VEB Deutscher Verlag der Wissenschaften, Berlin, 1982. German. [4] I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations. I, vol. 53 of Operator Theory: Advances and Applications, Birkh¨ auser Verlag, Basel, 1992. Introduction. [5] I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations. Vol. II, vol. 54 of Operator Theory: Advances and Applications, Birkh¨ auser Verlag, Basel, 1992. General theory and applications. [6] R. Hagen, S. Roch, and B. Silbermann, Spectral theory of approximation methods for convolution equations, vol. 74 of Operator Theory: Advances and Applications, Birkh¨ auser Verlag, Basel, 1995. [7] P. Junghanns and R. Kaiser, On a collocation-quadrature method for the singular integral equation of the notched half-plane problem, in: Large truncated Toeplitz matrices, Toeplitz operators, and related topics, vol. 259 of Oper. Theory Adv. Appl., pp. 413–462, Birkh¨ auser/Springer, Cham, 2017.

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[8] P. Junghanns and A. Rathsfeld, On polynomial collocation for Cauchy singular integral equations with fixed singularities, Integral Equations Operator Theory 43 no. 2 (2002), 155–176. [9] J.E. Lewis and C. Parenti, Pseudodifferential operators of Mellin type, Comm. Partial Differential Equations 8 no. 5 (1983), 477–544. [10] E.C. Titchmarsh, Introduction to the theory of Fourier integrals, Chelsea Publishing Co., New York, 3rd ed., 1986. [11] D.V. Widder, The Laplace Transform, vol. 6 of Princeton Mathematical Series, Princeton University Press, Princeton, N. J., 1941. Peter Junghanns and Robert Kaiser Chemnitz University of Technology Faculty of Mathematics D-09107 Chemnitz Germany e-mail: [email protected] [email protected]

Towards a system theory of rational systems Jana Nˇemcov´a, Mih´aly Petreczky and Jan H. van Schuppen Dedicated to M.A. Kaashoek on the occasion of his 80th birthday and for his continued efforts to stimulate mathematics and the interaction of mathematicians.

Abstract. In this paper the reader finds a description of current research in the system theory of rational systems. The research is motivated by the needs of researchers in the life sciences and in engineering for control, estimation, and prediction of rational systems. An overview of existing results is provided. Successively discussed are solved and open problems of: canonical forms for rational systems, the problem of decomposing a rational system, the system approximation problem, control synthesis of rational systems, and computer algebraic problems for rational systems. The discussions are illustrated by academic examples. Mathematics Subject Classification (2010). Primary 93B25, secondary 93B10, 93B15, 93B17, 93C10. Keywords. Rational systems, system theory, canonical forms, decompositions, system approximation, computer algebra.

1. Introduction The purpose of this paper is to inform the reader of the investigations in control and system theory of rational systems. The motivation of the current investigation in rational systems is the occurence of these systems in biochemical reaction systems, in engineering, and in economics. A rational system is understood to be, as defined in system theory, a rational control system with an input and an output signal. In such a system the system equations are defined by rational functions each component of which is a quotient of two polynomials in the components of the state and of the input. Associate with any rational system its response map which maps any input trajectory on a finite interval to an output value at the end of the time interval. The problem of obtaining a minimal realization of its response map then arises. In addition, the problem of identifiability arises: can the parameters of the system be uniquely determined from the response map. Further, the problem of control synthesis of a rational system may be © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_13

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J. Nˇemcov´ a, M. Petreczky and J.H. van Schuppen

considered: Determine a control law, which maps at any time a past output trajectory to a current input value, such that the closed-loop system meets prespecified control objectives. These are problems of control and system theory. These problems are motivated by the research areas mentioned above. The investigation into control and system theory of rational systems requires the use of polynomial algebra and of the theory of rational functions. What is actually needed for the problems of control and system theory either is computationally complex like computing a transcendence basis or is related to one of Hilbert’s open problems. The aim of this paper is to show to the readers: how problems of control and system theory for rational systems can be formulated and how they can be rephrased into problems of commutative algebra. It should be clear that the investigation of the authors is a long-term research program that will take many more years to complete. Several of the problems of control and system theory of rational systems have been solved by the authors. A brief summary follows. In Subsection 4.2 the reader may find a more detailed overview. The realization problem of rational systems is solved meaning that there exist necessary and sufficient conditions on a response map for which there exists a rational system whose response map equals the considered response map. Necessary and sufficient conditions for structural identifiability of a rational system have also been published. A reduction procedure from a nonrationally-observable rational system to a rationally-observable system has been formulated. Approaches to check whether a rational system is algebraically controllable have been proposed but more research is required for this research issue. Rational observers for rational systems have been formulated. A system identification procedure has been proposed for a discrete-time polynomial system. The motivation by problems of the life sciences, engineering, and economics requires further investigation into problems of control and system theory of rational systems. Most urgent seem the problems of system identification, canonical forms, and control synthesis. In addition, procedures using tools of commutative algebra and the extensions of the existing computer algebra packages require investigation. The authors plan to contribute to this development. A summary of the remainder of the paper follows. The next section provides a more detailed problem formulation. Section 3 shows examples of rational systems from the life sciences and from engineering. The subsequent section provides concepts of rational systems including those of realization theory. Section 5 contains a definition of an observable-canonical form for a minimal rational system and discusses extensions. Section 6 has a problem formulation on decompositions of an arbitrary rational system into rational subsystems with a particular structure. The problem of obtaining system approximations of rational systems is discussed in Section 7. Finally control synthesis for rational systems is discussed in Section 8.

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2. Problem Formulation In this section the reader finds the motivation of the authors for developing a system theory of rational systems and a list of major open problems requiring attention of researchers. Polynomial and rational systems, formally defined in Section 4, occur in engineering, biochemistry, and economics. The dynamics is different from that of linear systems, of mechanical systems, or of electric systems. Hence the class of rational systems requires a separate investigation. The classes of polynomial and of rational systems also require the use of commutative algebra which goes much beyond linear algebra. The subarea of nonlinear systems in differential-geometric structures does not fully satisfy the demands of the problems of rational systems. The expectation is that the developments for computer algebra may assist researchers with the investigation of system theory of rational systems. The authors have investigated several major problems of the system theory of rational systems. Results obtained for rational systems include: controllability and observability and their characterizations, existence of a rational realization [44], minimality of a rational realization [45], and structural identifiability [43]. In addition, there is a synthesis procedure for rational observers of rational systems [49] and a system approximation procedure for a discrete-time polynomial system [47]. Due to experience with modeling rational systems of biochemical reaction systems and of engineering systems, it has become clear that many other problems of system theory of rational systems need to be formulated and solved at the system theoretic level. Below attention is focused on the class of rational systems while keeping in mind its relations with other classes. The class of rational systems contains the class of polynomial systems. The class of rational systems is contained in the class of Nash systems defined in [46]. A Nash system is like a rational system in which any rational function is replaced by a Nash fuction. A Nash function is defined as an analytic function which satisfies an algebraic equation. For biochemical reaction systems the subclass of positive rational systems is of particular interest. Problem 2.1 (Problems of system theory of rational systems). Investigate the following research issues of polynomial and of rational systems: 1. Canonical forms of rational systems for the response-map equivalence relation and for the feedback-and-response-map equivalence relation; 2. Decompositions of (positive) rational systems; 3. System approximation of rational systems; 4. System identification of rational systems; 5. Control synthesis of rational systems; 6. Computer algebra for rational systems. The issues mentioned above are partly to be discussed in the subsequent sections of this paper. In each section then a motivation and approach

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is discussed for the particular problem issue. The problem issue of system identification is not discussed. The problem issues are motivated by the plan of the authors to develop a system theory for rational systems for biochemical reaction systems and for control-engineering systems. Stability of polynomial and of rational systems is of interest also to the life sciences and to engineering. But the stability problems may not be directly treatable by algebraic methods only. There is extensive work on stability of biochemical reaction systems, see the lecture notes and papers of Feinberg, [22, 23, 28, 29, 30], and the paper [59]. There is an extensive literature on stability, persistence, and permanence of polynomial reaction systems, see [4, 7, 5, 6, 9, 14, 18, 21, 24, 26, 31, 54]. Recent reports on stability of biochemical reaction systems include, [12, 13]. Most of the above publications address the stability of polynomial systems only. Stability of rational systems requires a different approach because of the singularities. A rational system has a singularity at a particular state if one or more of the components of the right-hand side of the system of differential equations is not defined at that state. The singularities partition the state set into a finite number of domains and the stability of each domain of the system has to be analyzed separately. Singularities will occur in rational systems but will in general not occur in biochemical reaction systems due to the denominator polynomials being strictly positive.

3. Examples of Rational Systems 3.1. Example Glycolysis in Yeast This example is motivated by the need of life scientists of mathematical models for phenomena in the cell, in physiological systems, or in communities of plants and animals. Mathematicians can assist researchers in the life sciences by developing concepts, theory, algorithms, and computer programs for the analysis of such mathematical models. Though mathematical models for phenomena in the life sciences have been developed since the early twentieth century, only a small part of the needs of life scientists have been satisfied. In particular, life scientists need algorithms to determine the parameters of mathematical models from data, the area is called system identification. In addition, they need assistance with understanding the behavior of systems of equations and their control. The development of medicines needs concepts and theory on the effect of medicines on the behavior of parts of the cell or on a physiological organ. These needs have motivated the authors to develop system theory of rational systems. They expect that the concepts and theory will eventually be used by a group of researchers in the life sciences for the advancement of medical treatment and of understanding the processes of life. In this subsection a very simple model is described of the phenomenon of glycolysis in yeast. The model comes from a standard text book on systems

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biology but it will show to mathematicians how such a model is described and how it originates. A tutorial on biochemical reaction systems is [8]. Baker’s yeast (Saccharomyces cerevisiae) is used by bakers to produce bread. The life sciences model of yeast is regarded as a model system in the life sciences. A model system means that it has been extensively described and investigated. Much of its behavior is known. A description of the model system may be found in, for example, [61]. Glycolysis is how the cell chemically processes its food. The end product of glycolysis is the chemical species ADP (= Adenosine-diphosphate). There is a chain of reactions which takes glucose and decomposes this chemical species into other chemical species. The model is described below. The example of glycolysis in baker’s yeast is taken from the book, [37, pp. 138–139]. The reaction diagram is displayed in Figure 1.

ATP ADP

v2

 6 ADP

ATP ADP ATP v3 - Fruc6P Glucose Gluc6P Fruc1,6P   v1 v4 v5 Figure 1. Reaction diagram of glycolysis in yeast.

Table 1 relates the symbols to names of the states, the input, the reaction speeds, and of several constants. A state represents the concentration of chemical species, the input is the concentration of the chemical species glucose, and the output is the observation of the material flow out of the system at the end of the reaction scheme. The constants are of the three phosphates concentrations. In distinction with the example of the book [37], these phosphate concentrations are assumed constant because they are available in large quantities in a cell and therefore change little over time. The differential equations of the biochemical reaction system follow in the format specified by the biologists. There is one differential equation for each state variable. The equations contain on the right-hand side the reaction speeds further specified below. dGluc6P (t)/dt = v1 − v2 − v3 , dFruc6P (t)/dt = v3 − v4 , dFruc1, 6P2 (t)/dt = v4 − v5 .

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Symbol Gluc6P Fruc6P Fruc1, 6P2 Glucose ATP ADP AMP v1 v2 v3 v4 v5

Name glucose-6-phosphate fructose-6-phosphate fructose-1,6-biphosphate glucose adenosine-triphosphate adenosine-diphosphate adenosine-monophosphate hexokinase consumption of glucose-6-phosphate phosphoglucoisomerase phosphofructokinase aldolase

Represents in system state state state input constant constant constant reaction speed reaction speed reaction speed reaction speed reaction speed

Table 1. The relation of symbols to names for the states, the input, the reaction speeds, and of several constants.

The rates of these differential equations are specified in terms of the states, the input, and the constants as,

v1 =

1+

Vmax,1 AT P Glucose , Glucose + KGlucose + KATP ATP,1 KGlucose,1 Glucose,1

ATP KATP,1

v2 = k2 ATP Gluc6P (t), v3 =

f r Vmax,3 Vmax,3 KGluc6P,3 Gluc6p(t) − KFruc6P,3 Fruc6P (t) , (t) Fruc6P (t) 1 + Gluc6P KGluc6P,r + KFruc6P,3



v4 = KFruc6P,4

Vmax,4 (Fruc6P (t))2 ,  ATP 2  1 + κ AMP + (Fruc6P (t))2

v5 = k5 Fruc1, 6P2 (t). Below the above differential equations in the form written by the biologists are transformed to the standard form used in control and system theory. Define the state variables, the input of Glucose, and the output as respectively, x1 (t) = Gluc6P (t), x2 (t) = Fruc6P (t), x3 (t) = Fruc1, 6P2 (t), u(t) = Glucose(t), y(t) = h(x(t)) = v5 (x(t)) = c11 x3 (t).

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The reaction speeds as functions of the states and the inputs are then, after the transformation, the following rational functions, c1 u c5 x1 − c 6 x2 , v2 (x) = c4 x1 , v3 (x) = , v1 (u) = c2 + c3 u 1 + c 7 x1 + c 8 x2 c9 x22 v4 (x) = , v5 (x) = c11 x3 . c10 + x22 The differential equations of the state variables and the output equation are then, dx1 (t)/dt = v1 (u(t)) − v2 (x(t)) − v3 (x(t)), x1 (0) = x1,0 , dx2 (t)/dt = v3 (x(t)) − v4 (x(t)), x2 (0) = x2,0 , dx3 (t)/dt = v4 (x(t)) − v5 (x(t)), x3 (0) = x3,0 , y(t) = h(x(t)) = v5 (x(t)) = c11 x3 (t). The stoichiometric matrix is a matrix which maps the vector of reaction speeds to the differential of the state vector of chemical species. Define successively the stoichiometric matrix N , the reaction speed vector v(x, u), the state vector x, and the initial state x0 , as, ⎛ ⎞ 1 −1 −1 0 0 0 1 −1 0 ⎠ ∈ Z3×5 , N =⎝ 0 0 0 0 1 −1 ⎞ ⎛ v1 (u) ⎛ ⎛ ⎞ ⎞ ⎜ v2 (x) ⎟ x1 1 ⎟ ⎜ 5 ⎟ ⎝ ⎠ ⎝ 0 ⎠ ∈ R3 . (x) x v ∈ R , x = = , x v(x, u) = ⎜ 2 0 ⎟ ⎜ 3 ⎝ v4 (x) ⎠ x3 0 v5 (x) With these definitions, the biochemical reaction system in terms of the standard form of control and system theory becomes the rational system, dx(t)/dt = N v(x(t), u(t)), x(0) = x0 , y(t) = h(x(t)). The values of the constants are displayed in Table 2. The definition of the newly defined constants in terms of the original constants is then, c1 = Vmax,1 ATP, c2 = 1 +

ATP 1 , c3 = c2 , KATP,1 KGlucose,1

f r Vmax,3 Vmax,3 , c6 = , KGluc6P,3 KFruc6P,3 1 1 c7 = , c8 = , c9 = Vmax,4 , KGluc6P,3 KFruc6P,3 " 2 #  ATP c10 = KFruc6p,4 1 + κ , c11 = k5 . ADP

c4 = k2 ATP, c5 =

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Symbol vmax,1 KATP,1 KGlucose,1 k2 f Vmax,3 r Vmax,3 KGluc6P,3 KFruc6P,3

Value 1398 0.10 0.37 2.26 140.282 140.282 0.80 0.15

Dimension mM min−1 mM mM mM −1 min−1 mM min−1 mM min−1 mM mM

Symbol vmax,4 KFruc6P,4 κ k5 AT P ADP AM P

Value 44.7287 0.021 0.15 6.04662 1.9 1.8 0.4

Dimension mM min−1 mM 2 min−1 mM mM mM

Table 2. Values of the reaction constants.

The values and dimensions of the new constants are then provided in Table 3. Symbol c1 c2 c3 c4 c5 c6

Value 2656.20 20.00 54.05 4.29 175.35 935.21

Dimension mM 2 min−1 mM −1 min−1 min−1 min−1

Symbol c7 c8 c9 c10 c11

Value 1.2500 6.6667 44.7287 0.0921 6.0466

Dimension mM mM mM min−1 mM min−1

Table 3. Values of the newly defined reaction constants.

Next consider the problem of determining the steady state of this rational system. Fix the value of the glucose concentration to a particular value us = 12.8174 mM . Then one wants to prove existence and to determine the steady state, xs ∈ (0, ∞)3 , such that 0 = N v(xs , us ). Note that by inspection of the matrix follows that rank(N ) = 3, dim(ker(N )) = 5 − rank(N ) = 5 − 3 = 2, and a basis for the kernel of N is

⎛

⎜ ⎜ ker(N ) = span{w1 , w2 }, w1 = ⎜ ⎜ ⎝

1 1 0 0 0

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ , w2 = ⎜ ⎟ ⎜ ⎠ ⎝

1 0 1 1 1

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

The procedure to determine the steady state has to take account of the kernel of the stoichiometric matrix. For that kernel it is directly clear that, in steady state, the values v3,s = v4,s = v5,s and that v1,s = v2,s + v3,s . This is also seen

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335

from the diagram of the biochemical reactions. Then one can solve for the steady state xs components by solving a set of equations. Once the user has the biochemical reaction system then further questions can be investigated. Often simulations are made of the system. The resulting trajectories of the state variables and of observable variables are compared with measurements of the cell in various circumstances. The parameter values may be adjusted based on these results. How the biochemical reaction system behaves if the amount of glucose is reduced or increased can be investigated in the mathematical model and be compared with observations of the cells. In addition, the effect of diminishing one or more of the many reactions by introducing other chemical species can be investigated first in the mathematical model. Questions for researchers of control and system theory for biochemical reaction systems include the following. Is the system rationally observable from a particular set of observable variables? Is the system controllable from the enzyme concentrations of the reactions? Which reactions have to be blocked to prevent the biochemical reaction system to produce particular chemical species? Is the parameterized biochemical reaction system structuraly identifiable? How to estimate the parameter values of the biochemical reaction system? These questions have motivated the authors of this paper to develop the system theory of rational systems as described in this paper.

3.2. A Rational System of an Economic Control Problem In this subsection the reader finds an economic control problem which is formulated as a control problem of control theory in terms of a polynomial system. The problem has been borrowed from the paper [34]. Consider a producer of a product, an economic good. The aim of the producer is to maximize the profit or reward of the sales of the product. For this aim, the price of the product over a horizon has to be determined. For the price determination, the consumers are arbitrarily divided into a focus group and the remainder group. The price of the product is initially very high so that only members of the focus group can afford to buy the product. The remainder group cannot afford the product at that price. This way ownership of the product yields a social status to the members of the focus group. Over time the price of the product drops. If the price has dropped below a particular price then the members of the remainder group will also start to buy the product. But at a lower price it becomes less attractive to the members of the focus group to buy the product, possession of the product looses part of its social status. How to determine a price trajectory for the product so that the profit or rewards are maximized over time? The above formulated economic control problem can be formulated as a problem of control theory.

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The economic model described above is reformulated as a control system. The mathematical model is described and subsequently explained as T = [0, ∞), t1 ∈ (0, ∞], x1 : T → R+ , x2 : T → R+ , u : T → R+ ,   x1 (t) x(t) = , x2 (t) c ∈ [0, ∞), a1 ∈ (0, ∞), a2 ∈ (0, ∞), a2 < a1 , M ∈ (0, ∞), S ∈ [0, ∞), u(t) ∈ [0, a1 ], the constraint on the price; where x1 (t) represents the fraction of the focus group who has acquired the product at time t ∈ T , x2 (t) represents the fraction of the remainder group who has acquired the product at time t ∈ T , u(t) represents the price of the product at time t ∈ T , c represents the production costs of one product, a1 represents the price above which focus group members will not buy the product, a2 represents the price above which remainder group members will not buy the product, M represents the relation of the population sizes of the remainder group/the focus group, S represents the price per fraction of the focus group at the end of horizon. The dynamics of the state variables x1 and x2 is assumed to be described by the differential equations 1 dx1 (t) = (1 − x2 (t))(a1 − u(t)), x1 (0) = 0, 1 − x1 (t) dt 1 dx2 (t) = x1 (t) max{0, (a2 − u(t))}, x2 (0) = 0. 1 − x2 (t) dt

(3.1) (3.2)

The first differential equation states that of the focus group a fraction 1/(1 − x1 (t)) which has not yet acquired a product may in an interval of length dt acquire a product with rate (1 − x2 (t))(a1 − u(t)) which is proportional to the fraction of the remainder group who have not yet bought the product multiplied by the difference of the maximal price a1 minus the actual price u(t). The corresponding rate for the remainder group is the multiplication of the proportion of the focus group who have bought the product already at the time due to the attraction of the project as a social status symbol, and the price difference (a2 − u(t)). The differential equations may be transformed to the form of control theory, dx1 (t) = (1 − x1 (t))(1 − x2 (t))(a1 − u(t)), (3.3) dt dx2 (t) = x1 (t)(1 − x2 (t)) max{0, a2 − u(t)}, (3.4) dt dx(t) = p(x(t), u(t)), x(0) = 0. (3.5) dt The control system defined above is a polynomial system though with two domains because of the max operator. Note that it is not a bilinear system due to the presence of a term x1 x2 u in the first differential equation.

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It is simple to prove that if u(t) ∈ [0, a1 ] for all t ∈ T then [0, 1]2 ⊂ R2+ is an invariant set of the differential equation. An invariant set means that if x(0) ∈ [0, 1]2 then for all t ∈ T , x(t) ∈ [0, 1]2 . The control objective is to maximize the profit. The profit is formulated as the reward function which is to be maximized by a price setting, 

 t1  dx1 (s) dx2 (s) (u(s) − c) ds + Sx1 (t1 ) J(t1 ) = +M (3.6) ds ds 0

 t1 (1 − x1 (s))(1 − x2 (s))(a1 − u(s))+ = ds + Sx1 (t1 ). +M x1 (s)(1 − x2 (s)) max{0, a2 − u(s)} 0 (3.7) The rate of the cost function is a polynomial in the variables (x1 , x2 , u) but with two domains due to the max operator. The solution of the optimal control problem is the price trajectory over time. The solution depends on the relations of the parameters of the problem, the product cost, the maximal prices where the focus group will start to buy and the remainder group will start to buy, the numerical relation between the two groups, and the sale price at the end of the horizon. The general form of a price trajectory is that the initial price is at or near the maximal price of the focus group, u(t) ≈ a1 . Over time it drops continuously. Then, at a particular time depending on the parameters of the problem, the price drops with a jump such that u(t) < a2 hence the members of the remainder group will also start to buy the product. The dynamics of the price depend on the parameters of the problem and have to be computed via the Hamiltonian and the co-state differential equations. That the control system is polynomial is due to the assumptions on the dynamics and on how the price affects the dynamics. The multiplication of the fraction of buyers times the price yields the polynomial of the state and of the input. The resulting optimal control problem has by assumption a polynomial cost rate.

4. Rational Systems 4.1. Concepts Terminology and notation of commutative algebra is used from the books [27, 10, 11, 66, 67]. References on algebraic geometry include, [19, 33, 35, 57, 58]. The notation of the paper is standard. The set of the integers is denoted by Z and the set of the strictly positive integers by Z+ = {1, 2, . . .}. For n ∈ Z+ define Zn = {1, 2, . . . , n}. The set of the natural numbers is denoted by N = {0, 1, 2, . . .} and, for n ∈ Z+ , Nn = {0, 1, 2, . . . , n}. The set of the real numbers is denoted by R and that of the positive and the strictly positive real numbers respectively by R+ = [0, ∞) and Rs+ = (0, ∞). The vector space of n-tuples of the real numbers is denoted by Rn .

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A subset X ⊂ Rn for n ∈ Z+ is called a variety if it is determined by a finite set of polynomial equalities, X = {x ∈ Rn | 0 = pi (x), ∀ i ∈ Zk }, n, k ∈ Z+ . Such a set is also called an algebraic set. A variety is called irreducible if it cannot be written as a union of two nonempty varieties. A canonical form for a variety can be formulated based on a decomposition of the ideal of polynomials defining the variety as a union of prime ideals. This will not be detailed in this paper. Denote the algebra of polynomials in n ∈ Z+ variables with real coefficients by R[X1 , . . . , Xn ]. For a variety X, denote by I(X) the ideal of polynomials of R[X1 , . . . , Xn ] which vanish on the variety X. The elements of R[X1 , . . . , Xn ]/I(X) are referred to as polynomials on the variety X. The ring of all such polynomials on the variety is denoted by AX which is also an algebra. If the variety X is irreducible then the ring AX is an integral domain hence one can define the field of rational functions on the variety X as a field of fractions QX of the algebra AX . A polynomial and a rational function for n variables are denoted respectively by the representations (assumed to be defined over a finite sum) p(x) =



cp (k)

k∈Nn

n M

k(i)

xi

=

i=1



cp (k)xk ∈ R[X1 , . . . , Xn ],

k∈Nn

(∀ k ∈ Nn , cp (k) ∈ R), r(x) =

p(x) ∈ R(X1 , . . . , Xn ), p, q ∈ R[X1 , . . . , Xn ], q = 0. q(x)

A rational function can be transformed into another rational function (1) by cancellation of common factors of the numerator and the denominator polynomial, and (2) by multiplication of the numerator and the denominator by a real number. If one defines an equivalence relation on the set of rational functions based on the two transformations defined above, then one may define a canonical form. For a rational function the following canonical form is defined: (1) there are no common factors in the numerator and the denominator; and (2) the constant factor in the denominator polynomial, assumed to be present, is set to one by multiplication of the numerator and the denominator by a real number. r(x) =

p(x) , q(x) = 1 + q(x)



cq (k)

k∈Nn \{0}

QX,can = {r(x) ∈ QX , as defined above}.

n M

k(i)

xi

,

(4.1)

i=1

(4.2)

The transcendence degree of a field F , denoted by trdeg(F ), is defined to be the greatest number of algebraically-independent elements of F over R, [10, Section 7.1, p. 293, p. 304] and [66, Ch. 2, Sections 3 and 12]. For the detailed definitions of the concepts introduced below, the reader is referred to the papers, [45, 48].

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Definition 4.1. An affine-input rational system on a variety (also called a rational control system on a variety) is defined as a control system as understood in control theory with the representation, X ⊆ Rn , an irreducible variety, dx(t)/dt = f0 (x(t)) +

m 

f1,j (x(t))uj (t), x(0) = x0 ,

(4.3) (4.4)

j=1

y(t) = h(x(t)),

(4.5)

s = (X, U, Y, f0 , f1 , h, x0 ) ∈ Sr ; where n, m, p ∈ Z+ , X ⊆ Rn is an irreducible nonempty variety called the state set, U ⊆ Rm is called the input set, it is assumed that {0} ⊆ U and that U contains at least two distinct elements, Y = Rp is called the output set, x0 ∈ X is called the initial state, f0,1 , . . . , f0,n , f1,1,1 , . . . f1,m,n ∈ QX and h1 . . . hp ∈ Qx are rational functions on the variety, u : [0, ∞) → U is a piecewise-constant input function, and Sr denotes the set of rational systems as defined here. Call the system a single-input-single-output affineinput rational system if m = 1 and p = 1. One defines the set of piecewise-constant input functions which are further restricted by the existence of a solution of the rational differential equation of the system. Denote for a rational system s ∈ Sr as defined above, the admissible set of piecewise-constant input functions by Upc (s). Further, for u ∈ Upc (s), tu ∈ R+ denotes the life time of the solution of the differential equation for x with input u thus [0, tu ) is the interval of existence of the solution. For any u ∈ Upc (s) and any t ∈ [0, tu ) denote by u[0, t) the restriction of the function u to the interval [0, t). By assumption, u[0, t) ∈ Upc for all t ∈ [0, tu ). Denote the solution of the differential equation (4.4) by x(t; x0 , u[0, t)), ∀ t ∈ [0, tu ). Define the dimension of this rational system s ∈ Sr as the transcendence degree of the set of rational functions on X, dim(X(s)) = trdeg(QX ). In the remainder of the paper, a rational system will refer to a inputaffine rational system on a variety as defined above. Definition 4.2. In the remainder of this subsection, attention is restricted to a single-output system, thus with p = 1. Associate with any rational system in the considered set, its response map as, rs : Upc → Y = R, rs ∈ A(Upc → R), where A(Upc → R) denotes the algebra generated by all polynomial maps from Upc to R, such that for all u ∈ Upc , if x : [0, tu ) → X is a solution of the rational system s ∈ Sr for u and for x0 ∈ X, i.e. x satisfies the differential equation (4.4) and the output (4.5) is well defined, then rs (u[0, t)) = y(t) = h(x(t; x0 , u[0, t))) ∈ Y = R, ∀ t ∈ [0, tu ). The realization problem for the considered set of rational systems is, when considering an arbitrary response map, to determine whether there

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exists a rational system whose response map equals the considered response map. See for the literature the papers [44, 45] and the references quoted there, and in particular, [63]. Definition 4.3. Consider a response map, r : Upc → R. Call the system s ∈ Sr a realization of the considered map r if ∀ u ∈ Upc , ∀ t ∈ [0, tu ), r(u[0, t)) = rs (u[0, t)). One can define the observation algebra of the response map, Qobs (r), and its transcendence degree trdeg(Qobs (r)), see [44, 45]. Call the system a minimal realization of the response map if dim(X(s)) = trdeg(Qobs (r)). Define a minimal rational system to be a rational system which is a minimal realization of its own response map. In general, a realization is not unique. Attention will be restricted to minimal realizations characterized by a condition of controllability and of observability defined next. Definition 4.4. Consider a rational system as defined in Def. 4.1. The observation algebra Aobs (s) ⊆ QX of a rational system s ∈ Sr is defined as the smallest subalgebra of QX which contains the R1 -valued components of the output map h and is closed under taking Lie derivatives with respect to the vector fields of the system: n m   ∂a(x) Lfα a(x) = [fi,0 (x) + f1,i,j (x)αj ] , a ∈ QX , (4.6) ∂xi i=1 j=1 ⎫⎞ ⎛⎧ ⎨ h1 , . . . , hp , Lfα h1 , . . . , Lfα hp , ⎬ ⎠. Aobs (s) = R ⎝ Lfαk · · · Lfα1 hj ∈ QX , (4.7) ⎭ ⎩ ∀ α1 , . . . , αk ∈ U, ∀ j ∈ Zp , ∀ k ∈ Z+ Denote by Qobs (s) ⊆ QX the field of fractions of Aobs (s) and call this set the observation field of the system. Call the system s ∈ Sr rationally observable if the observation field of the system equals the set of all rational functions, Qobs (s) = {p/q ∈ QX | p, q ∈ Aobs (s)}, QX = Qobs (s).

(4.8) (4.9)

Definition 4.5. Consider a rational system as defined in Def. 4.1. Call the system algebraically controllable or algebraically reachable if X = Z − cl({x(t) ∈ X| t ∈ [0, tu ), u ∈ Upc }), where Z − cl(W ) of a set W ⊂ Rn denotes the smallest variety containing the set W , also called the Zariski-closure of W . See [51] for procedures how to check that algebraic controllability holds. It follows from the existing realization theory that if a rational system is algebraically controllable and rationally observable then it is a minimal realization of its response map, [45, Proposition 6]. A minimal realization of a response map is not unique. It has been proven that any two minimal rational realizations are birationally equivalent

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if the condition holds that the elements of QX \Qobs (s) are not algebraic over Qobs (s) for both systems s ∈ Sr . 

Definition 4.6. Let X ⊆ Rn and X  ⊆ Rn be two irreducible varieties. A birational map from X to X  is a map which has n components which are all rational functions of QX and for which an inverse exists such that it is a map which has n components which are all rational functions of QX  . A birational map transforms a rational system on a variety to another rational system on another variety, see [45]. A reference on birational geometry is [38]. 4.2. Overview of Existing Theory The realization problem of rational systems was solved. An equivalent condition has been proven for the existence of a finite-dimensional realization in the form of a rational system of a response map, see [44]. The equivalent condition is that there exists a finitely generated field which contains the observation field of the response map. A characterization of rational observability and of algebraic controllability, is provided in [45]. Necessary and sufficient conditions for structural identifiability of a structured rational system are published in [43]. A reduction procedure for a non-observable rational system to a rationally-observable rational system was formulated and proven to be correct in [48]. An exploration of checkable conditions for algebraic controllability was published in the conference paper [51]. An introduction to modeling biochemical reaction systems as rational systems was provided in [50]. Rational observers for rational systems were synthesized in [49]. A system identification algorithm for identification of a discrete-time polynomial system based on a subalgebraic approach was published in [47]. Realization of biochemical reaction systems was developed in [2, 3, 1].

5. Canonical Forms of Rational Systems There is a need for canonical forms of rational systems to be used in system theory, in system identification, and in control synthesis. The problem of the formulation of a canonical form requires an introduction. Consider the problem of obtaining, for a response map, a realization in the form of a rational system. As stated in Section 2, this problem has been solved. There exists a set of minimal rational realizations of the considered response map. This subset in general has many members. It is of interest to have only one member representing the complete subset. For this purpose consider the subset of rational realizations each of which is a minimal realization of its own response map. The equivalence relation is defined on this considered subset of rational systems by the condition that two systems are equivalent if they have the same response map. A

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canonical form is to be defined on the set of minimal rational systems with the considered equivalence relation. A canonical form is needed for system identification. In system identification one is assumed to be provided a set of input-output measurements. The problem of system identification is to find the values of the parameters of a rational system of which the response map matches approximately the considered input-output measurements. Because the set of minimal realizations of a response map has many members, it is necessary to define a canonical form for the set of rational systems having the same response map. If a canonical form of the set of rational systems has been chosen then that form can be used for system identification. A canonical form is such that the parameters of this form are uniquely determined by the response map. Hence an approximation of the response map in the form of input-output measurements yields an approximation of the parameters of a rational system. The concept of a canonical form is defined in the area of universal algebra. For a set with an equivalence relation, the concept of a canonical form has been formulated, see [36, Section 0.3], or [64, Subsection 2.2.1], or [10, Section 4.5 Reduction relations]. Consider the set of time-invariant linear systems each of which is a minimal realization of its own response map. The equivalence relation considered most in the literature is the feedback-andresponse-map equivalence defined by the conditions that (1) the response map of two equivalent systems are identical and (2) the response maps of the two equivalent systems may differ by a linear state-feedback control law. A canonical form for the considered set of time-invariant linear system and the above defined equivalence relation was formulated independently by P. Brunovsky, by M. Popov, and by W.M. Wonham and A.S. Morse, in the papers, [15, 55, 65]. For system identification a weaker form of equivalence has to be considered, only response-map equivalence of the systems. The controllable canonical form and the observable canonical form of a time-invariant linear system for response-map equivalence only are defined but not proven in for example, [16, Section 9.2] and [53]. For a subclass of nonlinear systems in a differential-geometric structure, W. Respondek and A. Tall have investigated a canonical form for that class of systems for the equivalence relation of feedback-and-response-map equivalence, [60]. Below the attention is directed to a canonical form for a subset of rational systems. Of interest to system theory is also the approach of determining invariant varieties of a polynomial or a rational system and their controlledinvariant varieties. See [68, 56]. The approach of those papers does not cover the canonical forms as discussed in this subsection. Definition 5.1 (Canonical form). Consider a set X and an equivalence relation E ⊆ X × X defined on it. A canonical form of this set and this equivalence relation consists of a set Xc ⊆ X such that for any x ∈ X there exists a unique member xc ∈ Xc such that (x, xc ) ∈ E.

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Definition 5.2. Consider the subset Sr,min ⊆ Sr of the set of rational systems Sr each of which is a minimal realization of its response map. Restrict further attention to those rational systems for which the set QX \Qobs (s) is not algebraic over Qobs (s). Denote the resulting set by Sr,min,na . Define the relation Erm ⊆ Sr,min,na × Sr,min,na by the condition that each (s1 , s2 ) ∈ Erm is such that the response maps of the systems s1 and s2 are identical. It is then easily proven that the relation is an equivalence relation. Problem 5.3. Consider the subset of rational systems Sr,min,na and the equivalence relation Erm on that set. Formulate a canonical form for this subset of rational systems. Prove that the candidate canonical form is actually a proper canonical form. The obstacles of Problem 5.3 are in the choice of the canonical form. A canonical form is almost never unique for classes of systems. For the class of time-invariant finite-dimensional linear systems there exist the control canonical form and the observable canonical form. For rational system the observable canonical form is defined next. Definition 5.4. Consider the subset of single-input-single-output affine-input rational systems. Restrict further attention to those rational systems in the considered subset for which the set QX \Qobs (s) is not algebraic over Qobs (s). Define the observable canonical form as the algebraic structure, X = {x ∈ Rn |0 = pi (x), ∀ i ∈ Zk }, an irreducible nonempty variety, d = trdeg(Qobs (s)) ∈ Z+ , n, k ∈ Z+ , dx1 (t)/dt = x2 (t) + f1,1 (x(t)) u(t), x1 (0) = x1,0 ,

(5.1)

.. . dxn (t)/dt = fn,0 (x(t)) + fn,1 (x(t)) u(t), xn (0) = xn,0 , y(t) = x1 (t),

(5.2) (5.3)

{fi,1 , fn,0 ∈ QX,can , ∀ i ∈ Zn }, (∀ x ∈ X\Ae , fn,1 (x) = 0); Sr,ocf ⊂ Sr , denotes the subset of rational systems, in the observable canonical form. See for the definition of QX,can equation (4.2). Every rational system in the observable canonical form is assumed to be algebraically controllable. Comments on the definition follow. The subset of rational systems is below argued to be a proper canonical form. Note that any system of the form is not defined as a minimal realization of its response map. However, it can be proven that any system in the observable-canonical form is rationally observable. Because by assumption, a system in the observable-canonical form is algebraically controllable, it follows from [45, Proposition 6] that the system is a minimal realization of its response map. The assumption of algebraic

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controllability should not be a concern to the readers. Even in the case of the observable canonical form for a time-invariant linear system, the assumption of controllability has to be imposed to obtain a minimal realization. The particular form of the rational systems of the observable-canonical form is due to the emphasis on observability. Note that y = x1 and that in the differential equation for xi the variable xi+1 occurs. The form has still several rational functions which parametrize the form. The condition of nonsingularity of the function fn,1 is conjectured to be a necessary condition for algebraic controllability. Theorem 5.5. [52, Th. V.4]. Consider the setting of Def. 5.2 for the subset Sr,min,na of minimal rational systems and the equivalence relation Erm of response-map equivalence. The observable canonical form of Def. 5.4 is a well-defined canonical form for the subset of minimal rational systems Sr,min,na and the equivalence relation Erm . Thus, each system in the form is a minimal realization of its response map and each minimal rational system is equivalent to a unique element of the observable-canonical form. Hence Sr,ocf ⊆ Sr,min,na . Example 5.6. Consider the following rational system of single-input-singleoutput form, affine in the input, with the particular representation dx1 (t)/dt = x2 (t)3 − c1 x1 (t)2 u(t), x1 (0) = x1,0 , c2 x1 (t) − x2 (t) + u(t), x2 (0) = x2,0 , 1 + x2 (t) 1 + x2 (t) y(t) = x1 (t), c1 , c2 ∈ (0, ∞).

dx2 (t)/dt =

Assume that the system is algebraically controllable. Then this system belongs to the observable canonical form. A consequence is that the system is rationally observable. How to use the observable-canonical form for modeling a rational system? In system identification this canonical form can be used. Suppose one wants to model observations by a minimal rational system. Assume further that there is no apriori information on the structure of the system. Consider then the equivalence relation defined by two systems having the same response map. Then the observable canonical form can be used. That form guarantees that a rational system in the observable canonical form represents all rational systems which have the same response map and are minimal realizations of their own response map. Because a rational system in the observable canonical form still has several rational functions or polynomial functions, one may impose assumptions on the degrees of the polynomial for the various components of the state. For example, a polynomial in two indeterminates with each monomial of the polynomial having the sum of the degrees bounded by 2, is of the form f (x) = c(0) + c(1, 0)x1 + c(0, 1)x2 + c(2, 0)x21 + c(1, 1)x1 x2 + c(0, 2)x22 .

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With this parametrization the rational system in the observable canonical form has a relatively small number of parameters. The set of single-inputsingle-output rational systems in the observable canonical form with the above parametrization of the polynomials then parametrizes all minimal rational systems in that restricted set. A canonical form can also be used to synthesize a control law for a rational system. See Section 8 for the approach. But the approach requires that the original system is made rationally-observable, algebraically controllable, hence a minimal realization of its response map, and transformed to the observable canonical form. The extension of the observable-canonical form to rational systems with dimension of the output vector of two or more is rather straightforward. The dependencies of the outputs on the state components become more complex but the structure is analogous. The observable-canonical form for timeinvariant linear systems provides the framework for this generalization, see [16]. A control-canonical form for time-invariant linear systems has been defined. The corresponding control-canonical form for rational systems with a single input has not been derived yet though the authors have carried out research on this. The form will be useful for control synthesis, see Section 8. The concept of the rational controllability indices will have to be defined also.

6. Decompositions and Subclasses of Rational Systems In biochemical reaction networks or in engineering, a rational system often has a particular structure such as a line network, a circle network, or a ladder network, or is a combination of several of such structures. This structure can then be used in the control and system theoretic problems considered. This approach is underdeveloped for rational systems. Problem 6.1 (The decomposition problem of a rational control system). Consider a large rational control system with several inputs and several outputs. Investigate the following subproblems. 1. Unique network determination. Can one uniquely determine from tuples of inputs and outputs of this system, or from the response-map, a network of rational systems and the relations of the various subsystems? It may be assumed that the overall system is a minimal realization of its response map. The main issue here is whether the network structure is uniquely determinizable from the response map. For many other classes of systems it has been proven that it is not possible to determine uniquely the network structure. The problem remains as to what part of the network structure is uniquely determinable with which assumptions. 2. Elementary subnetworks. What are appropriate subsets of rational control systems based on their network structure, so that every rational control system can be decomposed as an interconnection of rational

346

3.

4.

5.

6.

7.

J. Nˇemcov´ a, M. Petreczky and J.H. van Schuppen systems in these structures. Appropriate conditions have to be imposed to have the problem make sense. One expects the subsets to include a line structure and a ring structure as defined below. Possibly also a ladder structure, because of its role in the DNA structure. Sufficient conditions for preservation of minimality in a network of rational systems. If each of the subsystems of a rational control system is a minimal realization of its response map is then the network of those subsystems a minimal realization of its response map? If not, can one formulate sufficient conditions on the subsystems which will imply that the entire system is a minimal realization? It seems that one must impose a condition that the output behavior of a subsystem is not in the subset of inputs of a subsequent system which result in null outputs of that system. Determination of dominant subsystems. Can one identify subsystems which have a dominant role on the behavior of the complete network of rational systems? It seems likely that a particular subsystem in a line network is comparatively slow and hence influences dominantly the behavior of the entire network of rational systems. System reduction of a network of rational systems. Is it possible to reduce the network of rational control systems to a new network with a smaller number of subsystems and of smaller complexity than the original network? This is of interest to modeling of biochemical reaction systems which are very large. For the behavior of the interconnection, a smaller subsystem may suffice. Synthesis of a network of rational systems from its response map. Considering a response-map of a rational control system and a particular class of network structures, can one synthesize a network of rational control systems in the considered network structures so that the responsemap of the synthesized system equals the considered response map? This is of interest to synthetic biology. Control synthesis for particular subsystem. How can one control individual subsystems of a network of rational control systems without directly affecting other subsystems to a major degree? A detailed description of two networks of rational systems follows.

6.1. A Line Network of Rational Control Systems Consider then a line network of rational control systems. The line consists of nodes numbered consecutively, from 1 to n ∈ Z+ . Each node i ∈ Z+ is connected to the state variables of node i − 1 and of node i + 1, except for the first node 1 which is only connected to node 2 and the last node n which is only connected to node n − 1. Each node corresponds to a rational system of state-space dimension one. The dynamics of the rational system involves the state of the two nearest neighbors with the exceptions of the rational systems at the first and the last node. The input of the rational control system is only

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present at the rational system of the first node while the output depends only on the state of the rational system of the last node. Such a line network of rational systems occurs often as a subnetwork in biochemical reaction systems but also in physiological systems and in engineering. The example of glycolysis in yeast of Subsection 3.1 is not a line network because it has branch at the second node of the line when going downstream from the initial node. A mathematical formulation follows. Definition 6.2 (A line network of rational control systems). Define a line network of rational control systems to consist of a rational system with the structure n ∈ Z+ , {xi , ∀ i ∈ Zn }, x = (x1 , x2 , . . . , xn−1 , xn )T , dx1 (t) = f1 (x1 (t), x2 (t), u(t)), x1 (0) = x1,0 , dt dxi (t) = fi (xi (t), xi−1 (t), xi+1 (t)), xi (0) = xi,0 , i = 2, 3, . . . , n − 1, dt dxn (t) = fn (xn (t), xn−1 (t)), xn (0) = xn,0 , dt y(t) = h(xn (t)),

(6.1) (6.2) (6.3) (6.4)

{f1 ∈ R(x, u), fi , h ∈ R(x), ∀ i ∈ Zn \{1}}. Define a line network of biochemical reaction rational control systems as a subset of a line network of rational systems with the representation n ∈ Z+ , {xi , ∀ i ∈ Zn }, x = (x1 , x2 , . . . , xn−1 , xn )T , dx1 (t) = r1+ (x1 (t), u(t)) − r1− (x1 (t), x2 (t)), x1 (0) = x1,0 , dt dxi (t) = ri+ (xi (t), xi−1 (t)) − ri− (xi (t), xi+1 (t)), xi (0) = xi,0 , dt i = 2, 3, . . . , n − 1, dxn (t) = rn+ (xn (t), xn−1 (t)) − rn− (xn (t)), xn (0) = xn,0 , dt y(t) = h(xn (t)), {r1+

∈

R+ (x, u), ri+ ,

rj− ,

(6.5)

(6.6) (6.7) (6.8)

h ∈ R+ (x), ∀ i ∈ Zn \{1}, ∀ j ∈ Zn }.

The reader is alerted of the fact that R is a field while R+ is not a field but has the algebraic structure of a semi-ring. The set R+ is closed with respect to addition and to multiplication but no inverse exists with respect to addition while only the strictly positive elements in Rs+ = (0, ∞) ⊂ R+ admit an inverse with respect to multiplication. 6.2. A Ring Network of Rational Systems In many types of cells there occurs a reaction network with a ring structure. Examples are the Krebs cycle and the Calvin cycle of biochemical reaction

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networks. See [17]. It is of interest for the understanding of biochemical reaction systems to investigate such a ring network of rational systems or of polynomial systems. In the literature a ring is also called a cycle or a cycle network. Consider a ring network with eight nodes, say consecutively numbered from 1 to 8. There are assumed to be four material inflows at all of the odd numbered nodes. There are assumed to be material outflows at all of the even numbered nodes. The mathematical notation of the system follows. Definition 6.3 (A ring network of rational control systems). The network may be associated with a graph which is a ring with eight nodes, numbered consecutively from 1 to 8. The inflows are only at the odd-numbered nodes and the outflows are only even-numbered nodes. The rational control system is defined by the representation, {xi , ∀ i ∈ Z8 }, x = (x1 , x2 , . . . , x7 , x8 )T , dx1 (t) = r1+ (x1 (t), x8 (t), u1 (t)) − r1− (x1 (t), x2 (t)), x1 (0) = x1,0 , dt dx2 (t) = r2+ (x2 (t), x1 (t)) − r2− (x2 (t), x3 (t)) − h1 (x2 (t), x3 (t)), dt x2 (0) = x2,0 , y1 (t) = h1 (x2 (t), x3 (t)), dx3 (t) = r3+ (x3 (t), x2 (t), u2 (t)) − r3− (x3 (t), x4 (t)), x3 (0) = x3,0 , dt dx4 (t) = r4+ (x4 (t), x3 (t)) − r4− (x4 (t), x5 (t)) − h1 (x4 (t), x5 (t)), dt x4 (0) = x4,0 , y2 (t) = h2 (x4 (t), x5 (t)), dx5 (t) = r5+ (x5 (t), x4 (t), u3 (t)) − r5− (x5 (t), x6 (t)), x5 (0) = x5,0 , dt dx6 (t) = r6+ (x6 (t), x5 (t)) − r6− (x6 (t), x7 (t)) − h1 (x6 (t), x7 (t)), dt x6 (0) = x6,0 , y3 (t) = h3 (x6 (t), x7 (t)), dx7 (t) = r7+ (x7 (t), x6 (t), u4 (t)) − r7− (x7 (t), x8 (t)), dt x7 (0) = x7,0 , dx8 (t) = r8+ (x8 (t), x7 (t)) − r8− (x8 (t), x1 (t)) − h1 (x8 (t), x1 (t)), dt x8 (0) = x8,0 , y4 (t) = h4 (x8 (t), x1 (t)),

(6.9) (6.10)

(6.11) (6.12) (6.13)

(6.14) (6.15) (6.16)

(6.17) (6.18) (6.19) (6.20)

(6.21)

Towards a system theory of rational systems ⎞ u1 (t) ⎜ u2 (t) ⎟ ⎟ u(t) = ⎜ ⎝ u3 (t) ⎠ , u4 (t) ⎛ ⎞ ⎛ y1 (t) ⎜ y2 (t) ⎟ ⎜ ⎟ ⎜ y(t) = ⎜ ⎝ y3 (t) ⎠ = ⎝ y4 (t)

349

⎛

⎞ h1 (x2 (t), x3 (t)) h2 (x4 (t), x5 (t)) ⎟ ⎟ = h(x(t)), h3 (x6 (t), x7 (t)) ⎠ h4 (x8 (t), x1 (t))

((∀ i ∈ Z8 ), ri+ , ri− ∈ R+ (x, u)), ((∀ i ∈ Z4 ), hi ∈ R+ (x)); dx(t) = f (x(t), u(t)), x(0) = x0 , dt y(t) = h(x(t)).

(6.22) (6.23)

There are several questions for ring networks requiring investigation. Can the behavior of a ring network become blocked due to lack of an input or of the block of an output? How can the cell optimize the behavior of the ring network by achieving the largest throughput considering fixed inputs?

7. System Approximation In this section the reader finds a discussion, a problem formulation, and a sketch of an approach to approximate the response map of a rational system by the response map of another rational system of lower complexity. The motivation of this problem of system approximation is to reduce the complexity of a rational system. In biochemical reaction networks, the models in the form of biochemical reaction systems contain many terms consisting of a coefficient multiplying a monomial. It is not clear how many monomials are useful in the model. If a coefficient of a monomial term is relatively small then the effect of that monomial term on the overall dynamics of the rational system may be small. Of course, the relative contribution depends on the degrees of the monomial and on the size of the values of the state variables. The complexity is often a combination of the state-space dimension, of the degrees of the polynomials, and of the form of the rational functions. The formulation of a complexity criterion is part of the problem. For the discussion which follows, the reader may think that the complexity is represented only by the state-space dimension, which is n ∈ Z+ for the rational system of Def. 4.1. As defined in Def. 4.2, one may associate with each rational system its response map which maps an input trajectory to an output value of the system. The problem of system approximation is then, considering a rational system, to determine another rational system of lower state-space dimension such that the two response maps of the two systems are close with respect to an approximation criterion. The motivation of the systems approximation problem is further discussed below. Researchers with experience in handling examples of rational

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systems will recognize the usefulness of a solution to the system approximation problem. A line network of rational systems could be reduced in regard to the length of the line of systems. Consequently, the state-space dimension may be reduced. The actual reduction depends on the coefficients of the monomial terms of a rational system. In case of a ring network, a system reduction of the form of shortening the number of systems in the ring or even of collapsing the ring, could be an aproximation. In case of a ladder network, another approximate rational system may appear. The system approximation problem is not a full fledged algebraic problem because the approximation criterion is based on analysis rather than on algebra. Yet, there are algebraic aspects of the problem which require investigation. The case of the subclass of polynomial systems is more suitable for system reduction than the subclass of rational systems. The global dynamics of a rational system is quite restricted and these restrictions have to be taken care of in the approximation. It is not yet clear how to handle the domain restrictions in the approximation problem. The formal problem follows. Problem 7.1 (Problem of rational system approximation). Consider a rational system of state-space dimension n0 ∈ Z+ and class of input functions U0 . Determine a second rational system of state-space dimension n1 ∈ Z+ with n1 < n0 , such that the approximation criterion between the response map of the two systems is small for a particular class of input functions belonging to Ur ⊆ U0 . Instead of the dimension of the state space another complexity criterion can be considered. Example 7.2. The following example shows how in a simple way a rational system can be reduced from a system of state-space dimension two to one of state-space dimension one. Consider a rational system, single-input-single-output and affine in the input, with representation dx1 (t)/dt = f1,0 (x(t)) + f1,1 (x(t)) u(t), x1 (0) = x1,0 , dx2 (t)/dt = −c1 x2 (t) + c2 x1 (t), x2 (0) = x2,0 , y(t) = x2 (t), c1 , c2 ∈ (0, ∞), c1 * 0. Note that the state x1 is the output of the first differential equation of the system and then the input of the second differential equation of the system. The rational system is thus a line network. With an assumption on the first differential equation it is simple to prove that the domain R2+ = [0, ∞)2 is an invariant set for this system. Assume that the dynamics of the differential equation for x2 is fast compared to that of the differential equation for x1 . Then the second differential equation can be regarded in steady state most of the time, relative to the transient behavior of the first system. In the domain (0, ∞), there exists a

Towards a system theory of rational systems

351

unique solution for x2 if x1 is considered fixed, of the equation 0 = −c1 x2 + c2 x1 ⇒ x2 = (c2 /c1 )x1 = h(x1 ). Note that c1 x2 is an increasing function of x2 and c2 x1 is a constant function of x2 on the domain (0, ∞). Hence for any x1 ∈ (0, ∞) there exists a unique x2 which is a solution of the equation. The approximate system is then dx1 (t)/dt = f1,0 (x(t)) + f1,1 (x(t)) u(t), x1 (0) = x1,0 , y(t) = h(x1 (t)) = (c2 /c1 )x1 (t). The approximate system has state-space dimension one while the original rational system has state-space dimension two. The approximation is due entirely to the assumption that the differential equation for x2 is comparatively faster than that for x1 .

8. Control Synthesis of Rational Systems In this section the reader finds a formulation of how to find a rational control law such that the closed-loop system meets predescribed control objectives. A simple example is shown for a special case. The problem formulation is preceded by an introduction to the framework of control synthesis. This framework is well known to researchers of control theory but may not be known to the mathematically-interested readers of this paper. Consider a rational control system with inputs and outputs. A statefeedback-based rational control law is a rational function g which maps states to inputs, u = g(x). A rational control system combined with a rational control law is again a rational system, called the closed-loop control system. The closed-loop control system has particular properties which may be valuable to a user. Control synthesis is the opposite procedure of the preceding paragraph. One considers a rational control system. One specifies the control objectives, for example the behavior of the control-objective system. The control problem is to determine a rational control law such that the closed-loop system meets the control objectives, in particular the closed-loop system equals the controlobjective system or it is birationally related to the control-objective system. Control theory is well developed for several classes of control systems, for example for time-invariant finite-dimensional linear systems and for nonlinear systems in differential-geometric structures. Novel in the above control problem is the restriction to rational control laws and in the description of the closed-loop system by the control-objective system. The restriction to only rational control laws is motivated by the algebraic framework considered. General control objectives are stability properties, performance properties, suppression of disturbances or noise, robustness for small variations in the parameters of the system, and adaptation to structural changes of the environment.

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The motivation of the control problem for a rational system varies with the domain of engineering and sciences considered. In the life sciences, control problems arise to stimulate the behavior of the cell to carry out particular actions or to block particular reactions of enzymes by medicines. A problem of drug design has been reformulated as a control problem, [62]. In engineering, control problems arise to make a machine carry out operations accurately. Problem 8.1. Consider a rational control system without output function, of the form dx(t)/dt = f (x(t), u(t)), x(0) = x0 . Consider the control objective of attaining the dynamics of the closed-loop system of the form dx(t)/dt = fcl (x(t)), x(0) = x0 , fcl ∈ QX . The dynamics could also be obtained after a birational state-space transformation. Does there exist a control law g ∈ G in the set of rational state-feedback control laws G such that the closed-loop system equals the prescribed rational system of the control objective, G = {g : X → U, g ∈ Q(x)}, g ∈ G, dx(t) = f (x(t), g(x(t)), x(0) = x0 , dt u(t) = g(x(t)), either f (x, g(x)) = fcl (x),

(8.1) (8.2) (8.3) (8.4)

or ∃ s ∈ QX , such that, fcl (s(x)) =

n  ∂s−1 (x) i=1

∂xi

|x=s(x) fi (s(x), g(s(x)).

(8.5)

Below a simple example is presented to show how control synthesis of rational control systems could be carried out. The form of the rational system below is based on a conjecture of the control-canonical form of single-input rational systems. Example 8.2. Consider the rational system with state-space dimension n = 3 and with a single input, in the particular form dx1 (t)/dt = x2 (t), dx2 (t)/dt = u(t), y(t) = h(x(t)). The control-objective system is taken to be of the form dx1 (t)/dt = x2 (t), dx2 (t)/dt = f2 (x(t)), y(t) = h(x(t)).

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353

Then the control law g(x) = f2 (x), results in a closed-loop system which equals the control-objective system. The approach for rational systems which are multi-input should be similar. The application of the approach sketched here depends on the transformation of an arbitrary rational system to a control-canonical form which is not known yet. The research of this topic is in a very elementary stage. Experience with formulation of the control-objective system is needed.

9. Computer Algebra for Rational Systems For a rational system the reader may want to check whether the system is observable and controllable. A rational system of a low dimension can be checked by hand calculations for these properties. For systems of state-space dimension 4 or 5 or higher, hand made calculations are cumbersone and prone to errors. There is thus a need for procedures and algorithms to calculate with polynomials and with rational functions of rational systems. The research area for this is called computational algebra and the basis of this is commutative algebra and algebraic geometry. Problem 9.1 (Problems of computer algebra for rational systems). Formulate computer-algebraic algorithms or procedures for the following system theoretic properties of a considered rational system: 1. Rational observability of a rational system. 2. Algebraic controllability of a rational system. 3. System reduction of an uncontrollable rational system to a controllable rational system. 4. Transformation of a rational system to the observable-canonical form of a rational system or to the control-canonical form once it has been formulated. 5. Decomposition of a rational system to a particular form. 6. Control synthesis of rational systems. To clarify the above problem, there exist characterizations of observability and of controllability of a rational system. Based on these characterizations one can formulate simple computer-algebraic calculations. But to detail these calculations and to structure these calculations so that they are fast and efficient, is a difficulty. There is a need for efficient calculation procedures for observability and controllability that is the focus of this section. A check for rational observability is quite straightforward. But a check on algebraic controllability is far from simple, see the paper [51] for a discussion of the difficulties of determining algebraic controllability of a rational system.

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Example 9.2. In this example it is shown how to check for rational observability of a rational system. The calculations are carried out step by step. For such calculations it would be useful to have a program in a computer algebra package which automatically checked for rational observability. Consider the rational system of single-input-single-output form, affine in the input, represented by the equations dx1 (t)/dt = x2 (t) + f1 (x(t))u(t), dx2 (t)/dt = f2,0 (x(t)) + f2,1 (x(t))u(t), y(t) = x1 (t) = h(x(t)). This is actually an element of the observable canonical form. It will be shown how the observation field is calculated. For such a calculation a procedure is needed which using a computer algebra program, can automatically carry out these calculations. The observation field is calculated according to ∂ ∂ Lfα = [x2 + f1 (x)α] + [f2,0 (x) + f2,1 (x)α] , ∂x1 ∂x  0, if A ≥ 0 and A is invertible. As usual, we will write A ≤ B and A < B whenever B − A ≥ 0 and B − A > 0, respectively. The authors were supported by grants from ARRS, Slovenia, Grants No. P1-0288, N1-0061, J1-8133.

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_14

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362

By E(H) we denote the effect algebra, that is, the set of positive operators on H that are bounded by the identity, E(H) = {A ∈ S(H) : 0 ≤ A ≤ I}. In other words, the effect algebra is the operator interval [0, I]. If H is finitedimensional, dim H = n, then E(H) can be identified with En , the set of all n × n hermitian matrices whose eigenvalues belong to the unit interval [0, 1]. Effect algebras are important in mathematical foundations of quantum mechanics, see for example [2], [4], [25], and [27]. There are quite a few operations and relations defined on E(H) which play a significant role in different aspects of quantum theory. Besides the usual partial ordering ≤ defined above we will be mainly interested in sequential product, orthocomplementation, and coexistency. If A, B ∈ E(H), then their sequential product A ◦ B is defined to be A ◦ B = A1/2 BA1/2 , which is easily seen to be an effect again. For A ∈ E(H) we write A⊥ = I − A and call it the orthocomplement of A. Two effects A, B ∈ E(H) are said to be coexistent, A ∼ B, if there exist effects E, F, G such that A = E + G,

B = F + G,

and

E + F + G ∈ E(H).

Mathematical physicists are interested in symmetries, that is, bijective maps on quantum structures that preserve certain relations and/or operations in both directions. For example, a symmetry on the effect algebra with respect to the partial order is a bijective map φ : E(H) → E(H) such that for all A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B), and a symmetry with respect to the sequential product is a sequential automorphism, that is, a bijective map φ : E(H) → E(H) such that for all A, B ∈ E(H) we have φ(A ◦ B) = φ(A) ◦ φ(B). Let us formulate here three classical results in the area. We will start with the famous Ludwig’s theorem describing the general form of ortho-order automorphisms of effect algebras. To avoid trivialities we will assume from now on that the dimension of the underlying Hilbert space H is at least two. Theorem 1.1. Let φ : E(H) → E(H) be a bijective map such that for any A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B) and φ(A⊥ ) = φ(A)⊥ .

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Then there exists either a unitary operator U : H → H, or an antiunitary operator U : H → H such that φ(A) = U AU ∗ for every A ∈ E(H). Ludwig [27, Section V.5] formulated this theorem under the additional assumption that dim H ≥ 3. His proof has been clarified in [7] and it was shown in [37] that the statement remains valid when dim H = 2. We continue with sequential automorphisms. Theorem 1.2. Let φ : E(H) → E(H) be a bijective map such that for any A, B ∈ E(H) we have   φ A1/2 BA1/2 = φ(A)1/2 φ(B)φ(A)1/2 . Then there exists either a unitary operator U : H → H, or an antiunitary operator U : H → H such that φ(A) = U AU ∗ for every A ∈ E(H). Again, this result had been first proved in [10] under the additional assumption that dim H ≥ 3, and then later the two-dimensional case was resolved in [34]. In 2001, Moln´ar proved that the ortho-preservation in Ludwig’s theorem can be replaced by the assumption of preserving coexistency in both directions [30]. More precisely, we have the following. Theorem 1.3. Let φ : E(H) → E(H) be a bijective map such that for any A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B) and A ∼ B ⇐⇒ φ(A) ∼ φ(B). Then there exists either a unitary operator U : H → H, or an antiunitary operator U : H → H such that φ(A) = U AU ∗ for every A ∈ E(H). We recommend Moln´ ar’s book [34] as a quite recent survey on symmetries of effect algebras, other quantum structures, as well as many other related results on preservers (maps preserving certain relations, operations, or properties) on operator and function spaces. But especially in the case of effect algebras a significant progress has been made since the appearance of that monograph and in the next section we will briefly survey some recent results related to the above three theorems. Then we will explain some of the most interesting ideas that have been used in the study of symmetries

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of effect algebras. We will continue by formulating several open problems. In the last section a solution to one of these open problems will be given. Let us conclude the introduction with a remark on related results. In the literature one can find results on other symmetries of effect algebras like automorphisms with respect to partial addition [6, 23, 27], automorphisms with respect to generalized sequential products and Jordan triple product [13, 28], maps preserving order and some additional properties [32, 35], mixture preserving maps [39], spectral order-preserving maps [11, 36, 38], isometries [24], and also on symmetries of effect algebras of von Neumann algebras [11, 29, 33, 39]. There are also related results on local automorphisms of effect algebras [1, 23].

2. A brief survey of recent results In mathematical foundations of quantum mechanics linear bounded selfadjoint operators correspond to bounded observables. The partial order ≤ defined on S(H) is important in mathematical physics. Namely, in the language of quantum mechanics, the bounded observable A is below the bounded observable B if and only if the mean value of A in any state is less or equal to the mean value of B in the same state. Bijective maps on S(H) preserving order in both directions were characterized by Moln´ar in [31]. He proved the following theorem. Theorem 2.1. Assume that φ : S(H) → S(H) is a bijective map such that for all pairs A, B ∈ S(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B). Then there exist a bounded bijective linear or conjugate-linear operator T : H → H and S ∈ S(H) such that φ(A) = T AT ∗ + S for every A ∈ S(H). Let S(H)≥0 be the set of all positive operators on H. The crucial step in Moln´ ar’s proof of the above theorem (see [34]) is the following structural result for order automorphisms of the set of all positive operators. Theorem 2.2. Assume that φ : S(H)≥0 → S(H)≥0 is a bijective map such that for all pairs A, B ∈ S(H)≥0 we have A ≤ B ⇐⇒ φ(A) ≤ φ(B). Then there exists a bounded bijective linear or conjugate-linear operator T : H → H such that φ(A) = T AT ∗ for every A ∈ S(H)≥0 .

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365

Comparing the above two statements with Theorem 1.1 we immediately notice a significant difference. While in the above two theorems we consider bijective maps preserving order in both directions, the assumptions in Ludwig’s theorem are stronger: besides bijectivity and preserving order in both directions we additionally assume that the maps preserve orthocomplementation. And then in view of the above two theorems it looks natural to conjecture that Ludwig’s theorem can be improved by removing the assumption of preserving orthocomplementarity. More precisely, it would be tempting to believe that every bijective map φ : E(H) → E(H) satisfying A ≤ B ⇐⇒ φ(A) ≤ φ(B), A, B ∈ E(H), is of the form φ(A) = U AU ∗ for some unitary or antiunitary operator U . The following example given in [35] shows that this conjecture is wrong. For any fixed invertible operator T ∈ E(H), the transformation  −1/2 −1/2    T2 T2 2 −1 −1 A → (I − T + T (I + A) T ) − I 2I − T 2 2I − T 2 (2.1) is a bijective map of E(H) onto itself and it preserves order in both directions. The main difficulty here is that it is not at all obvious that this map is a bijection of E(H) onto itself and that it preserves order in both directions. But surprisingly, the explanation is quite easy. Recall first the wellknown fact that if A, B ∈ S(H) are strictly positive, then we have A ≤ B if and only if B −1 ≤ A−1 . For two self-adjoint operators A, B such that B − A is strictly positive, we define the operator interval [A, B] = {C ∈ S(H) : A ≤ C ≤ B}. If B − A and D − C are both strictly positive, then a bijective map φ : [A, B] → [C, D] is called an order isomorphism if for every X, Y ∈ [A, B] we have X ≤ Y ⇐⇒ φ(X) ≤ φ(Y ). It is called an order anti-isomorphism if for every X, Y ∈ [A, B] we have X ≤ Y ⇐⇒ φ(Y ) ≤ φ(X). We are now ready to present a few simple examples of order isomorphisms and order anti-isomorphisms of operator intervals. Let T ∈ S(H) be any self-adjoint operator. Then the translation map X → X + T is an order isomorphism of [A, B] onto [A + T, B + T ]. Further, if T : H → H is any bounded invertible linear or conjugate-linear map, then the transformation X → T XT ∗ is an order isomorphism of [A, B] onto [T AT ∗ , T BT ∗ ]. And finally, if A, B ∈ S(H) with both A and B − A being strictly positive, then the map X → X −1 is an order anti-isomorphism of [A, B] onto [B −1 , A−1 ].

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It is now trivial to understand the above “complicated” example. Namely, the map −1  A → ξ(A) = I − T 2 + T (I + A)−1 T −I is a product of several order isomorphisms and two order anti-isomorphisms, A → I + A → (I + A)−1 → T (I + A)−1 T → I − T 2 + T (I + A)−1 T → (I − T 2 + T (I + A)−1 T )−1 → (I − T 2 + T (I + A)−1 T )−1 − I, and is therefore an order isomorphism of E(H) onto [ξ(0), ξ(I)]. Clearly, ξ(0) = 0 and ξ(I) = T 2 (2I−T 2 )−1 . Composing ξ with the suitable congruence transformation we obtain the order automorphism of E(H) given above. Having this insight the next natural step is to observe that the set of all order automorphisms of an effect algebra is a group. Indeed, if φ, ψ : E(H) → E(H) are order automorphisms, then both φψ and φ−1 are order automorphisms of E(H) as well. At first glance the above “wild” example seems to indicate that there is no nice description of the general form of order automorphisms of E(H), but the above explanation shows that even such a complicated map can be decomposed into simple maps that are either order isomorphisms or order anti-isomorphisms of operator intervals. Hence, this seems to imply that it would be possible to find a simple generating set of the group of order automorphisms of E(H). Of course, besides the examples of simple order isomorphisms and order anti-isomorphisms of operator intervals given above in order to understand example (2.1) we have one more example that we should take into account. Namely, it is obvious that the orthocomplementation A → A⊥ is an order anti-automorphism of E(H). In our attempt to get a better understanding of the group of order automorphisms of E(H) by finding a simple generating set we arrive at a surprising result – it turned out that contrary to our expectation there is a rather nice description of the general form of order automorphisms of E(H). In order to formulate a structural theorem for order automorphisms of E(H) [43, 45] we need to introduce a certain set of functions on the unit interval. For any real number p < 1 we denote by fp the bijective monotone increasing function of the unit interval [0, 1] onto itself defined by x fp (x) = , x ∈ [0, 1]. px + 1 − p Then we have the following. Theorem 2.3. Assume that φ : E(H) → E(H) is a bijective map such that for every pair A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B). Then there exist real numbers p, q, 0 < p < 1, q < 0, and a bijective linear or conjugate-linear bounded operator T : H → H with T  ≤ 1 such that   −1/2 −1/2 fp (T AT ∗ ) (fp (T T ∗ )) φ(A) = fq (fp (T T ∗ )) (2.2)

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for all A ∈ E(H). Conversely, for any pair of real numbers p, q, 0 < p < 1, q < 0, and any bijective linear or conjugate-linear bounded operator T : H → H with T  ≤ 1 the map φ defined by (2.2) is an order automorphism of E(H). At this point the case of the effect algebra looks much more complicated than corresponding results for S(H) and S(H)≥0 , see Theorems 2.1 and 2.2. In particular, there was no need to formulate the converse statements in Theorems 2.1 and 2.2 because they are trivial. But in the case of the effect algebra the fact that each map of the form (2.2) is an order automorphism is not entirely trivial. To verify this we need to apply the highly nontrivial theory of operator monotone functions to conclude that for every p < 1 the map A → fp (A), A ∈ E(H), is a bijective map of E(H) onto itself with the property that for every pair A, B ∈ E(H) we have A ≤ B ⇐⇒ fp (A) ≤ fp (B) (later on we will see that this statement can be proved in an elementary way without involving the theory of operator monotone functions). It is then clear that the map A → fp (T AT ∗ ), A ∈ E(H), is an order isomorphism of E(H) onto [0, fp (T T ∗ )]. It follows that the map A → (fp (T T ∗ ))

−1/2

fp (T AT ∗ ) (fp (T T ∗ ))

−1/2

,

A ∈ E(H),

is an order automorphism of E(H), and consequently, each map φ of the form (2.2) is a bijective mapping of E(H) onto itself preserving order in both directions. There is something mysterious about the above statement. Namely, if we take two maps φ, ψ : E(H) → E(H) both of the form (2.2), then the product φψ is an order automorphism of E(H) and is therefore again of the form (2.2). It looks impossible to verify this by a direct computation. An alternative description of order automorphisms of the operator interval E(H) was given in [47]. It reads as follows. Theorem 2.4. Assume that φ : E(H) → E(H) is a bijective map such that for every pair A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B). Then there exist a negative real number p and an invertible bounded linear or conjugate-linear operator T : H → H such that     φ(A) = fp (I + (T T ∗ )−1 )1/2 I − (I + T AT ∗ )−1 (I + (T T ∗ )−1 )1/2 (2.3) for all A ∈ E(H). Conversely, for any real number p < 0 and any invertible bounded linear or conjugate-linear operator T : H → H the map φ defined by (2.3) is an order automorphism of E(H). This result can make us even more confused when trying to understand the structure of order automorphisms of the effect algebra. It is not at all obvious that each map of the form (2.3) can be rewritten in the form (2.2) and vice versa. In [47] a detailed explanation of the fact that the set of all

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maps of the form (2.2) coincides with the set of all maps of the form (2.3) was given but it is too long to be repeated here. Let us get back to the group G of all order automorphisms of S(H)≥0 . We denote by GL (H) the generalized linear group on a Hilbert space H, that is, the set of all invertible bounded linear or conjugate-linear operators on H equipped with the usual product – the composition of operators. For each T ∈ GL (H) we say that the map A → T AT ∗ is the order automorphism of S(H)≥0 induced by T . Clearly, if we have two order automorphisms of S(H)≥0 induced by S and T , respectively, then the product of these two automorphisms is induced by ST . Moreover, order automorphisms of S(H)≥0 induced by S and T are the same if and only if S = zT for some complex number z of modulus one. It follows easily that G = GL (H)/S 1 , where S 1 is the multiplicative group of all complex numbers of modulus one, S 1 = {z ∈ C : |z| = 1}. Thus, it is easy to understand what is the group G of all order automorphisms of S(H)≥0 , while Theorems 2.3 and 2.4 seem to imply that there is no easy way to understand the structure of the group of all order automorphisms of E(H). But again the first impression turned out to be wrong. The “complicated” case of the effect algebras was clarified in [48] by showing that the group of order automorphisms of E(H) is not more complicated than the group G of all order automorphisms of S(H)≥0 ! Let us explain this briefly. It can be proved that for every order automorphism φ : E(H) → E(H) we have φ([0, I) ) = [0, I), where [0, I) = {A ∈ E(H) : 0 ≤ A < I}. Thus, if φ : E(H) → E(H) is any order automorphism, then the restriction φ|[0,I) is an order automorphism of [0, I). It can be further proved that the map φ → φ|[0,I) is a bijection from the set of all order automorphisms of E(H) onto the set of all order automorphisms of [0, I), and therefore this map is an isomorphism of the group of order automorphisms of E(H) onto the group of order automorphisms of [0, I). The next important observation is that the operator interval [0, I) is order isomorphic to S(H)≥0 . Indeed, the map X → I − X is an order anti-isomorphism of [0, I) onto (0, I], the map X → X −1 is an order antiisomorphism of (0, I] onto [I, ∞) = {A ∈ S(H) : A ≥ I} and the translation X → X − I is an order isomorphism of [I, ∞) onto S(H)≥0 . Hence, ϕ : [0, I) → S(H)≥0 defined by ϕ(A) = (I − A)−1 − I,

A ∈ [0, I),

(2.4)

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is an order isomorphism of [0, I) onto S(H)≥0 . Clearly, its inverse is ψ(A) = I − (I + A)−1 ,

A ∈ S(H)≥0 .

(2.5)

It follows that the map φ → ϕ φ ψ is an isomorphism of the group of order automorphisms of [0, I) onto the group of order automorphisms of S(H)≥0 . This together with the previous paragraph implies that the group of order automorphisms of the effect algebra is isomorphic to G, the group of order automorphisms of S(H)≥0 . We can get something more from the above idea. Let φ : E(H) → E(H) be an order automorphism. Then ϕ ◦ φ|[0,I) ◦ ψ : S(H)≥0 → S(H)≥0 , where ϕ and ψ are given by (2.4) and (2.5), respectively, is an order automorphism, and by Theorem 2.2 we can find a bijective bounded linear or conjugate-linear operator T : H → H such that ϕ(φ(ψ(A))) = T AT ∗ for every A ∈ S(H)≥0 . Consequently, for every A ∈ [0, I) we have φ(A) = ψ(T ϕ(A)T ∗ )  −1 = I − I − T T ∗ + T (I − A)−1 T ∗ .

(2.6)

Of course, formula (2.6) cannot be applied to the effects whose spectrum contains 1. But it is well known that every effect A ∈ E(H) is equal to the supremum of all rank-1 effects R ∈ E(H) satisfying R ≤ A, see [3, Corollary 3]. And then, after taking care of some minor technical details (see [48]), we arrive at yet another description of order automorphisms of the effect algebra. Theorem 2.5. Assume that φ : E(H) → E(H) is a bijective map such that for every pair A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B). Then there exists a bounded bijective linear or conjugate-linear operator T : H → H such that  −1  , A ∈ [0, I), I − I − T T ∗ + T (I − A)−1 T ∗ φ(A) = sup{φ(R) : R ∈ [0, I) and rank R = 1 and R ≤ A}, A ∈ [0, I). (2.7) It is much easier to guess that the group of all automorphisms of the effect algebra is isomorphic to G using the representation (2.7) than using Theorems 2.3 and/or 2.4, see [48]. An additional insight into the structure of order automorphisms of the effect algebra has been recently made by Drnovˇsek [9] who used the above idea to present very short and elegant proofs of Theorems 2.3 and 2.4. He showed that both results can be deduced rather quickly from Theorem 2.2. We will now turn to Theorems 1.1 and 1.3. Are they optimal? We start with Ludwig’s theorem. It is trivial to see that any map φ : E(H) → E(H)

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with the property that for every pair A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B) is injective. Hence, a slight improvement is possible: the bijectivity assumption in Theorem 1.1 can be replaced by the weaker surjectivity assumption. Do we get the same conclusion if we replace the assumption of preserving order in both directions by the weaker assumption that the order is preserved in one direction only? Recall that a map φ : E(H) → E(H) preserves order in one direction if for every pair A, B ∈ E(H) we have A ≤ B ⇒ φ(A) ≤ φ(B).

(2.8)

In [46] a bijective nonstandard map φ : E(H) → E(H) satisfying (2.8) and φ(A⊥ ) = φ(A)⊥ , A ∈ E(H), was constructed, showing that the answer to the above question is negative. Here, by a standard map we mean any map of the form A → U AU ∗ , where U is a unitary or antiunitary operator. It was also shown that the surjectivity assumption in Ludwig’s theorem is indispensable. And we have seen above that the conclusion in Ludwig’s theorem changes substantially if we omit the assumption of preserving orthocomplementation. Hence, Theorem 1.1 is optimal (with the bijectivity assumption replaced by the surjectivity assumption). On the other hand, it was proved in [46] that Moln´ar’s theorem can be improved in two directions. We say that a map φ : E(H) → E(H) preserves coexistency in both directions if for every pair A, B ∈ E(H) we have A ∼ B ⇐⇒ φ(A) ∼ φ(B),

(2.9)

and it preserves coexistency in one direction only if for every pair A, B ∈ E(H) we have A ∼ B ⇒ φ(A) ∼ φ(B). (2.10) Theorem 2.6. Let H be a Hilbert space, dim H ≥ 3. Assume that φ : E(H) → E(H) is a surjective map satisfying (2.8) and (2.9). Then there exists an either unitary or antiunitary operator U on H such that φ(A) = U AU ∗ for every A ∈ E(H). Theorem 2.7. Let H be a Hilbert space, dim H ≥ 3. Assume that φ : E(H) → E(H) is a surjective map preserving order in both directions and satisfying (2.10). Then there exists an either unitary or antiunitary operator U on H such that φ(A) = U AU ∗ for every A ∈ E(H). It was demonstrated by counterexamples that both theorems are optimal. In particular, it was somewhat surprising to discover that there exists a nonstandard bijective map φ : E(H) → E(H) satisfying (2.8) and (2.10). Let us conclude this brief survey of recent results with the following characterization of continuous sequential endomorphisms of the effect algebras over finite-dimensional Hilbert spaces [8]. For an n × n matrix A we denote by At and adj A the transpose of A and the adjugate matrix of A,

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that is, the transpose of the matrix of cofactors, respectively. For real numbers t1 , . . . , tn we denote by diag (t1 , . . . , tn ) the n × n diagonal matrix whose diagonal entries are t1 , . . . , tn . In the theorem below we will define 00 to be 1. Theorem 2.8. Let n ≥ 3 and assume that a continuous map φ : En → En satisfies φ(A1/2 BA1/2 ) = φ(A)1/2 φ(B)φ(A)1/2 ,

A, B ∈ En .

Then we have one of the following seven possibilities: • there exist a unitary n × n matrix U and a nonnegative real number c such that φ(A) = (det A)c U AU ∗ ,

A ∈ En ;

• there exist a unitary n × n matrix U and a nonnegative real number c such that φ(A) = (det A)c U At U ∗ ,

A ∈ En ;

• there exists a unitary n × n matrix U such that φ(A) = U (adj A)U ∗ ,

A ∈ En ;

• there exists a unitary n × n matrix U such that φ(A) = U (adj A)t U ∗ ,

A ∈ En ;

• there exist a unitary n × n matrix U and a real number c > 1 such that  (det A)c U A−1 U ∗ , A ∈ En is invertible, φ(A) = 0, otherwise; • there exist a unitary n × n matrix U and a real number c > 1 such that  (det A)c U (A−1 )t U ∗ , A ∈ En is invertible, φ(A) = 0, otherwise; • there exist a nonnegative integer m ≤ n, nonnegative real numbers c1 , . . . , cm , and a unitary matrix U such that φ(A) = U diag ((det A)c1 , . . . , (det A)cm , 0, . . . , 0)U ∗ ,

A ∈ En .

The main novelty of this result is that the authors drop the bijectivity assumption. The price they had to pay is that they needed to restrict to the finite-dimensional case and to add the continuity assumption. They presented a few examples showing that without the continuity assumption there are a lot of “wild” sequential endomorphisms of En . It is not clear what happens when n = 2. The reason for assuming that n ≥ 3 was the use of Gleason’s theorem in the proof of the above statement. The authors of [8] have also explained why at present the infinite-dimensional analogue of the above theorem seems to be out of reach.

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3. Some interesting proof techniques In order to understand all the proofs of the results mentioned in this survey paper one needs to read quite a few papers, some of them rather long. To make this job easier for an interested reader we are going to present here some of the crucial ideas. Of course, we will not explain them in full detail. We will just outline some of the approaches used in recent papers. Once the reader knows these general ideas he/she can start to think of open problems mentioned in the next section without paying too much attention to technical details in the already published papers where the full proofs were given. The proof of Theorem 2.1 given in [31] is quite involved. It depends on several deep results including Rothaus’s theorem [42] on the automatic linearity of bijective maps between closed convex cones preserving order in both directions, Vigier’s theorem [40, Theorem 4.1.1], and the Kadison’s wellknown structural theorem for bijective linear positive unital maps on C ∗ algebras [22, Corollary 5]. A much simpler approach based on a reduction to adjacency preserving maps has been developed later. Let us recall that two operators A, B ∈ S(H) are called adjacent if rank (B − A) = 1. The main idea is the following. Take any two adjacent operators A, B ∈ S(H). Then B − A is of rank 1, that is, B − A = tP for some nonzero real number t and some rank-1 projection (self-adjoint idempotent) P . Clearly, P ≥ 0, and therefore, B ≥ A if t > 0 and A ≥ B if t < 0. Assume we have the first possibility. Then we can consider the operator interval [A, B] = [A, A + tP ] and it is easy to verify that [A, A + tP ] = {A + sP : 0 ≤ s ≤ t}. Hence, if we take any two elements C, D ∈ [A, B], then they are of the form C = A + s1 P and D = A + s2 P , and consequently, they are comparable, that is, we have C ≤ D or D ≤ C depending on whether s1 ≤ s2 or s2 ≤ s1 . We summarize: if A, B ∈ S(H) are adjacent, then they are comparable, and moreover, any two self-adjoint operators C, D that are in between A and B are comparable as well. Let us assume now that A and B, A = B, are not adjacent. It can happen that they are not comparable. But if they are comparable, then we have A ≤ B or B ≤ A. We will treat just one of the two possibilities, say the first one. Then B − A is positive and its image is at least twodimensional. Therefore we can find two rank-1 positive self-adjoint operators R1 , R2 ≤ B − A such that neither R1 ≤ R2 nor R2 ≤ R1 . Hence, A ≤ A + R 1 , A + R2 ≤ B and A + R1 and A + R2 are not comparable. This together with the previous paragraph shows that for any pair A, B ∈ S(H), A = B, the operators A and B are adjacent if and only if

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they are comparable, and any pair C, D ∈ S(H) that is in between A and B is also comparable. If we have a bijective map on S(H) preserving order in both directions, then it clearly preserves comparability in both directions, and by the above, it preserves adjacency in both directions. Bijective maps on various spaces of matrices preserving adjacency in both directions were characterized by Hua in a series of papers [14]–[21]. In particular, he proved that if n is an integer ≥ 2 and φ is a bijection of Hn , the space of all n × n hermitian matrices, onto itself preserving adjacency in both directions, then there exist an invertible complex matrix T and a hermitian n × n matrix S such that either φ(A) = T AT ∗ + S for every A ∈ Hn , or φ(A) = −T AT ∗ + S for every A ∈ Hn , or φ(A) = T At T ∗ + S for every A ∈ Hn , or φ(A) = −T At T ∗ + S for every A ∈ Hn . Of course, in the first and the third case the map φ is an order automorphism of Hn , while in the remaining two cases φ is an order anti-automorphism of Hn . This together with the previous paragraph completes the proof of Theorem 2.1 in the finite-dimensional case and it is not that difficult to extend it to the infinite-dimensional case, see [44] for the details. Let us mention here a connection with geometry. For the sake of simplicity we will again restrict ourselves to the finite-dimensional case only. We are interested in bijective maps φ : Hn → Hn preserving adjacency in both directions. We call a subset S ⊂ Hn an adjacent set if any two elements A, B ∈ S, A = B, are adjacent. We will use the language of geometry and call each maximal adjacent set a line. Since φ is a bijection preserving adjacency in both directions, it maps every line onto some line. Thus, it is interesting to characterize lines in Hn . If S ⊂ Hn is any line and A ∈ S is any point on this line, then the set S − A = {B − A : B ∈ S} is again a line. But this line contains the zero matrix. So, any nonzero element of this new line is adjacent to the zero matrix. In other words, it is of rank 1. It follows that any nonzero element of the line S − A is of the form tP , where t is a nonzero real number and P a projection of rank 1. Take any two nonzero elements tP, sQ ∈ S − A, tP = sQ. Here, t, s ∈ R \ {0} and P, Q ∈ Hn are projections of rank 1. Since S − A is a line, the points tP and sQ must be adjacent, or equivalently, rank (tP − sQ) = 1. It is an elementary linear algebra problem to show that this is possible only if P = Q. It follows easily that S − A = {tP : t ∈ R} for some projection P of rank 1. We have shown that each line in Hn is a subset of the form {A + tP : t ∈ R}, where A ∈ Hn and P is a projection of rank 1. But this is just a line in the usual Euclidean geometry passing through the point A with the direction vector P . And we know that the map φ maps each such line onto some line. But of course, when we want to describe the general form of bijective adjacency preserving maps φ we cannot use the fundamental theorem of affine geometry. Namely, the set of lines in our geometry does not coincide with the set of all lines in the Euclidean geometry. The set of lines in our geometry

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is the subset of only those lines in the Euclidean geometry whose direction vectors are matrices of rank 1. In fact, we have a rather exotic point-line geometry. For example, if we take any line {A + tP : t ∈ R} ⊂ Hn and any point B ∈ Hn such that B ∈ {A + tP : t ∈ R}, then there is at most one line passing through B and intersecting the line {A + tP : t ∈ R}. To verify this we need to check that there is at most one point on the line {A + tP : t ∈ R} that is adjacent to B, or in other words, we need to verify that there is at most one real number t such that A − B + tP is of rank 1. This is again a rather easy linear algebra exercise. It turns out that such “exotic” properties of our point-line geometry are of great help when studying bijective maps preserving adjacency in both directions (=maps that send lines onto lines). With this geometrical insight it is much easier to follow the arguments in the paper [44] where Theorem 2.1 and many other results describing the general form of various symmetries of bounded observables were proved using the approach based on the reduction to adjacency preservers. When dealing with maps on effect algebras preserving order in both directions one of the ideas that might be used is the reduction to the twodimensional case. It is rather easy to prove that an effect R ∈ E(H) is the zero effect or an effect of rank 1 if and only if for every pair of effects A, B ∈ E(H) satisfying A, B ≤ R we have A ≤ B or B ≤ A. Hence, if φ : E(H) → E(H) is a bijective map preserving order in both directions then for every R ∈ E(H) the effect φ(R) is of rank at most 1 if and only if R is of rank at most 1. Noticing that rank-1 projections are maximal elements of the set of all effects of rank at most 1, we conclude that φ maps the set of projections of rank 1 onto itself. And using the order preserving property one can then conclude that φ maps the set of projections of rank 2 onto itself. Let now P be any projection of rank 2. Then φ(P ) = Q is again a projection of rank 2. And φ maps the set {A ∈ E(H) : A ≤ P } = P E(H)P onto QE(H)Q. But obviously, P E(H)P can be identified with the set of all effects defined on the image of P which is two-dimensional. Thus, both P E(H)P and QE(H)Q can be identified with E2 , and therefore, the restriction of φ to P E(H)P considered as a map from P E(H)P onto QE(H)Q can be considered as an order automorphisms of E2 . Consequently, if we can describe the general form of order automorphisms of E2 , then we know how φ behaves on subsets RE(H)R ⊂ E(H), where R is any projection of rank 2. Once we have a nice behaviour of φ on these “small pieces” of the effect algebra E(H) we can try to get a nice behaviour of φ on the whole effect algebra E(H). The transition from the local nice behaviour to global nice behaviour is not entirely trivial. However, it does not require any surprising ideas – there is just a lot of technical work to be done.

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One important fact that we need to keep in mind is that when studying bijective maps preserving order in both directions we actually deal with bijections preserving adjacency in both directions. The explanation above has been given for maps defined on the set of all self-adjoint bounded linear operators (the set of all hermitian matrices in the finite-dimensional case) but the same arguments work for effect algebras. The main advantage of reducing the general case to the two-dimensional case is that this low-dimensional case has been studied a lot. Again, the motivation came from physics, more precisely, from mathematical foundations of relativity. We will denote by M the usual Minkowski space of all space-time events, that is, M = {(x, y, z, t) : x, y, z, t ∈ R}. The first three coordinates of each space-time event are spatial, and the last one represents time. Two space-time events (x1 , y1 , z1 , t1 ) and (x2 , y2 , z2 , t2 ) are said to be lightlike if the distance in space is equal to the distance light travels in the time interval between t1 and t2 , that is, (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 = c2 (t2 − t1 )2 . Here, c is the speed of light, and as it is usual in mathematical physics we will take c = 1. We will now identify M with the set of all 2 × 2 hermitian matrices. The identification is given by the linear map

t − z x + iy r = (x, y, z, t) → = A. x − iy t + z We have det A = t2 − x2 − y 2 − z 2 and from here we immediately get that two space-time events r1 = (x1 , y1 , z1 , t1 ) and r2 = (x2 , y2 , z2 , t2 ), r1 = r2 , are lightlike if and only if det(A1 − A2 ) = 0. Here, of course, A1 and A2 are the 2 × 2 hermitian matrices that correspond to r1 and r2 , respectively. But the determinant of a nonzero 2 × 2 matrix is equal to zero if and only if this matrix has rank 1. Thus, the space-time events r1 and r2 , r1 = r2 , are lightlike if and only if the corresponding hermitian matrices are adjacent. Hence, if we want to study bijective maps on effect algebras preserving order in both directions we can reduce the problem to the two-dimensional case. Then we observe that such maps preserve adjacency. The effect algebra E2 is a subset of H2 and after applying the above identification we arrive at a certain subset of Minkowski space and a bijective map on this subset preserving lightlikeness in both directions. It is well known that bijective maps on Minkowski space preserving lightlikeness in both directions are Poincar´e similarities. But such maps were studied also on certain subsets of Minkowski space, see, for example, [26] and [41], and the ideas developed in these papers can be used to solve our starting problem, that is, the problem of describing the general form of order automorphisms of effect algebras.

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Another important observation that helps a lot is that every order automorphism φ : E(H) → E(H) satisfying the additional property that   1 1 φ I = I (3.1) 2 2 is a standard map. Hence, when dealing with order automorphisms of E(H) the first step might be to reduce the general case to the special case where the additional property (3.1) is satisfied. If we apply this simplification together with the reduction to the two-dimensional case mentioned before we end up with a problem that has been recently solved by entirely elementary methods [47]. To reduce the general case to the special case when (3.1) holds we first need to observe that for every order automorphism φ : E(H) → E(H) we have φ((0, I)) = (0, I), where (0, I) = {A ∈ E(H) : 0 < A < I}. This is easy to guess and also not very difficult to prove, see [47] and the references therein for the details. And then all we need to do is to find for every A ∈ (0, I) an order automorphism ξ of E(H) such that   1 I = A. (3.2) ξ 2 Indeed,   if φ : E(H) → E(H) is any order automorphism and if we denote φ 21 I = A, then we know that A ∈ (0, I). If ξ : E(H) → E(H) is an automorphism satisfying (3.2), then clearly ξ −1 ◦ φ is an automorphism sending 1 2 I into itself. Thus, for every A ∈ (0, I) we need to construct an order automorphism ξ of E(H) such that (3.2) holds true. This can be done in an elementary way. First we need to check that for every real p < 1 the function fp is a bijection of the unit interval [0, 1] with the property that for every pair X, Y ∈ E(H) we have X ≤ Y ⇐⇒ fp (X) ≤ fp (Y ). In other words, we want to check that the map X → fp (X), X ∈ E(H), is an order automorphism of the effect algebra. Of course, this can be verified using the theory of operator monotone functions. However, it can be done in a purely elementary way. Indeed, it is straightforward to check that for every p < 1 we have p , fp−1 = f p−1 p and since p−1 < 1 we need to verify only that for every pair X, Y ∈ E(H) we have X ≤ Y ⇒ fp (X) ≤ fp (Y ). This is certainly true when p = 0. When p = 0 this follows from the observation that for every X ∈ E(H) and every p, 0 < p < 1, we have  −1   1 1−p 1 −1 I , X+ fp (X) = I − 2 p p p

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while for every X ∈ E(H) and every p < 0 we have   −1 1 1−p 1 fp (X) = I + 2 1− . I −X p p p We are now ready to find ξ as above. Let S be any strictly positive operator and p any negative real number. We already know that then the map ξ : E(H) → E(H) defined by     ξ(X) = fp (S + S 2 )1/2 S −1 − (S + X)−1 (S + S 2 )1/2 , X ∈ E(H), is an order automorphism – it is a composition of order isomorphisms and two order anti-isomorphisms of operator intervals. Clearly,     1 ξ I = fp (I + S)(I + 2S)−1 . 2 We know that for every real p < 0 there is a real number q, 0 < q < 1, such p that the function fq is the inverse of fp (more precisely, q = p−1 ). Then we have    1 (I + S)(I + 2S)−1 = fq ξ I . 2 Let A ∈ (0, I). In order to complete our construction of a map ξ we need to find q, 0 < q < 1, and a strictly positive operator S such that (I + S)(I + 2S)−1 = fq (A). The spectrum of A is compact and is therefore contained in some closed subinterval of (0, 1), σ(A) ⊂ [a, b], 0 < a ≤ b < 1. It is straightforward to verify that there exists q ∈ (0, 1) such that fq (a) > 1/2. It follows that fq ([a, b]) ⊂ ((1/2), 1). Therefore, the spectrum of fq (A) is contained in ((1/2), 1), and consequently, the spectrum of S = (I − fq (A))(2fq (A) − I)−1 is contained in the set of positive real numbers. A direct calculation verifies that (I + S)(I + 2S)−1 = fq (A). On the one hand, the above calculations show that the study of order automorphisms of effect algebras can be reduced to the study of only those order automorphisms that map 12 I into itself. On the other hand, a careful reader has noticed that the above arguments explain the “mysterious formula” (2.3) in the conclusion of Theorem 2.4.  Indeed, if φ : E(H) → E(H) is any order automorphism and if we set φ 12 I = A and then choose ξ : E(H) → E(H) as above, then the order automorphism ξ −1 ◦φ : E(H) → E(H) sends 12 I into 1 −1 (φ(X)) = U XU ∗ for some unitary 2 I, and is therefore standard, that is ξ or antiunitary operator U . From here one can easily conclude that φ is of the form (2.3). In a similar way one can get a better insight into the statement of Theorem 2.3. It turns out that quite often one can solve the problem of describing the general form of certain symmetries by reducing the given problem to the same problem for some other type of symmetries. Let us illustrate this with

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proving Theorem 1.2 by such a reduction technique. All we need to observe is that for any pair of effects A, B ∈ E(H) we have A ≤ B ⇐⇒ ∃C ∈ E(H) : A = B 1/2 CB 1/2 . It follows that every sequential automorphism preservers order in both directions. Hence, we can use the structural results for order automorphisms of effect algebras when proving Theorem 1.2. Theorems 1.1, 1.2, 1.3, 2.6, and 2.7 can be considered as characterizations of standard maps. When trying to prove that every surjective map φ on an effect algebra having certain preserving properties is of the form A → U AU ∗ , A ∈ E(H), for some unitary or antiunitary operator U , then quite often the first step is to prove that we have φ(P ) = U P U ∗ for every projection P of rank 1. Once we know that φ has the desired form when restricted to the set of all projections of rank 1, we are usually quite close to the solution of the problem - it remains to use the preserving properties of φ to show that we actually have φ(A) = U AU ∗ for all effects A. When using this approach one of the main tools is Uhlhorn’s theorem. Before formulating this statement we introduce some more notation. We denote by P ⊂ E(H) the set of all projections, and by P1 ⊂ P the subset of all projections of rank 1. Two projections P, Q ∈ P are said to be orthogonal, P ⊥ Q, if and only if P Q = 0. Of course, P Q = 0 is equivalent to QP = 0. A map τ : P1 → P1 preserves orthogonality in both directions if for every pair P, Q ∈ P1 we have P ⊥ Q ⇐⇒ τ (P ) ⊥ τ (Q). The Uhlhorn’s theorem states that when dim H ≥ 3, every bijective map τ : P1 → P1 preserving orthogonality in both directions is of the form τ (P ) = U P U ∗ for some unitary or antiunitary operator U . It should be mentioned that this theorem can be deduced rather quickly from the fundamental theorem of projective geometry. The main weakness of this method is that it works only under the additional assumption that dim H ≥ 3, and then the twodimensional case must be treated separately. Let us illustrate this approach by outlining the first step in the proof of Ludwig’s theorem. So, let φ be an ortho-order automorphism of E(H) with dim H ≥ 3. Since φ is bijective and it preserves order in both directions we have φ(0) = 0 and φ(I) = I. It is easy to see that E ∈ E(H) is not a projection if and only if there exists a nonzero effect F such that F ≤ E and F ≤ I − E. This together with the fact that φ preserves orthocomplementation and order in both directions yields that φ(P) = P. Since φ restricted to P is a bijection preserving order in both directions, it preserves minimal nonzero elements. Thus, φ(P1 ) = P1 . Further, for arbitrary P, Q ∈ P1 we have P Q = 0 if and only if P ≤ Q⊥ which is equivalent φ(P ) ≤ φ(Q)⊥ . Hence, P ⊥ Q if and only if φ(P ) ⊥ φ(Q), and we can

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apply Uhlhorn’s theorem to deduce that there exists an either unitary or antiunitary operator U such that φ(P ) = U P U ∗ for every P ∈ P1 . The second part of the proof of Ludwig’s theorem is much longer than the first step above. This happens quite often when using this approach to study symmetries of effect algebras. Nevertheless, this first step of the proof, that is, verifying that the map φ is of the desired form on the set of all projections of rank 1, is always crucial, and the rest is usually easier although it can be technically quite involved.

4. Open problems We believe that the most interesting open question in this area is to describe the general form of bijective maps φ : E(H) → E(H) preserving coexistency in both direction. Of course, each standard map is a bijective map preserving coexistency in both directions. It is easy to verify that for every A ∈ E(H) and every real number t ∈ [0, 1] the effects A and tI are coexistent. Hence, if g : [0, 1] → [0, 1] is any bijective function of the unit interval, then the bijective map ξ : E(H) → E(H) defined by ξ(A) = A whenever A ∈ E(H) \ {tI : t ∈ [0, 1]} and ξ(tI) = g(t)I, t ∈ [0, 1], preserves coexistency in both directions. It was proved in [46, Proposition 2.7] that for any pair of effects A, B ∈ E(H) we have A ∼ B ⇐⇒ A⊥ ∼ B. Therefore, a bijective map η : E(H) → E(H) which maps every pair {A, A⊥ } bijectively onto itself (for each A ∈ E(H) we have either η(A) = A and η(A⊥ ) = A⊥ , or η(A) = A⊥ and η(A⊥ ) = A), preserves coexistency in both directions. Of course, any composition of a standard map with such maps ξ and η is again a bijection preserving coexistency in both directions. The question is whether all bijections preserving coexistency in both directions are of this form. It is believed that this is a quite difficult problem. One of the reasons is that we do not have a good understanding of coexistent pairs of effects. Even in the simplest two-dimensional case a full understanding of this relation seems to be out of reach [5]. We have three assumptions in Theorem 1.1: 1. the surjectivity, 2. A ≤ B ⇐⇒ φ(A) ≤ φ(B), and 3. φ(A⊥ ) = φ(A)⊥ . We have already mentioned that Theorem 1.1 is optimal. Let us give a detailed explanation. First, the injective map φ : E(H) → E(H) defined by 1 1 (4.1) φ(A) = A + I, A ∈ E(H), 2 4 satisfies the second and the third assumption but is not of the standard form. This shows that the first assumption is indispensable.

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We know that there exist nonstandard maps φ : E(H) → E(H) satisfying the first two assumptions. Hence, the third assumption is essential in Ludwig’s theorem. Nevertheless, we were able to describe the general form of maps on effect algebras satisfying only the first two assumptions, see Theorems 2.3, 2.4, and 2.5. As far as the optimality of Theorem 1.1 is concerned there is only one question left to be answered. Namely, if we assume that φ : E(H) → E(H) satisfies the first and the third property and a weaker version of the second property, that is, if we assume that φ preserves order in one direction only, do we still get as a conclusion that φ must be standard? It was shown in [46] that the answer is negative. The conclusion is that Theorem 1.1 is indeed optimal. Still, in view of the previous paragraph we can ask the following question: What is the general form of surjective (bijective) maps φ : E(H) → E(H) with the property that for every pair of effects A, B ∈ E(H) we have φ(A⊥ ) = φ(A)⊥ and A ≤ B ⇒ φ(A) ≤ φ(B)? And since the counterexample given in [46] is discontinuous, we may ask a simpler question: Is it true that every continuous surjective (bijective) map φ : E(H) → E(H) with the property that for every pair of effects A, B ∈ E(H) we have φ(A⊥ ) = φ(A)⊥ and A ≤ B ⇒ φ(A) ≤ φ(B) is of the form φ(A) = U AU ∗ for some unitary or antiunitary operator U : H → H? Or an even easier question: Is this true in the case that the underlying Hilbert space H is finite-dimensional? One can ask the same kind of questions related to Moln´ ar’s theorem (Theorem 1.3) and its optimal versions, that is, Theorems 2.6 and 2.7. In particular, is it true that every continuous surjective (bijective) map φ : E(H) → E(H) with the property that for every pair of effects A, B ∈ E(H) we have A ∼ B ⇒ φ(A) ∼ φ(B) and A ≤ B ⇒ φ(A) ≤ φ(B) is of the form φ(A) = U AU ∗ for some unitary or antiunitary operator U : H → H? And a more general question: What is the general form of surjective (bijective) maps φ : E(H) → E(H) with the property that for every pair of effects A, B ∈ E(H) we have A ∼ B ⇒ φ(A) ∼ φ(B) and A ≤ B ⇒ φ(A) ≤ φ(B)? Is it possible to answer these two questions at least in the finite-dimensional case? Motivated by Theorems 1.2 and 2.8 one may ask how essential is the assumption of bijectivity when trying to describe the general form of maps on effect algebras preserving certain operations and/or relations. Mathematicians working in this area are aware that in the infinite-dimensional case the assumption of bijectivity (or at least surjectivity) is usually indispensable. But Theorem 2.8 indicates that in the finite-dimensional case the problems of this kind may be solved in the absence of the bijectivity assumption. In the last section we will present a new result of this type confirming that when looking for optimal results on preservers on effect algebras it is natural to consider such maps without assuming bijectivity or surjectivity whenever the underlying Hilbert space is finite-dimensional. We will conclude this section by repeating that in this paper we have restricted our attention to symmetries with respect to four operations or

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relations: the usual partial order, the orthocomplementation, the coexistency, and the sequential product. As mentioned at the end of the Introduction, one can find in the literature results on symmetries of effect algebras with respect to other operations and relations, and then of course, one can study the problem of the optimality of known results in the same way as we did for the above four relations/operations.

5. A new result In this section we will prove a theorem that will be related to Theorem 1.1 in exactly the same way as Theorem 2.8 is related to Theorem 1.2. Hence, we will take the assumptions of Theorem 1.1, omit the bijectivity assumption, but add the continuity and the finite-dimensionality. In such a way we get the following statement. Theorem 5.1. Let H be a Hilbert space with 2 ≤ dim H < ∞ and φ : E(H) → E(H) a continuous map such that for any pair of effects A, B ∈ E(H) we have A ≤ B ⇐⇒ φ(A) ≤ φ(B)

(5.1)

φ(A⊥ ) = φ(A)⊥ .

(5.2)

and Then there exists a bijective linear or conjugate-linear operator T : H → H satisfying T  ≤ 1 such that φ(A) =

1 (I − T T ∗ ) + T AT ∗ . 2

(5.3)

The converse is trivial. Note that the counterexample (4.1) is a special case of (5.3) with T = √12 I. Effect algebras and ortho-order endomorphisms of effect algebras are important in mathematical foundations of quantum mechanics. But when studying these maps from purely mathematical point of view it is convenient to slightly reformulate the assumptions as follows. Obviously, the map ϕ : [0, I] → [−I, I] defined by ϕ(X) = 2X − I is an order isomorphism whose inverse is given by ϕ−1 (X) = 12 (I + X). It is then trivial to check that a map φ : E(H) → E(H) satisfies (5.1) and (5.2) if and only if the map ξ : [−I, I] → [−I, I] defined by ξ = ϕ ◦ φ ◦ ϕ−1 has the property that for every pair A, B ∈ [−I, I] we have A ≤ B ⇐⇒ ξ(A) ≤ ξ(B)

(5.4)

ξ(−A) = −ξ(A).

(5.5)

and

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And in order to prove Theorem 5.1 we need to show that for every continuous map ξ : [−I, I] → [−I, I] satisfying (5.4) and (5.5) there exists a bijective linear or conjugate-linear operator T : H → H satisfying T  ≤ 1 such that ξ(A) = T AT ∗ . The verification that Theorem 5.1 follows from the above statement is done by a trivial straightforward computation. But actually we will prove slightly more. Theorem 5.2. Let H be a Hilbert space with 2 ≤ dim H < ∞ and ξ : [−I, I] → S(H) a continuous map such that for any pair A, B ∈ [−I, I] we have (5.4) and (5.5). Then there exists a bijective linear or conjugate-linear operator T : H → H such that ξ(A) = T AT ∗ for every A ∈ [−I, I]. Of course, if we add the assumption that the image of ξ is contained in the operator interval [−I, I] then the operator T appearing in the conclusion must be a contraction. Before proving Theorem 5.2 we will make one simple observation and prove an elementary lemma. We have seen that considering ortho-order endomorphisms of effect algebras is (via the above identification) the same as considering order endomorphisms ξ of the operator interval [−I, I] having the additional property (5.5). And then, of course, Theorem 1.1 has the following reformulation. Let H be a Hilbert space with dim H ≥ 2 and ξ : [−I, I] → [−I, I] a bijective map such that for any pair A, B ∈ [−I, I] we have (5.4) and (5.5). Then there exists a unitary or antiunitary operator U : H → H such that ξ(A) = U AU ∗ for every A ∈ [−I, I]. We will now slightly generalize this statement. If A, B ∈ S(H) are both strictly positive then the maps X → A1/2 XA1/2 and X → B 1/2 XB 1/2 , X ∈ [−I, I], are order isomorphisms of [−I, I] onto [−A, A] and [−B, B], respectively. Both maps satisfy the condition (5.5). Therefore, the following statement is a direct consequence of Ludwig’s theorem. Proposition 5.3. Let H be a Hilbert space with dim H ≥ 2, A, B ∈ S(H) strictly positive operators, and ξ : [−A, A] → [−B, B] a bijective map such that for any pair X, Y ∈ [−A, A] we have X ≤ Y ⇐⇒ ξ(X) ≤ ξ(Y ) and ξ(−X) = −ξ(X). Then there exists a bijective linear or conjugate-linear operator T : H → H satisfying T AT ∗ = B such that ξ(X) = T XT ∗ for every X ∈ [−A, A].

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Lemma 5.4. Let ε be a positive and t a nonnegative real number. Then for every nonnegative real number s we have √



s 0 0 tε sE11 = ≥ √ = At,ε ⇐⇒ s ≥ t. 0 0 tε −ε Proof. The trace of the matrix sE11 − At,ε is equal to s+ε, while its determinant is (s−t)ε. Because the trace is positive, this matrix is positive if and only if its determinant is nonnegative, which is obviously equivalent to s ≥ t.  Proof of Theorem 5.2. From (5.5) we get that ξ(0) = 0. If for some X, Y ∈ [−I, I] we have ξ(X) = ξ(Y ), then by (5.4) we conclude that X ≤ Y and Y ≤ X, and hence X = Y . Thus, ξ is injective. The operator interval (−I, I) = {X ∈ S(H) : X < 1} is an open subset of S(H) that is a finite-dimensional real vector space. Recall that all norms on a finite-dimensional vector space are equivalent. Since ξ is injective and continuous, we can apply the invariance of domain theorem to conclude that ξ( (−I, I) ) is an open subset, and since it contains 0, we can find a positive real number a such that the operator interval [−aI, aI] is contained in the image of ξ. Hence, there is A ∈ [−I, I] such that ξ(A) = aI, and then, of course, ξ(−A) = −aI. We claim that A > 0 and that ξ restricted to the operator interval [−A, A] is a homeomorphism of [−A, A] onto [−aI, aI]. Indeed, because ξ(A) = aI ≥ 0 = ξ(0) we have A ≥ 0. We know that for each B ∈ [−aI, aI] there exists C ∈ [−I, I] such that ξ(C) = B. But then, by (5.4) we have −A ≤ C ≤ A, and thus, C actually belongs to the operator interval [−A, A]. Hence, the restriction of ξ to [−A, A] is a bijective continuous map of the compact set [−A, A] onto the Hausdorff space [−aI, aI] and is therefore a homeomorphism. It follows that A > 0. Indeed, assume that this is not true. Then A is not invertible, that is, if we identify A with a hermitian matrix, then rank A = r < n = dim H. Without loss of generality we can assume that A is a diagonal matrix with the first r diagonal elements positive and all other diagonal entries equal to zero. But then each matrix in the operator interval [−A, A] has nonzero entries only in the upper left r × r corner. We denote by τ the restriction of ξ to [−A, A] considered as a bijection of [−A, A] onto [−aI, aI]. We know that τ −1 maps the open set (−aI, aI) continuously and injectively into [−A, A] that can be considered as a subset of an r2 -dimensional real normed space. This contradicts the invariance of domain theorem, and therefore, we conclude that A > 0. Hence, we can apply Proposition 5.3 to the map ξ|[−A,A] : [−A, A] → [−aI, aI] to conclude that there exists an invertible linear or conjugate-linear operator T : H → H such that ξ(X) = T XT ∗ for every X ∈ [−A, A]. We can further replace the map ξ by the map X → T −1 ξ(X)(T −1 )∗ without affecting the assumptions of our theorem.

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for every X ∈ [−A, A] and we need to prove that we have ξ(X) = X for all X ∈ [−I, I]. Take any R ∈ [0, I] of rank 1. We claim that ξ(R) = sR for some positive real number s. Equivalently, we claim that the image of R coincides with the image of ξ(R). If this was not true, then by an elementary linear algebra argument it would be possible to find a rank-1 positive operator S such that S ≤ A, and either S ≤ R and S ≤ ξ(R), or S ≤ R and S ≤ ξ(R). In both cases we get a contradiction with the fact that ξ(S) = S. In particular, for every projection P of rank 1 there is an increasing function fP : [0, 1] → [0, 1] such that ξ(tP ) = fP (t)P for every t ∈ [0, 1]. We will show that fP (t) = t for every projection P of rank 1 and every real number t belonging to the unit interval. First we identify operators with matrices and then observe that there is no loss of generality in assuming that P = E11 , the n × n matrix having all entries zero but the (1, 1)-entry that is equal to 1. Then ξ(tE11 ) = fP (t)E11 . Fix t ∈ [0, 1], denote s = fP (t), and observe that if ε > 0 is small enough then the matrix Bt,ε , whose upper left 2 × 2 corner equals At,ε while all other entries are zero, as well as the matrix Bs,ε belong to the operator interval [−A, A], and therefore we have ξ(Bt,ε ) = Bt,ε and ξ(Bs,ε ) = Bs,ε . Since sE11 = ξ(tE11 ) ≥ ξ(Bt,ε ) = Bt,ε , Lemma 5.4 yields that fP (t) = s ≥ t. If fP (t) = s > t, then applying Lemma 5.4 once more, we see that fP (t)E11 ≥ Bs,ε which yields ξ(tE11 ) ≥ ξ(Bs,ε ) and so tE11 ≥ Bs,ε , a contradiction with Lemma 5.4. Hence, fP (t) = t for every t ∈ [0, 1]. It follows that ξ(tP ) = tP for every projection P of rank 1 and every real number t ∈ [−1, 1]. For any positive real numbers t, s ∈ [0, 1] and any rank-1 projection P we have tP ≤ sI if and only if t ≤ s. Hence, tP ≤ ξ(sI) if and only if t ≤ s. It follows easily that ξ(sI) = sI for every s ∈ [0, 1). Of course, this has to be true for every s ∈ [−1, 1]. Consequently, ξ maps the operator interval [−I, I] into itself and satisfies (5.4) and (5.5). Assume for a moment that we know that ξ maps the operator interval [−I, I] onto itself. Then the desired conclusion follows directly from Proposition 5.3. Hence, we will complete the proof once we verify the surjectivity of the map ξ : [−I, I] → [−I, I]. Observing that for every A ∈ [−I, I] and every u ∈ [0, 1] we have −uI ≤ A ≤ uI ⇐⇒ A ≤ u, we conclude that ξ(A) = A. Thus, for every positive u ≤ 1 the map ξ maps the set Su = {A ∈ S(H) : A = u} injectively and continuously into

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itself. We need to verify that all these maps are surjective. This follows from the Borsuk–Ulam theorem. Namely, the set Su , where u > 0, is homeomorphic to (n2 − 1)-dimensional sphere. If ξ(Su ) = Su for some u ∈ (0, 1], then ξ would be an injective continuous map from Su into Su \ {Z} for some 2 point Z ∈ Su . However, Su \ Z is homeomorphic to Rn −1 . By the Borsuk– Ulam theorem every continuous map from a k-dimensional sphere to a kdimensional Euclidean space takes the same value at two antipodal points. In particular, it is not injective. This contradiction completes the proof.  Let us remark that Theorem 5.1 cannot be extended to the infinitedimensional case. We can easily find counterexamples that are even linear. Namely, if H is an infinite-dimensional Hilbert space then we can identify H with the orthogonal direct sum of two copies of H, and hence any map from E(H) into itself can be considered as a map from E(H) into E(H ⊕ H). The effects in E(H ⊕ H) can be represented by 2 × 2 operator matrices. The map φ : E(H) → E(H ⊕ H) defined by

A 0 φ(A) = , A ∈ E(H), 0 ϕ(A) where ϕ : E(H) → E(H) is a bounded linear unital positive map clearly satisfies (5.1) and (5.2). As it is well known, the description of the general form of linear unital positive maps is out of reach even in the low-dimensional matrix cases. Hence, the assumption of finite-dimensionality of H is essential. We do not know what happens in the absence of the continuity assumption.

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[33] L. Moln´ ar, Sequential isomorphisms between the sets of von Neumann algebra effects, Acta Sci. Math. (Szeged) 69 (2003), 755–772. [34] L. Moln´ ar, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lect. Notes Math. 1895, Springer-Verlag, 2007. [35] L. Moln´ ar and E. Kov´ acs, An extension of a characterization of the automorphisms of Hilbert space effect algebras, Rep. Math. Phys. 52 (2003), 141–149. [36] L. Moln´ ar and G. Nagy, Spectral order automorphisms on Hilbert space effects and observables: the 2-dimensional case, Lett. Math. Phys. 106 (2016), 535–544. [37] L. Moln´ ar and Zs. P´ ales, ⊥ -order automorphisms of Hilbert space effect algebras: The two-dimensional case, J. Math. Phys. 42 (2001), 1907–1912. ˇ [38] L. Moln´ ar and P. Semrl, Spectral order automorphisms of the spaces of Hilbert space effects and observables, Lett. Math. Phys. 80 (2007), 239–255. [39] L. Moln´ ar and W. Timmermann, Mixture preserving maps on von Neumann algebra effects, Lett. Math. Phys. 79 (2007), 295–302. [40] G.J. Murphy, C ∗ -algebras and Operator Theory, Academic Press, 1990. [41] I. Popovici and D.C. Rˇ adulescu, Sur les bases de la g´eom´etrie conforme minkowskienne, C. R. Acad. Sci. Paris S´er I. Math. 295 (1982), 341–344. [42] O.S. Rothaus, Order isomorphisms of cones, Proc. Amer. Math. Soc. 17 (1966), 1284–1288. ˇ [43] P. Semrl, Comparability preserving maps on Hilbert space effect algebras, Comm. Math. Phys. 313 (2012), 375–384. ˇ [44] P. Semrl, Symmetries on bounded observables - a unified approach based on adjacency preserving maps, Integral Equations Operator Theory 72 (2012), 7– 66. ˇ [45] P. Semrl, Symmetries of Hilbert space effect algebras, J. London Math. Soc. 88 (2013), 417–436. ˇ [46] P. Semrl, Automorphisms of Hilbert space effect algebras, J. Phys. A 48 (2015), 195301, 18pp. ˇ [47] P. Semrl, Order isomorphisms of operator intervals, Integral Equations Operator Theory 89 (2017), 1–42. ˇ [48] P. Semrl, Groups of order automorphisms of operator intervals, Acta Sci. Math. (Szeged) 84 (2018), 125–136. [49] U. Uhlhorn, Representation of symmetry transformations in quantum mechanics, Ark. Fysik 23 (1963), 307–340. ˇ Lucijan Plevnik and Peter Semrl Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19, SI-1000 Ljubljana Slovenia e-mail: [email protected] [email protected]

GBDT of discrete skew-selfadjoint Dirac systems and explicit solutions of the corresponding non-stationary problems Alexander L. Sakhnovich To Rien Kaashoek on the occasion of his 80th anniversary

Abstract. Generalized B¨ acklund–Darboux transformations (GBDTs) of discrete skew-selfadjoint Dirac systems have been successfully used for explicit solving of the direct and inverse problems of Weyl–Titchmarsh theory. During explicit solving of direct and inverse problems, we considered GBDTs of the trivial initial systems. However, GBDTs of arbitrary discrete skew-selfadjoint Dirac systems are important as well and we introduce these transformations in the present paper. The obtained results are applied to the construction of explicit solutions of the interesting related non-stationary systems. Mathematics Subject Classification (2010). Primary 34A05; Secondary 39A06, 39A12. Keywords. Discrete skew-selfadjoint Dirac system, generalized B¨ acklund–Darboux transformation, fundamental solution, non-stationary system, explicit solution.

1. Introduction We consider a discrete skew-selfadjoint Dirac system   i yk+1 (z) = Im + Ck yk (z), Ck = Uk∗ jUk z

(k ∈ I) ,

(1.1)

where Im is the m × m identity matrix, Uk are m × m unitary matrices,

0 Im 1 (1.2) (m1 , m2 ∈ N, m1 + m2 = m), j= 0 −Im2 This research was supported by the Austrian Science Fund (FWF) under Grant No. P29177.

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_15

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and I is either N0 or the set {k ∈ N0 : 0 ≤ k < N < ∞} (N ∈ N). Here, as usual, N denotes the set of natural numbers and N0 = {0} ∪ N. The relations Ck = Ck∗ ,

Ck2 = Im

(1.3)

are immediate from the second equality in (1.1). This paper is a certain prolongation of the papers [3, 10], where direct and inverse problems for Dirac systems (1.1) have been solved explicitly and explicit solutions of the isotropic Heisenberg magnet model have been constructed. We would like to mention also the earlier papers on the cases of the continuous Dirac systems (see, e.g., [8, 9]). The GBDT version of the B¨acklund–Darboux transformations have been used in [3, 8–10]. B¨ acklund–Darboux transformations and related commutation methods (see, e.g., [1, 2, 6, 7, 11, 14, 17] and numerous references therein) are well-known tools in the spectral theory and in the construction of explicit solutions. In particular, the generalized B¨acklund–Darboux transformations (i.e., the GBDT version of the B¨acklund–Darboux transformations) were introduced in [12] and developed further in a series of papers (see [14] for details). Whereas GBDTs of the trivial initial systems have been used in [3,8–10] (in particular, initial systems (1.1) where Ck ≡ j have been considered in [3,10]), the case of an arbitrary initial discrete Dirac system (1.1) is considered here. In Section 2, we introduce GBDT, construct the so-called Darboux matrix and give representation of the fundamental solution of the transformed system (see Theorem 2.2). Note that an explicit representation of the fundamental solutions of the transformed systems in terms of the solutions of the initial systems is one of the main features and advantages of the Darboux transformations. One of the recent developments of the GBDT theory is connected with its application to the construction of explicit solutions of important dynamical systems (see, e.g., [5, 13]). In Section 3 of this article, we use the same approach in order to construct explicit solutions of the non-stationary systems corresponding to the systems (1.1). In the paper, C stands for the complex plane and C+ stands for the open upper halfplane. The notation σ(α) stands for the spectrum of the matrix α and the notation diag{d1 , d2 , . . .} stands for the block diagonal matrix with the blocks d1 , d2 , . . . on the main diagonal.

2. GBDT of discrete skew-selfadjoint Dirac systems Each GBDT of the initial system (1.1) is determined by a triple {α, S0 , Λ0 } of some n × n matrices α and S0 = S0∗ and some n × m matrix Λ0 (n ∈ N) such that αS0 − S0 α∗ = iΛ0 Λ∗0 .

(2.1)

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The initial skew-selfadjoint Dirac system has the form (1.1) and the transformed (i.e., GBDT-transformed) system has the form   i  yk+1 (z) = Im + Ck yk (z) (k ∈ I) , (2.2) z k } (k ∈ I) is given by the relations where the potential {C Λk+1 = Λk + iα−1 Λk Ck , Sk+1 = Sk + α

−1

∗ −1

Sk (α )

(2.3) +α

−1

Λk Ck Λ∗k (α∗ )−1 ,

k = Ck + Λ∗k S −1 Λk − Λ∗k+1 S −1 Λk+1 , C k k+1

k ∈ I.

(2.4) (2.5)

Here and further in the text we assume that det α = 0,

(2.6)

and suppose additionally in (2.5) that det Sk = 0

(2.7)

for k ∈ N0 or for 0 ≤ k ≤ N depending on the choice of the interval I, on which the Dirac system is considered. Similar to the proof of [3, (3.7)], using the equality Ck2 = Im from (1.3) and relations (2.1)–(2.4) one easily proves by induction that αSk − Sk α∗ = iΛk Λ∗k .

(2.8)

k = C  ∗ . Further in the text, in PropoRemark 2.1. Clearly, Sk = Sk∗ and C k 2  = Im . In Theorem 2.5, we show that under sition 2.3 we show that C k conditions S0 > 0 and 0, i ∈ σ(α) (σ(α) is the spectrum of α) we have Sk > 0 and k = U k∗ j U k , C (2.9) k are unitary. The equality (2.9) where j is given in (1.2) and the matrices U means that the transformed system (2.2) is again a skew-selfadjoint Dirac system in the sense of the definition (1.1). Before Theorem 2.5 we consider the GBDT-transformed system (2.2) without the requirement (2.9). The fundamental solutions of (1.1) and (2.2) are denoted by w(k, z) and w(k,  z), respectively, and are normalized by the conditions w(0,  z) = w(0, z) = Im .

(2.10)

In other words, yk = w(k, z) and yk = w(k,  z) are m × m matrix solutions of the initial and transformed systems, respectively, which satisfy the initial conditions (2.10). The so-called Darboux matrix corresponding to the transformation of the system (1.1) into (2.2) is given by the transfer matrix function wα in Lev Sakhnovich form: wα (k, z) = Im − iΛ∗k Sk−1 (α − zIn )−1 Λk .

(2.11)

See [15, 16] as well as [14] and further references therein for the notion and properties of this transfer matrix function. The statement that the Darboux matrix has the form (2.11) may be formulated as the following theorem.

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Theorem 2.2. Let the initial Dirac system (1.1) and a triple {α, S0 , Λ0 }, which satisfies the relations (2.1), (2.6), (2.7) and S0 = S0∗ , be given. Then, the fundamental solution w of the initial system and fundamental solution w  of the transformed system (2.2) (determined by the triple {α, S0 , Λ0 } via relations (2.3)–(2.5)) satisfy the equality w(k,  z) = wα (k, −z)w(k, z)wα (0, −z)−1

(k ≥ 0),

(2.12)

where wα has the form (2.11). Proof. The following equality is crucial for our proof     i i  wα (k + 1, z) Im − Ck = Im − Ck wα (k, z). z z

(2.13)

(It is easy to see that the important formula [3, (3.16] is a particular case of (2.13).) In order to prove (2.13), note that according to (2.11) formula (2.13) is equivalent to the formula    i  1  Ck − Ck = Im − Ck Λ∗k Sk−1 (zIn − α)−1 Λk z z   i −1 ∗ −1 − Λk+1 Sk+1 (zIn − α) Λk+1 Im − Ck . (2.14) z Using the Taylor expansion of (zIn − α)−1 at infinity we see that (2.14) is in turn equivalent to the set of equalities: k − Ck = Λ∗ S −1 Λk − Λ∗ S −1 Λk+1 , C k k k+1 k+1 −1 Λ∗k+1 Sk+1 αp Λk+1

=

Λ∗k Sk−1 αp Λk

−

(2.15)

−1 iΛ∗k+1 Sk+1 αp−1 Λk+1 Ck

k Λ∗k S −1 αp−1 Λk − iC k

(p > 0).

(2.16)

Equality (2.15) is equivalent to (2.5) and it remains to prove (2.16). From (2.3), taking into account Ck2 = Im we have αΛk+1 − iΛk+1 Ck = αΛk+1 − iΛk Ck + α−1 Λk = αΛk + α−1 Λk .

(2.17)

Substituting (2.17) into the left-hand side of (2.16) and using simple transformations, we rewrite (2.16) in the form Zk αp−2 Λk = 0,

−1 k Λ∗ S −1 α. Zk := Λ∗k+1 Sk+1 (α2 + In ) − Λ∗k Sk−1 α2 + iC k k

Therefore, in order to prove (2.16) (and so to prove (2.13)) it suffices to show that −1 k Λ∗k S −1 α, Λ∗k+1 Sk+1 (α2 + In ) = Λ∗k Sk−1 α2 − iC k

(2.18)

that is, Zk = 0. Relation (2.18) is of interest in itself, since it is an analogue of (2.3) (more precisely of the relation adjoint to (2.3)) when Λ∗r Sr−1 is taken instead of Λ∗r . Such analogues are useful in continuous and discrete GBDT as well as in the construction of explicit solutions of dynamical systems (see, e.g., [5, 13, 14] and references therein).

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Taking into account (2.5) and (2.3), we rewrite (2.18) in the form −1 Λ∗k+1 Sk+1 (α2 + In ) − Λ∗k Sk−1 α2   −1 + i Ck + Λ∗k Sk−1 Λk − Λ∗k+1 Sk+1 Λk+1 Λ∗k Sk−1 α −1 −1 (α2 + In ) − iΛ∗k+1 Sk+1 (Λk + iα−1 Λk Ck )Λ∗k Sk−1 α = Λ∗k+1 Sk+1   − Λ∗k Sk−1 α2 + i Ck + Λ∗k Sk−1 Λk Λ∗k Sk−1 α = 0. (2.19)

Since (2.8) yields iΛk Λ∗k Sk−1 = α − Sk α∗ Sk−1 ,

(2.20)

we rewrite the third line in (2.19) and see that (2.19) (i.e., also (2.18)) is equivalent to −1 Λ∗k+1 Sk+1 (In + α−1 Λk Ck Λ∗k Sk−1 α + Sk α∗ Sk−1 α)   − Λ∗k Sk−1 α2 + i Ck + Λ∗k Sk−1 Λk Λ∗k Sk−1 α = 0.

(2.21)

Formula (2.4) implies that (In + α−1 Λk Ck Λ∗k Sk−1 α + Sk α∗ Sk−1 α) = Sk+1 α∗ Sk−1 α. Hence, (2.21) is equivalent to

  Λ∗k+1 α∗ Sk−1 α − Λ∗k Sk−1 α2 + i Ck + Λ∗k Sk−1 Λk Λ∗k Sk−1 α = 0.

Using again (2.20), we rewrite (2.22) as  ∗  Λk+1 − Λ∗k + iCk Λ∗k (α∗ )−1 α∗ Sk−1 α = 0.

(2.22) (2.23)

The equality (2.23) is immediate from (2.3), and so (2.18) is also proved. Thus, (2.13) is proved as well. Next, (2.12) is proved by induction. Clearly (2.10) yields (2.12) for k equal to 0. If (2.12) holds for k = r, using (2.12) for k = r and relations (1.1), (2.2) and (2.13) we write   i  w(r  + 1, z) = Im + C  z) r w(r, z   i  −1 = Im + C r wα (r, −z)w(r, z)wα (0, −z) z   i = wα (r + 1, −z) Im + Cr w(r, z)wα (0, −z)−1 z = wα (r + 1, −z)w(r + 1, z)wα (0, −z)−1 . Thus, (2.12) holds for k = r + 1, and so (2.12) is proved.

(2.24) 

Using (2.13) we prove the next proposition. Proposition 2.3. Assume that the matrices Ck satisfy the second equality in (1.1) and the triple {α, S0 , Λ0 } satisfies the relations (2.1), (2.6), (2.7), and k given by (2.3)–(2.5) have the S0 = S0∗ . Then, the transformed matrices C following property:  2 = Im . C (2.25) k

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Proof. It easily follows from (2.8) and (2.11) (see, e.g., [15] or [14, Corollary 1.13]) that wα (r, z)wα (r, z)∗ ≡ Im . Since Ck2 = Im , we have      i i 1 Im − Ck Im + C k = 1 + 2 Im . z z z

(2.26)

(2.27)

In view of (2.26) and (2.27) we derive      i i 1 wα (k + 1, z) Im − Ck Im + Ck wα (k + 1, z)∗ = 1 + 2 Im . z z z (2.28) On the other hand, (2.26) yields       i  i  i  ∗ Im + Im − Ck wα (k, z)wα (k, z) Im + Ck = Im − Ck z z z 1 2 = Im + 2 C . z k

i  Ck z



(2.29)

According to (2.13), the left-hand sides of (2.28) and (2.29) are equal, and so we derive   1 2 1 Im + 2 Ck = 1 + 2 Im , z z 

that is, (2.25) holds.

Now, we introduce the notion of an admissible triple {α, S0 , Λ0 } and show afterwards that the admissible triples determine Sk > 0 (k ∈ N0 ). The definition of an admissible triple differs somewhat from the corresponding definition in [3], and the proof that Sk > 0 uses an idea from [4]. Definition 2.4. The triple {α, S0 , Λ0 } is called admissible if 0, i ∈ σ(α), S0 > 0 and the matrix identity (2.1) is valid. Theorem 2.5. Let an admissible triple {α, S0 , Λ0 } and an initial Dirac system (1.1) be given. Then, the conditions of Theorem 2.2 are satisfied. Moreover, we have Sk > 0

(k ∈ N0 ),

(2.30)

and the transformed system (2.2) is skew-selfadjoint Dirac, that is, (2.9) is valid. Proof. In order to prove (2.30) consider the difference     Sk+1 − In − iα−1 Sk In + i(α−1 )∗ = Sk+1 − Sk − α−1 Sk (α−1 )∗   +i α−1 Sk − Sk (α−1 )∗ . (2.31)

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395

Using (2.4), (2.8) and the second equality in (1.1), we rewrite (2.31) and derive a useful inequality:     Sk+1 − In − iα−1 Sk In + i(α−1 )∗ = α−1 Λk Ck Λ∗k (α−1 )∗ + α−1 Λk Λ∗k (α−1 )∗ ≥ 0. (2.32)     −k −k Since 0, i ∈ σ(α), the sequence In −iα−1 Sk In +i(α−1 )∗ (k ∈ N0 ) is well defined. In view of (2.32), this sequence is nondecreasing. Hence, taking  −k  −k into account S0 > 0, we have In − iα−1 Sk In + i(α−1 )∗ > 0, and so (2.30) holds. Similar to [3, Lemma A.1] one can show that σ(α) ⊂ C+ .

(2.33)

That is, one rewrites (2.1) in the form    ∗ −1/2 1/2 −1/2 −1/2 −1/2 −1/2 S0 αS0 − S0 αS0 = iS0 Λ0 Λ∗0 S0 , −1/2

−1/2

−1/2

and from S0 Λ0 Λ∗0 S0 ≥ 0, the relation σ(α) = σ(S0 αS0 ) ⊂ C+ follows. Clearly, (2.33) yields −i ∈ σ(α). Therefore, we may set z = −i in (2.13) and (taking into account the second equality in (1.1) and formula (2.26)) we obtain 1/2

k = wα (k + 1, −i)(Im + Ck )wα (k, i)∗ Im + C

  ∗ I m1 Im1 0 Uk wα (k, i)∗ . = 2wα (k + 1, −i)Uk 0 In the same way, setting in (2.13) z = i we obtain

 0  ∗  0 Im2 Uk wα (k, −i)∗ . Im − Ck = 2wα (k + 1, i)Uk I m2

(2.34)

(2.35)

k correAccording to (2.34) and (2.35) the dimension of the subspace of C sponding to the eigenvalue λ = −1 is greater than or equal to m2 and the k corresponding to the eigenvalue λ = 1 is dimension of the subspace of C greater than or equal to m1 . Thus, the representation (2.9) is immediate.  k ≥ 0 and that Im + C k has Remark 2.6. It follows from (2.9) that Im + C rank m1 . Hence, (2.34) yields     Im1 0 Uk wα (k + 1, −i)∗ = q˘k Im1 0 Uk wα (k, i)∗ , q˘k > 0 (2.36) for some matrix q˘k . In the same way, formulas (2.9) and (2.35) imply that     0 Im2 Uk wα (k + 1, i)∗ = qˆk 0 Im2 Uk wα (k, −i)∗ , qˆk > 0. (2.37) Now, setting

⎡ 

1/2 1/2 k = Wk := diag{˘ U qk , qˆk } ⎣ 

Im1

 0 Uk wα (k, i)∗ 

0 Im2 Uk wα (k, −i)

⎤ ∗

⎦,

(2.38)

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k in the reprewe provide expressions for some suitable unitary matrices U  sentations (2.9) of the matrices Ck . Indeed, according to the definition of Wk in (2.38) and to the relations (2.34)– (2.37) we have   k = 1 (Im + C k ) − (Im − C k ) = W ∗ jWk , C (2.39) k 2 and it remains to show that Wk is unitary. In view of (2.26) and (2.38), it is easy to see that Wk Wk∗ is a block diagonal matrix : Wk Wk∗ = diag{˘ ρk , ρˆk } > 0.

(2.40)

Hence, for Rk > 0 from the polar decomposition Wk = Rk Vk (Vk Vk∗ = Im ) we have Rk2 = diag{˘ ρk , ρˆk } (and, in particular, Rk is block diagonal). Therefore, (2.39) may be rewritten in the form k = V ∗ diag{˘ C ρk , −ˆ ρk } V k k

(˘ ρk > 0, ρˆk > 0).

Comparing (2.9) with the formula above, we see that all the eigenvalues of ρ˘k and ρˆk equal 1, that is, Rk2 = Rk = Im , and so Wk = Vk . In other words, Wk is unitary.

3. Explicit solutions of the corresponding non-stationary systems Recall the equalities (2.18): −1 k Λ∗ S −1 α, Λ∗k+1 Sk+1 (α2 + In ) = Λ∗k Sk−1 α2 − iC k k

k ∈ N0 ,

(3.1)

which are basic for the construction of explicit solutions of non-stationary systems. Introduce the semi-infinite shift block matrix S and diagonal block  matrix C: ⎡ ⎤ 0 Im 0 0 ... ⎢0 0 Im 0 . . . ⎥  := diag{C 0 , C 1 , C 2 , . . .}. ⎥, S := ⎢ C (3.2) ⎣0 0 0 Im . . . ⎦ ... ... ... ... ... The semi-infinite block column Ψ(t) is given by the formula Ψ(t) = Y eitα ,

Y = {Yk }∞ k=0 ,

Yk = Λ∗k Sk−1 .

(3.3)

It is easy to see that equalities (3.1) and Theorem 2.5 yield the following result. Theorem 3.1. Let the initial Dirac system (1.1) and an admissible triple k are well defined via {α, S0 , Λ0 } be given. Then, the matrices Λk , Sk and C (2.3)–(2.5) for k ∈ N0 . Moreover, the block vector function Ψ(t) constructed in (3.3) satisfies the non-stationary system   + SΨ = 0, (I − S)Ψ + CΨ where I is the identity operator and Ψ =

d dt Ψ.

(3.4)

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We note that an analogue of α in the standard B¨ acklund–Darboux transformation is a scalar. However, in GBDT α is an n × n matrix, which is called a generalized eigenvalue. Since α is matrix, we obtain a much wider family of solutions h(t) = Ψ(t)g (where Ψ is given in (3.3) and g ∈ Cn ) of the system   + Sh = 0. (I − S)h + Ch

References [1] J.L. Cieslinski, Algebraic construction of the Darboux matrix revisited, J. Phys. A 42 (2009), Paper 404003. [2] P.A. Deift, Applications of a commutation formula, Duke Math. J. 45 (1978), 267–310. [3] B. Fritzsche, M.A. Kaashoek, B. Kirstein and A.L. Sakhnovich, Skewselfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations, Math. Nachr. 289 (2016), 1792–1819. [4] B. Fritzsche, B. Kirstein, I. Roitberg and A.L. Sakhnovich, Stability of the procedure of explicit recovery of skew-selfadjoint Dirac systems from rational Weyl matrix functions, Linear Algebra Appl. 533 (2017), 428–450. [5] B. Fritzsche, B. Kirstein, I. Roitberg and A.L. Sakhnovich, Continuous and discrete dynamical Schr¨ odinger systems: explicit solutions, J. Phys. A: Math. Theor. 51 (2018), Paper 015202. [6] F. Gesztesy, A complete spectral characterization of the double commutation method, J. Funct. Anal. 117 (1993), 401–446. [7] F. Gesztesy and G. Teschl, On the double commutation method, Proc. Amer. Math. Soc. 124 (1996), 1831–1840. [8] I. Gohberg, M.A. Kaashoek and A.L. Sakhnovich, Pseudocanonical systems with rational Weyl functions: explicit formulas and applications, J. Differential Equations 146 no. 2 (1998), 375–398. [9] I. Gohberg, M.A. Kaashoek and A.L. Sakhnovich, Scattering problems for a canonical system with a pseudo-exponential potential, Asymptotic Analysis 29 (2002), 1–38. [10] M.A. Kaashoek and A.L. Sakhnovich, Discrete pseudo-canonical system and isotropic Heisenberg magnet, J. Funct. Anal. 228 (2005), 207–233. [11] V.B. Matveev and M.A. Salle, Darboux transformations and solitons, Springer, Berlin, 1991. [12] A.L. Sakhnovich, Exact solutions of nonlinear equations and the method of operator identities, Linear Alg. Appl. 182 (1993), 109–126. [13] A.L. Sakhnovich, Dynamics of electrons and explicit solutions of Dirac–Weyl systems, J. Phys. A: Math. Theor. 50 (2017), Paper 115201. [14] A.L. Sakhnovich, L.A. Sakhnovich and I.Ya. Roitberg, Inverse problems and nonlinear evolution equations. Solutions, Darboux matrices and Weyl– Titchmarsh functions, De Gruyter Studies in Mathematics 47, De Gruyter, Berlin, 2013. [15] L.A. Sakhnovich, On the factorization of the transfer matrix function, Sov. Math. Dokl. 17 (1976), 203–207.

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[16] L.A. Sakhnovich, Factorisation problems and operator identities, Russian Math. Surv. 41 (1986), 1–64. [17] V.E. Zakharov, A.V. Mikhailov, On the integrability of classical spinor models in two-dimensional space-time, Commun. Math. Phys. 74 (1980), 21–40. Alexander L. Sakhnovich Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien Oskar-Morgenstern-Platz 1, A-1090 Vienna Austria e-mail: [email protected]

On the reduction of general Wiener–Hopf operators Frank-Olme Speck Dedicated to Rien Kaashoek on the occasion of his 80th birthday

Abstract. The aim of this work is to present criteria for the equivalent reduction of general Wiener–Hopf operators W = P2 A|P1 X where X, Y are Banach spaces, P1 ∈ L(X) , P2 ∈ L(Y ) are any projectors and A ∈ L(X, Y ) is a bounded linear operator, namely, to more special forms where X = Y and possibly P1 = P2 and/or A is invertible or even where A is a cross factor. This is carried out with the help of operator relations: equivalence, equivalence after extension, matricial coupling and further related relations. Examples are given for the occurrence of different operator relations in applications. Mathematics Subject Classification (2010). Primary 47A68; Secondary 47B35. Keywords. Wiener–Hopf operator, reduction, operator relation, equivalence, cross factor, generalized invertibility.

1. Introduction A general Wiener–Hopf operator is given by [34] W = P2 A|P1 X : P1 X → P2 Y

(1.1)

X, Y are Banach spaces,

(1.2)

P1 ∈ L(X) , P2 ∈ L(Y ) are projectors,

(1.3)

A ∈ L(X, Y ) is a bounded linear operator.

(1.4)

where

We investigate the following questions: Under which conditions can the general Wiener–Hopf operator W in (1.1) be “equivalently reduced” to a © Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_16

399

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F.-O. Speck

˜ of the form (1.1) where at least one of the following “simpler” operator W conditions is satisfied: X=Y , (1.5) P 1 = P2 ,

(1.6)

A ∈ GL(X, Y ),

(1.7)

i.e., in case (1.7) holds, the underlying operator A is an isomorphism (boundedly invertible) or, moreover, a cross factor with respect to the setting (1.2)– (1.3), i.e., A−1 P2 AP1 and AQ1 A−1 Q2 are projectors where Q = I − P with corresponding subscripts throughout the paper. Herein “equivalent reduction” of an operator T to an operator S is defined by the validity of an operator relation between T and S. We admit different kinds of relations, because the use of the term “equivalent reduction” is not unique in the literature (and sometimes not precisely indicated). First of all we consider operator equivalence between Banach space operators in a most popular sense, written as T ∼ S, which means that there exist isomorphisms E and F such that T = ESF . Hence we look for sufficient (and eventually necessary) conditions under which ˜ ˜ ˜ : P˜1 X ˜ → P˜2 Y˜ ˜ = P˜2 A| W ∼W P1 X

(1.8)

˜ (more precisely the new setting X, ˜ Y˜ , P˜1 , P˜2 , A) ˜ where the new operator W has certain prescribed properties. In various cases it is necessary to replace the relation T ∼ S by an equivalence after extension (EAE) relation abbreviated by ∗

T ∼S

(1.9)

which stands for the fact that there exist additional Banach spaces Z1 , Z2 and isomorphisms E and F acting between suitable spaces such that     T 0 S 0 =E F. (1.10) 0 I Z1 0 I Z2 This more general relation was, for instance, intensively studied in [1, 4, 23]. On the other hand, a more special relation (rather than T ∼ S) plays a ˜ holds, which means that W itself can fundamental role, namely that W = W be written in a “simpler” or somehow more convenient form (1.8). The advantages of these relations are well known, see, e.g., [4, 8, 10, 34, 36]. Most important in applications is that inverses or generalized inverses of T and S can be computed from each other provided E, F or E −1 , F −1 , respectively, are known. Various other relations appear in the literature and sometimes rather weak relations are used, e.g., defined by the transfer of the Fredholm property [6] and/or other “transfer properties”, see [36, Section 2]. Here we consider only relations which transfer at least the representation of generalized inverses. The operator (1.1) was first studied by Devinatz and Shinbrot in 1969 [17] in a symmetric space setting (1.5)–(1.6) in the case (1.7) and under

On the reduction of general Wiener–Hopf operators

401

the additional condition that the spaces are separable Hilbert spaces. They introduced the notation W = TP (A) = P A|P : P → P

(1.11)

where P was the image of an orthogonal projector P acting in X. The present version of a general Wiener–Hopf operator (WHO) as given by (1.1) in a symmetric and also in an asymmetric Banach space setting was first studied in [33, 34]. Early sources for the employment of operator relations in applications are [10, 13]. More recent applications are discussed in Section 7. We start in Section 2 with some basic known results and present direct extensions in Section 3 with proofs separated in Section 4. Then we focus on reduction by an EAE relation in Section 5. Section 6 illuminates the fact that the notion of a cross factor is quite natural and useful. The list of examples in Section 7 contains a mini-survey that guides the reader to further study of particular concrete applications. It also includes remarks on other relations and on the interaction with the previous. Section 8 contains a few open problems.

2. Review of some known results First we observe that the representation of W by (1.1) is not unique, e.g., one could replace P1 by another projector P˜1 ∈ L(X) with the same image and a different kernel (any other complement of the image) without changing W itself. On the other hand, if one replaces P2 by another projector P˜2 which has the same kernel as P2 , the operator W in general is changed to an equivalent ˜ . In formulas it reads: operator W 2 Proposition 2.1. Let (1.1)–(1.4) be satisfied. If P˜1 ∈ L(X) satisfies P˜1 = P˜1 and im P˜1 = im P1 , then (1.1) coincides with

W = P2 A|P˜1 X . 2 If P˜2 ∈ L(Y ) satisfies P˜2 = P˜2 and ker P˜2 = ker P2 , then we have

˜ = P˜2 A|P X : P1 X → P˜2 Y . W ∼W 1 Proof. The first is evident, the second results from P2 P˜2 = P2 and P˜2 P2 = P˜2 . ˜ and W ˜ = P˜2 W . Therefore W = P2 W  From a general point of view the assumption of A to be an isomorphism (boundedly invertible) is not a great restriction: Proposition 2.2. Let (1.1)–(1.4) be satisfied and A not invertible. Then there ˜ Y˜ ) such that ˜ Y˜ , P˜1 , P˜2 and an isomorphism A˜ ∈ GL(X, is a space setting X, ˜ ˜ ˜ : P˜1 X ˜ → P˜2 Y˜ . W ∼ P˜2 A| P1 X Proof. See [38, Proposition 5.1]. For a symmetric setting it is already known from [21, Section 4.17]. 

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F.-O. Speck

Note that here the complete space setting X, Y, P1 , P2 may be changed. We call it “space setting”, because a projector P is uniquely determined by two complemented subspaces: its image im P and its kernel ker P . The next results make use of the following definition [34]. Given a space setting (1.2)–(1.3), an operator C ∈ L(X, Y ) is called a cross factor with respect to this setting, if it is an isomorphism that splits the two spaces X, Y each into four complemented subspaces Xj and Yj (j = 0, 1, 2, 3), respectively, such that the following diagram holds with bijective restrictions to the indicated subspaces: X

=

H X1

P1 X

IF ⊕

G X0

⊕

↓ Y

=

Y1 F

H X2

Q1 X

IF ⊕

C . ⊕ GH

⊕

Y2 I

G X3 ↓

⊕ GH

Y0 F

P2 Y

(2.1)

Y3 . I

Q2 Y

One can identify immediately the kernel of WC = P2 C|P1 X , ker WC = X0 , the image im WC = Y1 etc. and (less immediately) a reflexive generalized inverse of WC as WC− = P1 C −1 |P2 Y : P2 Y → P1 X , WC WC− WC

WC− WC WC−

(2.2)

WC− .

i.e., there hold = WC and = It is not hard to prove that C ∈ GL(X, Y ) is a cross factor if and only if C −1 P2 CP1 and CQ1 C −1 Q2 are idempotent, i.e., projectors. For further properties of a cross factor see [34] and recent papers such as [8, 38]. We recall two results from factorization theory which can be considered as basic cases of equivalent reduction W ∼ P2 C|P1 X of a general WHO (1.1) to a truncation of a cross factor (2.12) provided A is an isomorphism. The following is taken from [38]. The first result is known from the 1980s [33, 34], the second was published only recently [8, 38]. To this end we need the following definition. Under the assumptions (1.1)–(1.4) let A be boundedly invertible. Then (an operator triple A− , C, A+ with) A=

A−

C

A+

(2.3)

:Y ←Y ←X ←X. is referred to as a cross factorization of A (with respect to X, Y, P1 , P2 ), if the factors A± and C possess the following properties: A+ ∈ GL(X),

A− ∈ GL(Y ),

A+ P1 X = P1 X,

A− Q2 Y = Q2 Y,

and C ∈ L(X, Y ) is a cross factor (2.1). Theorem 2.3 (Cross Factorization Theorem). Let (1.1)–(1.4) be satisfied and A boundedly invertible. Then W is generalized invertible if and only if a

On the reduction of general Wiener–Hopf operators

403

cross factorization (2.3) of A exists. In this case, a formula for a reflexive generalized inverse of W is given by a “reverse order law” −1 W − = A−1 P2 A−1 + P1 C − | P 2 Y : P2 Y → P1 X .

(2.4)

A crucial consequence of a cross factorization is the equivalence of W and P2 C|P1 X , briefly W ∼ P2 C|P1 X , namely, W = P2 A− |P2 Y P2 C|P1 X P1 A+ |P1 X = E P2 C|P1 X F,

(2.5)

where E, F are isomorphisms. It yields the explicit determination of their ker−1 nels and complements of the images provided the factor inverses A−1 are ± ,C known. More consequences on the Fredholm property, explicit presentation of solutions of the equation W f = g etc. are immediate [34]. Conversely, if V is a reflexive generalized inverse of W , a cross factorization of A is given by (using the equivalence of A with an operator matrix): A = ι−1 A˜ ι1 , 2

ι 1 : X = P 1 X ⊕ Q 1 X → P1 X × Q 1 X , ι 2 : Y = P 2 Y ⊕ Q 2 X → P2 Y × Q 2 X , and A˜ = A˜− C A˜+ : P1 X × Q1 X → P2 Y × Q2 Y, (2.6)   I|P2 Y 0 A˜− = , Q2 AP1 V |P2 Y I|Q2 Y   W P2 (A − AV P2 A)|Q1 X C= , Q2 (A − AV P2 A)|P1 X C22   I|P1 X (V P2 A − (P1 − V W P1 )A−1 (P2 − W V P2 )A)|Q1 X ˜ A+ = . 0 I|Q1 X where C22 = Q2 (A − AV P2 A + A(P1 − V W P1 )A−1 (P2 − W V P2 )A)|Q1 X . The variety of existing cross factorizations for the symmetric setting was already described in [34, Chapter 7]. Those results can be extended to the asymmetric setting by the technique of [39]. Alternatively to the splitting (2.3) we consider under the assumptions of (1.1)–(1.4) another splitting A = A−

C

A+

(2.7)

:Y ←Z ←Z ←X. It is referred to as a Wiener–Hopf factorization through an intermediate space Z (with respect to X, Y, P1 , P2 ) (FIS), if the factors A± and C possess the following properties: They are linear and boundedly invertible in the above setting with an additional Banach space Z called intermediate space. Further there is a projector P ∈ L(Z) such that A+ P1 X = P Z,

A− QZ = Q2 Y

(2.8)

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F.-O. Speck

with Q = IZ − P and such that C ∈ L(Z) splits the space Z twice into four subspaces with Z

=

H X1

PZ

IF ⊕

G X0

↓ Z

=

Y1 F

⊕

H X2

QZ

IF ⊕

G X3

C . ⊕ GH

Y2 I

⊕

↓ Y0 F

PZ

⊕ GH

(2.9)

Y3 I

QZ

where C maps each Xj onto Yj , j = 0, 1, 2, 3 , i.e., the complemented subspaces X0 , X1 , . . . , Y3 are images of corresponding projectors p0 , p1 , . . . , q3 , namely, X0 = p0 Z = C −1 QCP Z,

X1 = p1 Z = C −1 P CP Z, . . . . . . Y3 = q3 Z = CQC −1 QZ,

similarly as in (2.1) (with X and Y replaced by Z). Again A± are called strong WH factors and C is said to be a cross factor, now acting from a space Z onto the same space Z. Theorem 2.4 (Wiener–Hopf factorization through an intermediate space). Let (1.1)–(1.4) be satisfied where A is boundedly invertible. Then the following assertions are equivalent: (i) W is generalized invertible and P1 ∼ P2 ; (ii) A admits a FIS (2.7). In this case, a formula for a reflexive generalized inverse of W is given by a “reverse order law” −1 W − = A−1 P A−1 + PC − | P 2 Y : P2 Y → P1 X .

(2.10)

By analogy to (2.5) we have the equivalent reduction to the truncation of a cross factor TP (C) in symmetric space setting given by W = P2 A− |P Z P C|P Z P A+ |P1 X = E P C|P Z F. Conversely, if V is a reflexive generalized inverse of W and P1 ∼ P2 , then a FIS of A can be obtained briefly in the following way. First symmetrize the space setting (and therefore W ) by the help of the projector P onto im P1 along ker P2 acting in the intermediate space Z = im P1 ×ker P2 (see Theorem 2.4) and apply the scalar analogue of formula (2.6) using the symmetrized ˜ and V˜ . That formula was already discovered in [34, Formula operators W (6.7a)] in the context of the algebraic cross factorization theorem: Let R be a unital ring with unit e, p = p2 ∈ R, a ∈ GR, w = pap, v ∈ R, and wvw = w, vwv = v. Then the following factorization holds: a = a − c a+ = [e + qav][a − ava + w + a(p − vw)a−1 (p − wv)a] · [e + vaq − (p − vw)a−1 (p − wv)a] .

(2.11)

On the reduction of general Wiener–Hopf operators

405

Now we substitute R = L(Z), e = IZ , p = P , a = EAF = A˜ and w, v by the ˜ , V˜ ∈ L( im P ) to W ˆ , Vˆ ∈ L(Z), i.e., extensions of W ˜ , V˜ zero extensions of W by zero to ker P and linearly to the full space Z, cf. [34, 38]. In this way we ˆ of A, ˜ and finally of A, as well. obtain a FIS of A, Proposition 2.5. Let (1.1)–(1.4) be satisfied where A is an isomorphism. Then W is generalized invertible if and only if W can be represented as a truncation of a cross factor C in the same space setting: W = P2 C|P1 X .

(2.12)

Moreover, a representation (2.12) implies an explicit formula for a reflexive generalized inverse V of W , for instance, the operator given by (2.2). Proof. See [39, Corollary 2.4]. For a symmetric setting it is already proved in [34, Section 6, Corollary 1]. The crucial point is that the existence of a generalized inverse V of W implies that we can construct a particular cross factorization of A, see (2.6), such that W is a truncation of the corresponding cross factor. The inverse conclusion can be verified by the help of the diagram, see also p. 117 in [34].  Later we shall prove a generalization of these results that holds under weaker assumptions. It should be noted that cross factors play a fundamental role in general Wiener–Hopf factorization methods and its applications where they appear as “middle factors” in a classical Wiener–Hopf factorization, see [34, 38]. For the next results we need the following definition from [8]. The operator W in (1.1) is said to be symmetrizable if W is equivalent to a WHO in symmetric setting ˜ = P A| ˜ PX : PX → PX ˜ = TP (A) (2.13) W ∼W ˜ P = P 2 ∈ L(Z) in the following way: there exist isomorphisms where A, E ∈ L(Z, Y ) and F ∈ L(X, Z) such that ˜ F W =E W F (P1 X) = P Z,

(2.14)

E (P Z) = P2 Y.

This can be seen as a property of the space setting rather than a property of W , because the properties in the second line of (2.14) are independent of A and imply that the first line makes sense for any choice of A ∈ L(X, Y ). Therefore we also speak of a symmetrizable space setting (1.2)–(1.3) which just fulfills the second line of (2.14). The main result of [8] is the following: Theorem 2.6 (Symmetrization criterion). Given a space setting (1.2)–(1.3) where X ∼ = Y . Then the following statements are equivalent: 1. The setting X, Y, P1 , P2 is symmetrizable; 2. P1 X ∼ = P2 Y and Q1 X ∼ = Q2 Y ; 3. P1 ∼ P2 .

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F.-O. Speck

Proof. See the direct proof of Theorem 1.1 in [8].



Remark 2.7. Note that the condition X∼ = im P1 × ker P2 ∼ =Y

(2.15)

is necessary but not sufficient for a setting to be symmetrizable, cf. a counterexample in [8, Remark 2.2], which may be repeated here, since it will be needed later in this article. Let us consider X = Y = 2 (Z) and P1 : (. . . , x−2 , x−1 , x0 , x1 , x2 , . . .) → (. . . , 0, 0, 0, x1 , x2 , . . .),

(2.16)

P2 : (. . . , x−2 , x−1 , x0 , x1 , x2 , . . .) → (. . . , 0, 0, x0 , 0, 0, . . .). Here condition (2.15) holds because X, Y, P1 X, Q2 Y are infinite-dimensional separable Hilbert spaces, but P1 and P2 are clearly not equivalent. The example may be modified considering cases where ker P1 or im P2 are “significantly smaller” than the other two subspaces im P1 and ker P2 , e.g., separable vs. non separable subspaces. Remark 2.8. The question of equivalent reduction of the WHO (1.1) (by the above-mentioned relations) is much wider than the previous question of symmetrization (of a space setting) as discussed in [8]. Hence one of the reminder questions is: What can we say about equivalent reduction of W if the conditions in Theorem 2.4 are violated? This rather tough question will be discussed in a simplified setting in Section 5. Further reduction methods are obtained with the help of other operator relations such as equivalence after extension, one-sided equivalence after extension, matricial coupling, Schur coupling etc., see, e.g., [2, 4, 23]. Their role in the present context will be touched later in Section 7. However we first give some direct extensions of the known results in Section 3 and present the corresponding proofs separately in Section 4.

3. Direct extensions ˜ = Y˜ ) First we present a light strengthening of Proposition 2.2 (now with X and an extension of Theorem 2.6 (including further equivalent conditions). Proposition 3.1 (Reduction to the truncation of an isomorphism). Let (1.1)– ˜ two projectors P˜1 , P˜2 ∈ (1.4) be satisfied. Then there is a Banach space X, ˜ ˜ ˜ L(X) and an isomorphism A ∈ GL(X) such that ˜ ˜ ˜ W ∼ P˜2 A| (3.1) P1 X

˜∼ ˜∼ where P˜1 X = P1 X and P˜2 X = P2 Y . ˜ onto the same Note that now A˜ is a mapping from a Banach space X space. However, there are two in general different projectors P˜1 , P˜2 involved which are not necessarily equivalent. Theorem 3.2 (Extended symmetrization criterion). Given a space setting (1.2)–(1.3) such that X ∼ = Y . Then the following statements are equivalent:

On the reduction of general Wiener–Hopf operators 1. 2. 3. 4. 5.

407

The setting X, Y, P1 , P2 is symmetrizable; P1 X ∼ = P2 Y and Q1 X ∼ = Q2 Y ; P 1 ∼ P2 ; There exists an invertible WHO of the form (1.1) with A ∈ GL(X, Y ); There exists an invertible WHO of the form Q1 B|Q2 Y : Q2 Y → Q1 X where P1 + Q1 = I|X and P2 + Q2 = I|Y and B ∈ GL(Y, X).

Remark 3.3. It is known (and easily verified) that the two associated WHOs W = P2 A|P1 X and Q1 A−1 |Q2 Y are matricially coupled [1, 33] and therefore equivalent after extension [1, 4]. This has tremendous consequences in operator theory and important applications in mathematical physics [12, 23]. Only recently an explicit EAE relation between associated WHOs was given in the asymmetric case, which allows to derive a formula for a generalized inverse of W in a most convenient way from a formula of a generalized inverse of W∗ see [39, Formula (1.11)]: 

W 0

0 I|Q1 X

:



 =

−P2 A|Q1 X I|Q1 X

Q 1 X × P2 Y

 I|P2 Y − P2 AQ1 A−1 |P2 Y Q1 A−1 |P2 Y   W∗ 0 Q2 A|P1 X 0 I|P2 Y P2 A|P1 X

← Q 1 X × P2 Y

← P2 Y × Q 2 Y

(3.2) Q2 A|Q1 X P2 A|Q1 X



← P1 X × Q 1 X .

It is also known that the conditions of Theorem 3.2 characterize exactly the Banach space settings where the FIS Theorem holds: W is generalized invertible iff A admits a Wiener–Hopf factorization through an intermediate space, cf. Theorem 2.4. Hence our next interest is in the case where P1 ∼ P2 is violated, but A ∈ GL(X), asking: Under which conditions is W equivalently reducible to a WHO in symmetric setting, i.e., (2.13) holds (without demanding (2.14))? Remark 3.4. We recall that the expression “equivalently reducible” is not used in a uniform way in the existing literature. Hence it is necessary to mention precisely the relation that is behind this expression. See [36, Section 2], for a discussion of a variety of operator relations and their properties. Lemma 3.5. Let (1.1)–(1.4) be satisfied, X = Y a separable Hilbert space and ˜ in symmetric A ∈ GL(X). Then W is equivalently reducible to a WHO W ˜ ˜ ˜ setting: W ∼ W = P A|P Z where, moreover, A ∈ GL(Z). Theorem 3.6 (Symmetrization in separable Hilbert spaces). Let (1.1)–(1.4) be satisfied, X, Y separable Hilbert spaces of the same dimension. Then W is ˜ in symmetric setting: W ∼ W ˜ = P A| ˜ PZ equivalently reducible to a WHO W ˜ where, moreover, A ∈ GL(Z). Theorem 3.7 (Equivalent reduction to the truncation of a cross factor). Let (1.1)–(1.4) be satisfied and W be generalized invertible. Then W is equivalent

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F.-O. Speck

to a truncation of a cross factor acting in Z = X × Y : ˜ = P˜2 C| ˜ W ∼W

(3.3)

P1 Z

where C ∈ GL(Z). Now we present in turn some consequences of the equivalent reduction of W to a truncation of a cross factor, i.e., necessary conditions for the existence of such a relation. ˜ = P˜2 C|P˜1 X, ˜ Theorem 3.8. Let (1.1)–(1.4) be satisfied, W = P2 A|P1 X , W ˜ Y˜ ) a cross factor with respect to a setting X, ˜ Y˜ , P˜1 , P˜2 : and C ∈ GL(X,

˜ X

=

H X1

˜ P˜1 X

IF ⊕

G X0

↓ Y˜

=

Y1 F

⊕

H X2

˜ Q˜1 X

IF ⊕

C . ⊕ GH P˜2 Y˜

Y2 I

⊕

G X3 ↓

Y0 F

⊕ GH

(3.4)

Y3 . I

Q˜2 Y˜

˜ , then I. If W ∼ W • W is generalized invertible; ˜ • im W ∼ Y1 = P˜2 C P˜1 C −1 Y˜ ; = X1 ∼ = Y1 , X1 = P˜1 C −1 P˜2 C X, −1 ˜ • ker W ∼ Y0 = Q˜2 C P˜1 C −1 Y˜ . = X0 ∼ = Y0 , X0 = P˜1 C Q˜2 C X, ˜ ∗ = Q˜1 C −1 | ˜ ˜ has the properties II. Furthermore W Q2 Y ˜ • W∗ is generalized invertible; ˜∗ ∼ ˜ • im W Y3 = Q˜2 C Q˜1 C −1 Y˜ ; = X3 ∼ = Y3 , X3 = Q˜1 C −1 Q˜2 C X, ∼ ∼ ˜ • ker W∗ = X0 = Y0 , see above.

III. If, moreover, A ∈ GL(X, Y ), then W∗ = Q1 A−1 |Q2 Y is well defined and ˜ ∗ mentioned before. has the same properties as W ˜ ∗ ) together with Remark 3.9. These consequences (properties of W and W ∼ Y2 are characteristic for the relation W = P2 A|P1 X ∼ P˜2 C|P˜1 X. ˜ X2 = This follows if we combine the present results with what is known from [34]. Further so-called “transfer properties” (from W to W∗ and vice versa) were already discussed in [36].

4. Proofs Proof of Proposition 3.1. The idea of the proof goes back to the proof in the symmetric case, see [21, Section 4.17] and another asymmetric variant in [38, ˜ and Y˜ were involved). Proposition 5.1] (where two spaces X ˜ = P1 X ×P2 Y as a Banach We consider the topological product space X space and   μIP1 X 0 ˜ →X ˜ A˜ = : X (4.1) W μIP2 Y

On the reduction of general Wiener–Hopf operators where μ ∈ C, |μ| > A, and



˜ → P1 X × {0} , P˜1 : X ˜ → {0} × P2 Y, P˜2 : X

409

  x → , 0     x 0 → . y y x y



Then we have an invertible A˜ and the given operator W is equivalent to       x μx 0 ˜ ˜ P2 A|P˜1 X˜ : → → . (4.2) 0 Wx Wx ˜ ˜ ˜ in (4.2) with the natural isomorphisms I.e., we combine the operator P˜2 A| P1 X between X and X×{0} to the right and between Y and {0}×Y to the left.  Proof of Theorem 3.2. The equivalence of the first three conditions is known from [8, Theorem 1.1], as mentioned before in Theorem 2.4. Now let condition 1 be satisfied, i.e., the second line of (2.14) holds. Consider the operator B = EF , which obviously belongs to GL(X, Y ). Moreover B = EF represents a canonical Wiener–Hopf factorization (FIS) through the intermediate space Z of (2.13), see Theorem 2.4 with C = I. The operator P2 A|P1 X : P1 X → P2 Y is invertible according to [38, Theorem 2.1], i.e., condition 4 is fulfilled. Conversely, if there is an invertible WHO of this form, the underlying operator A admits a canonical FIS A = EF according to the same Theorem 2.1 of [38] and the factors enable a symmetrization of the setting (1.2)–(1.3). I.e., condition 1 is satisfied. The equivalence of the last two conditions 4 and 5 is a consequence of the fact that the two operators W = P2 A|P1 X : P1 X → P2 Y, W∗ = Q 1 A

−1

| Q2 Y : Q2 Y → Q1 X

(4.3) (4.4)

are simultaneously invertible or not, respectively, provided A is an isomorphism. See [34, Section 2, Theorem 1].  Proof of Lemma 3.5. We distinguish three cases. Case 1. If dim X < ∞, the underlying operator A can be represented by an invertible matrix of size n, say (using suitable bases of X and Y ). W is of finite rank r ≤ n and so is P that is found by linear algebra means. Case 2. If dim X = ∞ and im P1 or im P2 is finite dimensional, then W has finite rank again and a similar argument holds as before with a representation of W by a squared matrix of size rank P = rank W ≤ min{ rank P1 , rank P2 } according to the fact that finite-dimensional subspaces of a Banach space are complemented. Case 3. If dim X = ∞ and im Q1 or im Q2 is finite dimensional, then W∗ ∗ has finite rank and is therefore generalized invertible. Since W ∼ W∗ (see [39, Lemma 1.4]), W is generalized invertible as well. Moreover W is representable in a form (1.1) where the underlying operator is an isomorphism, see

410

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Proposition 3.1. By construction P˜1 ∼ P˜2 , since they have isomorphic images and isomorphic kernels. Case 4. If all the four previous subspaces are infinite-dimensional separable Hilbert spaces, then P1 ∼ P2 (by the same argument as before) and Theorem 2.4 applies.  Remark 4.1. The idea of the previous proof enables us to construct various ˜ P Z is obtained, cf. [8, examples where P1 ∼ P2 is violated, but W ∼ P A| Remark 2.2]. In other words: There exist classes of WHOs which are equivalent to a WHO in symmetric space setting although the space setting is not symmetrizable Proof of Theorem 3.6. First we conclude that W is representable as trunca˜ an isomorphism acting tion of a modified underlying operator A˜ ∈ GL(X), ˜ with the help of Proposition 3.1. Hence W satisfies the assumptions within X, of Lemma 3.5 and is therefore symmetrizable.  Proof of Theorem 3.7. Proposition 3.1 yields that W can be written in the ˜ is an isomorphism. According to Proposition form (3.1) where A˜ ∈ GL(X) ˜ ˜ = X × Y, P˜1 , P˜2 with a 2.3 A admits a cross factorization with respect to X ˜ cross factor C ∈ GL(X) and W is represented as a truncation (3.3) of this ˜ =X ×Y. cross factor C with Z = X  Proof of Theorem 3.8. All the consequences can be verified directly from the cross factor properties. The fact that they are characteristic, i.e., that the set of consequences implies the relation between the two operators W and ˜ = P˜2 C| ˜ ˜ ˜ , is a result of the Cross Factorization Theorem, see details W P1 X ˜ and W ˜ ∗ are in [34]. The other relations are concluded from the fact that W associated WHOs, which is also well-known from [34], i.e., they are matricially coupled, as well [4]. 

5. Equivalence after extension Let us extend the question if a WHO is equivalently reducible by operator equivalence to the question if a WHO is equivalently reducible by operator equivalence after extension (EAE). This relation has weaker but still important consequences, see [4, 13, 38] for instance. Most important in applications is the isomorphy of the defect spaces of the two related operators, in particular the transfer of the Fredholm property from one operator to the other preserving the defect numbers and the mutual computation of generalized inverses. So it is natural to study the case where the setting (1.1)–(1.3) is not ∗ symmetrizable but it satisfies a weaker condition, namely P1 ∼ P2 instead of P1 ∼ P2 . This embraces cases mentioned in Remark 2.7.

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∗

Theorem 5.1. Given a setting (1.1)–(1.3) let X = Y , P1 ∼ P2 , and im P1 ∩ ker P2 = {0}. Then Z = im P1 × ker P2 ∼ (5.1) = im P1 ⊕ ker P2 = X. Hence the projector P ∈ L(X) onto im P1 along ker P2 exists and W = P2 A|P1 X ∼ P A|P X

(5.2)

holds for any A ∈ L(X). ∗

Proof. We know from [4, Theorem 3], that P1 ∼ P2 implies ker P1 ∼ = ker P2 , coker P1 ∼ = coker P2 ,

(5.3)

since the images of P1 and P2 are closed. Hence we have Z = im P1 × ker P2 ∼ = im P1 × ker P1

(5.4)

On the other hand, the condition im P1 ∩ ker P2 = {0} implies ∼ im P1 ⊕ ker P2 Z=

(5.5)

∼ = im P1 ⊕ ker P1 = X.

where the isomorphism is given by the decomposition operator ι : X = P1 X ⊕ Q 1 X x = P1 x + Q 1 x

→ Z = P1 X × Q 2 X   P1 x → ιx = . Q2 x

(5.6)

Therefore the projector P onto P1 X along Q2 X exists and we conclude W = P2 A|P1 X = P2 A|P X ∼ P A|P X as observed in Proposition 2.1.

(5.7) 

Remark 5.2. In the foregoing theorem the nature of im P2 (and of ker P1 , as well) can be very different from the nature of the other subspaces, as seen in the example (2.16). Again we observe that P1 ∼ P2 is not necessary for a reduction by operator equivalence (5.2) to a WHO in symmetric setting. Another question is: When is W reducible by EAE to a truncation of a cross factor? Theorem 5.3. Let W be given by (1.1)–(1.4). Then W is generalized invertible if and only if ∗ ˜ → P˜2 Y˜ (5.8) W ∼ WC = P˜2 C|P˜1 X˜ : P˜1 X ˜ Y˜ , P˜1 , P˜2 and a cross factor C. with a suitable space setting X, Proof. If (5.8) is satisfied, we have a reflexive generalized inverse WC− = ∗ P˜2 C −1 |P˜1 X˜ and the relation W ∼ WC (see (1.9)–(1.10)) yields a reflexive generalized inverse of W by the formula     WC− 0 E −1 (5.9) W − = R11 F −1 0 I Z2

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where R11 denotes the restricted operator of the first matrix entry to the first component spaces, see [13, Theorem 2.5]. The inverse conclusion is known from Theorem 3.7 since operator equivalence can be seen as an EAE relation with trivial extension by Z1 = {0}.  Note that in this section the underlying operator A did not need to be invertible.

6. The case X = Y and A = I We study a very special WHO W = P2 |P1 X , i.e., where A = I ∈ GL(X), but the projectors P1 and P2 do not underlie further restrictions. Lemma 6.1. Let P1 , P2 be any projectors in a Banach space X. Then the subspaces X1 = P2 P1 X, X0 = Q2 P1 X,

(6.1)

X2 = P2 Q1 X, X3 = Q2 Q1 X form a set of complemented subspaces in X, i.e., X = (X1 ⊕ X0 ) ⊕ (X2 ⊕ X3 )

(6.2)

= X1 ⊕ X0 ⊕ X2 ⊕ X3 . Proof. By definition of the subspaces we have the mapping diagram X

=

H X1

P1 X

IF ⊕

G X0

↓ X

=

X F1

⊕

H X2

Q1 X

IF ⊕

I . ⊕ GH P2 X

X2 I

⊕

G X3 ↓

X0 F

⊕ GH

(6.3)

X3 . I

Q2 X

This means that the identity operator is a cross factor with respect to the setting X, P1 , P2 . In particular the subspaces Xj , j = 0, 1, 2, 3 are all complemented in X and (6.2) is satisfied.  Corollary 6.2. Let P1 , P2 be any projectors in a Banach space X. Then P2 |P1 X : P1 X → P2 X is generalized invertible and a reflexive generalized inverse is given by P1 |P2 X : P2 X → P1 X. Remark 6.3. The previous results give an answer to the question: How general can the space setting be if there exists a WHO W such that (1.1)–(1.4) is satisfied, A ∈ GL(X, Y ), and W is generalized invertible with any prescribed kind of image and defect spaces? The answer is given by Theorem 3.7 and Lemma 6.1, namely, by the condition ˜ ∼ X ∼ = X = Y ˜ decomposes as in (3.4), i.e., can be seen as a topological product of where X Banach spaces of the given type.

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7. Examples and some remarks on further relations Now let us see how the different kinds of relations appear in applications. We start with the simplest species of equivalence relation. 7.1. Equivalent operators This kind of relation appears typically in potential theory in the context of representation formulas, when the solution of an elliptic PDE is represented by volume and/or surface potentials or by fundamental solutions. The relation is given by substitution formulas which represent a one-to-one correspondence and can be interpreted as mappings E, F in T = ESF where T is the operator associated to a linear boundary value problem (BVP) and S stands for the potentials. For details see standard books such as [24, 42]. The prototype of an elliptic linear BVP is written in the form Au = f

in Ω

(pde in nice domain)

Bu = g

on Γ = ∂Ω

(boundary condition)

(7.1)

where the data f, g, the data space Y = Y1 × Y2 and the solution space X are assumed to be known. As a standard situation we work with Banach spaces. The operator associated with the BVP is therefore given by   A L= : X → Y = Y 1 × Y2 . (7.2) B It is clear that a linear BVP in the abstract setting (7.1) is well posed if and only if the operator L in (7.2) is boundedly invertible (an isomorphism). Thus the main problem is: Find (in a certain form) the inverse (resolvent) of the associated operator L or a generalized inverse etc. In this sense associated operators were systematically used, e.g., in [9, 15, 16, 20, 24, 29, 30, 42]. Operators that have the form of general WHOs appear frequently in the reduction of canonical BVPs [28, 35] or by localization of elliptic BVPs [11, 19]. Operator-theoretically interesting cases appear when E or F are not isomorphisms, for instance injective, but not boundedly invertible [11] or only Fredholm, see, e.g., [16] and [24, Chapter 5]. Another case of equivalent operators is present in the so-called lifting process for classical Wiener–Hopf, Bessel potential, and pseudo-differential operators. This consists in a factorization of an operator acting between Sobolev spaces of different order, say T : H r (Ω) → H s (Ω), T = ESF , S : L2 (Ω) → L2 (Ω) where E, F have the form of Bessel potential operators or certain generalizations, see [18, 20, 31, 39]. 7.2. Equivalence after extension (EAE) This relation, given by (1.9), is characteristic for the transfer of generalized invertibility, see [4], and appears in many applications such as integral equations and BVPs [13], as well as system theory [1]. A process commonly denoted by “equivalent reduction” is the step from a BVP of the form (7.1) to a semi-homogeneous BVP (where f = 0 or g = 0).

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This is not reflected by equivalence between the two associated operators but by equivalence after extension, see [36]. However, in this case we find a more special form of EAE, namely: 7.3. Equivalence after one-sided extension (EAOE) This is the special case of (1.9) where one of the two extensions is trivial, i.e., Z1 = {0} (or Z2 = {0}). Equivalence after one-sided extension occurs frequently in applications, as well. Another example would be the relationship between paired operators and general WHOs given by the well-known formula P AP + Q = (I − P AQ)(P A + Q) = (I − P AQ)(I + QA−1 P )(P + QA−1 Q)A.

(7.3)

for operators in a symmetric setting A, P, Q ∈ L(X), provided P 2 = P , Q = I − P and A is invertible (in the second line only). The first line can be seen as an EAOE between W = P A|P X and the paired operator P A + Q, replacing the entries (acting on X) by the corresponding truncated operators. This formula can even be extended to the asymmetric case (1.1), see (3.2), to prove some results in factorization theory in a modified or more efficient way. The relationship with paired operators is useful in the theory of singular integral operators of Cauchy type and Riemann boundary value problems, see further references in [39]. In [22, 23] it was shown that the two relations EAE and EAOE coincide (i.e., two operators satisfy the two relations only simultaneously) for certain classes of operators, but not in general. Particularly in the general Hilbert space case EAE and EAOE (and Schur coupling) coincide. That this is not the case in general was observed in [41]. 7.4. Delta-relations This generalization of an EAE was introduced in [13] were the operator I|Z1 in (1.10) was replaced by a so-called companion TΔ of T which is somehow related to T :     T 0 S 0 =E F. (7.4) 0 TΔ 0 I Z2 In particular, the simpler case of Z2 = {0} was considered, i.e., the matrix on the right replaced by S. This so-called Delta-relation between T and S, which in general is not an equivalence relation, plays a decisive role in the theory of singular integral operators with Carleman shift and Toeplitz plus/minus Hankel operators with a tremendous literature, see [7] for instance. The point is here that, if S is a WHO that admits a generalized inverse S − , then T has a generalized inverse, as well (together with TΔ ), which can be computed from S − by a formula similar to (5.9). Applications occur in diffraction theory in problems with certain symmetries, see [15, 16, 25] for instance.

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7.5. Matricial coupling (MC) Two bounded linear operators S, T acting in Banach spaces are said to be matricially coupled, if there exist operator matrices such that    −1 T ∗ ∗ ∗ = (7.5) ∗ ∗ ∗ S where the stars stand for suitable bounded linear operators and the operator matrices represent isomorphisms (in the corresponding Banach spaces). This operator relation was introduced by Bart, Gohberg and Kaashoek originally as “matrical coupling”, cf. [1]. It is well known that the two relations EAE and MC are equivalent to each other, see [4, Theorem 1] and [39]. If the two operators are generalized invertible, the validity of these relations is moreover equivalent to the fact that the two operators have isomorphic kernels and isomorphic cokernels, see [4, Theorem 3]. If they are generalized invertible and if the relation is explicitly known, it allows to determine generalized inverses from each other [13]. This fact can be used to reduce the problem of (generalized) inversion ˜, of a certain operator W of the form (1.1) to a simpler one denoted by W say. A very good example for the ease of a MC relation appears in diffraction theory in the context of diffraction of time-harmonic acoustic or electromagnetic waves from plane screens in R3 . It turns out that the operators associated to certain problems for complementary plane screens are matricially coupled, which was denoted as “abstract Babinet principle” in [37]. Therefore the construction of the resolvent operator for one of the problems implies a representation of the resolvent operator for the other one. Since the resolvent operators can be computed for certain convex screens, the method results in the explicit solution of diffraction problems for a much wider class of plane screens, the so-called “polygonal-conical screens” [12]. 7.6. Schur coupling (SC) Two operators T ∈ L(X) and S ∈ L(Y ) are said to be Schur coupled if there exists an operator   A B ∈ L(X ⊕ Y ) (7.6) C D where A and D are invertible and such that T = BD−1 C and S = D − CA−1 B. The equivalence after one-sided extension concept, being stronger than the equivalence after extension, is intimately related with the Schur coupling notion [5]. Schur coupled operators allow a more direct relationships between their kernels and images than in the equivalence after extension relation. See [2, Sections 2–3] and [22, 23, 41] . It was also observed in [41] that EAOE and SC do not coincide, while EAOE ⇒ SC does hold in general. So EAOE is a stronger relation than SC. For more details and still existing open problems within the Schur coupling theory see [23, 41].

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Schur coupling has great importance in the state space approach in system theory and in factorization theory, see [2, 3]. 7.7. Further relations Various techniques, which are important in applications, make essentially use of operator relations: • Localization of BVPs based on Simonenko’s approach for singular integral operators [32] applies to several classes of elliptic BVPs in [11, 18, 19, 42]. • Normalization of ill-posed BVPs can be carried out as a consequence of the stability of certain relations against a certain change of the spaces. See [11, 30, 31] for details of that technique. • Regularity results (like Weyl’s lemmas) can be obtained if the relation holds in a scale of Banach spaces. In various cases the relation transfers regularity properties from the solution of boundary equations (of WH type) to the solution of BVPs. See [14, 31] for instance.

8. Open problems We would like to point out three open problems. Problem 1. Let (1.1)–(1.4) be satisfied, but not P1 ∼ P2 . Under which necessary and sufficient conditions there exists a Banach space Z and operators ˜ P = P 2 ∈ L(Z) such that A, ˜ PZ ? W ∼ P A| Several modifications of this problem are open, as well: assume A ∈ GL(X, Y ) and/or X ∼ = Y or replace equivalence by EAE. Problem 2. Let (1.1)–(1.4) be satisfied, but not P1 ∼ P2 . Moreover let A ∈ GL(X, Y ) (without loss of generality) and W be generalized invertible. Can we conclude ∗ ˜ PZ W ∼ P A| ˜ P? with suitable Z, A, Problem 3. Given a space setting X, Y, P1 , P2 with X ∼ = Y such that ˜ PZ W = P2 A|P1 X ∼ P A| holds for a certain choice of Z, P and for all A˜ ∈ GL(X, Y ). Is then P1 ∼ P2 ? We know that the inverse conclusion is true, see Theorem 2.4. However it seems to me that here EAE is in the game.

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Acknowledgment The work was supported by FCT – Portuguese Foundation for Science and Technology through the Center for Functional Analysis, Linear Structures and Applications at Instituto Superior T´ecnico, Universidade de Lisboa, and by a voluntary agreement of the author with the Universidade de Lisboa. The author likes to express his gratitude to Professor M.A. Kaashoek and his co-authors H. Bart, I. Gohberg, S. ter Horst and A.C.M. Ran for helpful discussions about the present and related work over a long period of time.

References [1] H. Bart, I. Gohberg, and M. Kaashoek, The coupling method for solving integral equations, in: Oper. Theory Adv. Appl. 2, pp. 39–73, Birkh¨ auser, Basel, 1984. Addendum, Integral Equations Oper. Theory 8 (1985), 890–891. [2] H. Bart, I. Gohberg, M. Kaashoek, and A.C.M. Ran, Schur complements and state space realizations, Lin. Alg. Appl. 399 (2005), 203–224. [3] H. Bart, I. Gohberg , M. Kaashoek, and A.C.M. Ran, A state space approach to canonical factorization with applications, Oper. Theory Adv. Appl. 200, Birkh¨ auser, Basel 2010. [4] H. Bart and V.E. Tsekanovskii, Matricial coupling and equivalence after extension, in: Oper. Theory Adv. Appl. 59, pp. 143–160, Birkh¨ auser, Basel, 1991. [5] H. Bart and V.E. Tsekanovskii, Complementary Schur complements, Linear Algebra Appl. 197 (1994), 651–658. [6] M.A. Bastos, A.F. dos Santos, and R. Duduchava, Finite interval convolution operators on the Bessel potential spaces Hps , Math. Nachr. 173 (1995), 49–63. [7] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators, Springer, Berlin 2006. [8] A. B¨ ottcher and F.-O. Speck, On the symmetrization of general Wiener–Hopf operators, J. Operator Theory 76 (2016), 335–349. [9] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51. [10] L.P. Castro, Relations between Singular Operators and Applications, PhD thesis, Universidade T´ecnica de Lisboa, 1998. [11] L.P. Castro, R. Duduchava, and F.-O. Speck, Localization and minimal normalization of some basic mixed boundary value problems, in: Factorization, singular operators and related problems (S. Samko et al., eds.), pp. 73–100, Kluwer, Dordrecht, 2003. [12] L.P. Castro, R. Duduchava, and F.-O. Speck, Diffraction from polygonalconical screens, an operator approach, in: Oper. Theory Adv. Appl. 242, pp. 113–137, Birkh¨ auser, Basel, 2014. [13] L.P. Castro and F.-O. Speck, Regularity properties and generalized inverses of delta-related operators, Z. Anal. Anwend. 17 (1998), 577–598. [14] L.P. Castro and F.-O. Speck, Convolution type operators with symmetry in Bessel potential spaces, in: Oper. Theory Adv. Appl. 258, pp. 21–49, Birkh¨ auser, Cham, 2017.

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[15] L.P. Castro, F.-O. Speck, and F.S. Teixeira, On a class of wedge diffraction problems posted by Erhard Meister, in: Oper. Theory Adv. Appl. 147, pp. 211– 238, Birkh¨ auser, Basel, 2004. [16] L.P. Castro, F.-O. Speck, and F.S. Teixeira, Mixed boundary value problems for the Helmholtz equation in a quadrant, Integral Equations Oper. Theory 56 (2006), 1–44. [17] A. Devinatz and M. Shinbrot, General Wiener–Hopf operators, Trans. AMS 145 (1969), 467–494. [18] R. Duduchava and F.-O. Speck, Pseudodifferential operators on compact manifolds with Lipschitz boundary, Math. Nachr. 160 (1993), 149–191. [19] R. Duduchava and M. Tsaava, Mixed boundary value problems for the Laplace– Beltrami equation, Complex Var. Elliptic Equ. 63 (2018), 1468–1496. [20] G.I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, American Mathematical Society, Providence, Rhode Island, 1981. (Russian edition 1973). [21] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations I, II, Birkh¨ auser, Basel 1992. (German edition 1979, Russian edition 1973). [22] S. ter Horst, M. Messerschmidt, and A.C.M. Ran, Equivalence after extension for compact operators on Banach spaces, J. Math. Anal. Appl. 431 (2015), 136–149. [23] S. ter Horst and A.C.M. Ran, Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces, Linear Algebra Appl. 439 (2013), 793-805. [24] G.C. Hsiao and W.L. Wendland, Boundary Integral Equations, Springer, Berlin 2008. [25] G.S. Litvinchuk, Solvability theory of boundary value problems and singular integral equations with shift, Kluwer, Dordrecht 2000. [26] G.S. Litvinchuk and I.M. Spitkovsky, Factorization of Measurable Matrix Functions, Oper. Theory Adv. Appl. 25, Birkh¨ auser, Basel, 1987. [27] E. Meister, Some solved and unsolved canonical problems in diffraction theory, in: Differential Equations and Mathematical Physics (I.W. Knowles et al., eds.), Lect. Notes Math. 1285, pp. 320–336, Springer, Berlin, 1987. [28] E. Meister and F.-O. Speck, Modern Wiener–Hopf methods in diffraction theory, in: Ordinary and Partial Differential Equations 2 (B.D. Sleeman et al., eds.), pp. 130–171, Longman, London, 1989. [29] S.E. Mikhailov, Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficients, Math. Meth. Appl. Sciences 29 (2006), 715–739. [30] A. Moura Santos, F.-O. Speck, and F.S. Teixeira, Compatibility conditions in some diffraction problems, in: Pitman Research Notes in Mathematics Series 361, pp. 25–38, Pitman, London, 1996. [31] A. Moura Santos, F.-O. Speck, and F.S. Teixeira, Minimal normalization of Wiener–Hopf operators in spaces of Bessel potentials, J. Math. Anal. Appl. 225 (1998), 501–531.

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[32] I.B. Simonenko, A new general method of investigating linear operator equations of the type of singular integral equations, Soviet Math. Dokl. 5 (1964), 1323–1326. [33] F.-O. Speck, On the generalized invertibility of Wiener–Hopf operators in Banach spaces, Integral Equations Oper. Theory 6 (1983) 459–465. [34] F.-O. Speck, General Wiener–Hopf Factorization Methods, Pitman, London, 1985. [35] F.-O. Speck, Mixed boundary value problems of the type of Sommerfeld’s halfplane problem, Proc. Royal Soc. Edinburgh 104 A (1986), 261–277. [36] F.-O. Speck, On the reduction of linear systems related to boundary value problems, in: Oper. Theory Adv. Appl. 228, pp. 391–406, Birkh¨ auser, Basel, 2013. [37] F.-O. Speck, Diffraction from a three-quarter-plane using an abstract Babinet principle, Z. Angew. Math. Mech. 93 (2013), 485–491. [38] F.-O. Speck, Wiener–Hopf factorization through an intermediate space, Integral Equations Oper. Theory 82 (2015), 395–415. [39] F.-O. Speck, Paired operators in asymmetric space setting, in: Oper. Theory Adv. Appl. 259, pp. 681–702, Birkh¨ auser, Cham, 2017. [40] F.-O. Speck, A class of interface problems for the Helmholtz equation in Rn , Math. Meth. Appl. Sciences 40 (2017), 391–403. [41] D. Timotin, Schur coupling and related equivalence relations for operators on a Hilbert space, Linear Algebra Appl. 452 (2014), 106–119. [42] J. Wloka, Partial Differential Equations, University Press, Cambridge, 1987. Frank-Olme Speck Instituto Superior T´ecnico, Universidade de Lisboa Avenida Rovisco Pais, P 1049-001 Lisboa Portugal e-mail: [email protected]

Maximum determinant positive definite Toeplitz completions Stefan Sremac, Hugo J. Woerdeman and Henry Wolkowicz Dedicated to our friend Rien Kaashoek in celebration of his eightieth birthday.

Abstract. We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite replaced by positive semidefinite, and maximum determinant replaced by maximum rank. These results are used to determine the singularity degree of a family of semidefinite optimization problems. Mathematics Subject Classification (2010). 15A60, 15A83, 15B05, 90C22. Keywords. Matrix completion, Toeplitz matrix, positive definite completion, maximum determinant, singularity degree.

1. Introduction In this paper we study the positive definite completion of a partial symmetric Toeplitz matrix, T . The main contribution is Theorem 1.1, where we present a characterization of those Toeplitz patterns for which the maximum determinant completion is Toeplitz, whenever the partial matrix is positive definite completable. Part of this result answers a conjecture about the existence of a positive definite Toeplitz completion with a specific pattern. A consequence of the main result is an extension to the maximum rank completion in the positive semidefinite case, and an application to the singularity degree of a family of semidefinite programs (SDPs). In the following paragraphs we introduce relevant background information, state the main result, and motivate our pursuit. Research of S. Sremac and H. Wolkowicz supported by The Natural Sciences and Engineering Research Council of Canada. Research of H.J. Woerdeman supported by Simons Foundation grant 355645.

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_17

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A partial matrix is a matrix in which some of the entries are assigned values while others are unspecified, treated as variables. For instance, ⎤ ⎡ 6 1 x 1 1 ⎢ 1 6 1 y 1⎥ ⎥ ⎢ ⎥ (1.1) M := ⎢ ⎢u 1 6 1 z ⎥ ⎣ 1 v 1 6 1⎦ 1 1 w 1 6 is a real partial matrix, where the unspecified entries are indicated by letters. A completion of a partial matrix T is obtained by assigning values to the unspecified entries. In other words, a matrix T (completely specified) is a completion of T if it coincides with T over the specified entries: Tij = Tij , whenever Tij is specified. A matrix completion problem is to determine whether the partial matrix can be completed so as to satisfy a desired property. This type of problem has enjoyed considerable attention in the literature due to applications in numerous areas; see, e.g., [2, 28]. For example, matrix completion is used in sensor network localization [23, 24], where the property is that the completion is a Euclidean distance matrix with a given embedding dimension. Related references for matrix completion problems are, e.g., [1, 8, 14, 17, 18]. The pattern of a partial matrix is the set of specified entries. For example, the pattern of M is all of the elements in diagonals −4, −3, −1, 0, 1, 3, 4. Whether a partial matrix is completable to some property may depend on the values assigned to the specified entries (the data) and it may also depend on the pattern of specified entries. A question pursued throughout the literature is whether there exist patterns admitting certain completions whenever the data satisfy some assumptions. Consider, for instance, the property of positive definiteness. A necessary condition for a partial matrix to have a positive definite completion is that all completely specified principal submatrices are positive definite. We refer to such partial matrices as partially positive definite. Now we ask: what are the patterns for which a positive definite completion exists whenever a partial matrix having the pattern is partially positive definite? In [13] the set of such patterns is shown to be fully characterized by chordality of the graph determined by the pattern. In this work the desired property is symmetric Toeplitz positive definite. In particular, we consider the completion with maximum determinant over all (including non-Toeplitz) positive definite completions. Recall that a real symmetric n × n matrix T is Toeplitz if there exist real numbers t0 , . . . , tn−1 such that Tij = t|i−j| for all i, j ∈ {1, . . . , n}. A partial matrix is said to be partially symmetric Toeplitz if the specified entries are symmetric and consist of entire diagonals where the data is symmetric and constant over each diagonal. The pattern of such a matrix indicates which diagonals are known and hence is a subset of {0, . . . , n − 1}. Here 0 refers to the main diagonal, 1 refers to the superdiagonal and so on. The subdiagonals need not be specified in the pattern since they are implied by symmetry. In fact, since positive definite completions automatically exist when the main diagonal is

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not specified and the determinant is unbounded over the set of solutions, we will assume throughout that the main diagonal is specified. We therefore describe our patterns via subsets of {1, . . . , n − 1}. The pattern of M, for instance, is {1, 3, 4}. For a partial matrix T with pattern P and k ∈ P ∪ {0}, we let tk denote the value of T on diagonal k and we refer to {tk : k ∈ P ∪ {0}} as the data of T . For M the data is (t0 , t1 , t3 , t4 ) = (6, 1, 1, 1). We say that a partial Toeplitz matrix T is positive (semi)definite completable if there exists a positive (semi)definite completion of T . If T is positive definite completable, we denote by T  the unique positive definite completion of T that maximizes the determinant over all positive definite completions (for existence, see [13]). We now state the main contribution of this paper, a characterization of the Toeplitz patterns where the maximum determinant completion is itself Toeplitz, whenever the partial matrix is positive definite completable. Theorem 1.1. Let ∅ =  P ⊆ {1, . . . , n − 1} denote a pattern. The following are equivalent. 1. For every positive definite completable partial Toeplitz matrix T with pattern P , the matrix T  is Toeplitz. 2. There exist r, k ∈ N such that P has one of the three forms: • P1 := {k, 2k, . . . , rk}, • P2 := {k, 2k, . . . , (r − 2)k, rk}, where n = (r + 1)k, • P3 := {k, n − k}. The proof of Theorem 1.1 is presented in Section 2. Note that for the partial Toeplitz matrix M in (1.1), we can set all the unspecified entries to 1 and obtain a positive definite completion. However, the maximum determinant completion is given, to four decimal accuracy, when x = z = u = w = 0.3113 and y = v = 0.4247. But, this completion is not Toeplitz. Indeed, the pattern of M is not among the patterns of Theorem 1.1. Positive definite Toeplitz matrices play an important role throughout the mathematical sciences. Correlation matrices of data arising from time series, [26], and solutions to the trigonometric moment problem, [20], are two such examples. Among the early contributions to this area is the following sufficient condition and characterization, for a special case of pattern P1 . Theorem 1.2 ([10]). If T is a partially positive definite Toeplitz matrix with pattern P1 and k = 1, then T  exists and is Toeplitz. Theorem 1.3 ([20, Theorem 1.1]). Let P be a given pattern. Then all partially positive definite Toeplitz matrices with pattern P are positive definite Toeplitz completable if and only if P is of the form P1 . In these two results the assumption on the partial matrix is that it is partially positive definite, whereas in Theorem 1.1 we make the stronger assumption that a positive definite completion exists. As a consequence, our

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characterization includes the patterns P2 and P3 . In [27], sufficient and necessary conditions are provided for a partially Toeplitz matrix with pattern P2 and k = 1 to have a positive semidefinite completion. A special case of pattern P3 , with k = 1, is considered in [3], where the authors characterize the data for which the pattern is positive definite completable. In [15] the result is extended to arbitrary k and sufficient conditions for Toeplitz completions are provided. Moreover, the authors conjecture that whenever a partially positive definite Toeplitz matrix with pattern P3 is positive definite completable then it admits a Toeplitz completion. This conjecture is confirmed in Theorem 1.1 and more specifically in Theorem 3.4. Our motivation for the maximum determinant completion comes from optimization and the implications of the optimality conditions for completion problems (see Theorem 2.1). In particular, a positive definite completion problem may be formulated as an SDP. The central path of standard interior point methods used to solve SDPs consists of solutions to the maximum determinant problem. In the recent work [29] the maximum determinant problem is used to find feasible points of SDPs when the usual regularity conditions are not satisfied. A consequence of Theorem 1.1 is that when a partially Toeplitz matrix having one of the patterns of the theorem admits a positive semidefinite completion, but not a positive definite one, then it has a maximum rank positive semidefinite completion that is Toeplitz. This result, as well as further discussion on the positive semidefinite case, are presented in Section 3. The application to finding the singularity degree of a family of SDPs is presented in Section 4.

2. Proof of Main Result with Consequences To simplify the exposition, the proof of Theorem 1.1 is broken up into a series of results. Throughout this section we assume that every pattern P is a non-empty subset of {1, . . . , n − 1}, and T denotes an n × n partial symmetric Toeplitz matrix with pattern (or form) P . We begin by presenting the optimality conditions for the maximum determinant problem. Theorem 2.1. Let T be of the form P and positive definite completable. Then T  exists, is unique, and satisfies (T  )−1 / P. i,j = 0, whenever |i − j| ∈ Proof. This result is proved for general positive definite completions in [13]. See also [29].  For general positive definite completion problems, this result simply states that the inverse of the completion of maximum determinant has zeros in the unspecified (or free) entries. Since we are interested in Toeplitz completions, we may say something further using a permutation under which Toeplitz matrices are invariant. Let K be the symmetric n × n anti-diagonal matrix defined as: 1, if i + j = n + 1, (2.1) Kij := 0, otherwise,

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i.e., K is the permutation matrix that reverses the order of the sequence {1, 2, . . . , n}. Lemma 2.2. Let T be of the form P and positive definite completable. Let T  be the maximum determinant completion, and let K be the anti-diagonal permutation matrix in (2.1). Then the following hold. 1. T  = KT  K. 2. If P is of the form P2 with k = 1, i.e., P = {1, 2, . . . , n − 3, n − 1}, then T  is Toeplitz. Proof. For Item 1, it is a simple exercise to verify that the permutation reverses the order of the rows and columns and we have [KT  K]ij = T  n+1−i,n+1−j , ∀i, j ∈ {1, . . . , n}. Moreover, |n + 1 − i − (n + 1 − j)| = |i − j|. Therefore, it follows that [KT  K]ij = T  n+1−i,n+1−j = T  ij = t|i−j| , ∀|i − j| ∈ P ∪ {0}. Hence KT  K is a completion of T . Moreover, K · K is an automorphism of the cone of positive definite matrices. Hence KT  K is a positive definite completion of T , and since K is a permutation matrix, we conclude that det(KT  K) = det(T  ). By Theorem 2.1, T  is the unique maximizer of the determinant. Therefore T  = KT  K, as desired. For Item 2, we let T be as in the hypothesis and note that the only unspecified entries are (1, n−1) and (2, n), and their symmetric counterparts. Therefore it suffices to show that T  1,n−1 = T  2,n . By applying Item 1 we get T  1,n−1 = [KT  K]1,n−1 = T  n+1−1,n+1−(n−1) = T  n,2 = T  2,n , 

as desired.

The pattern {1, 2, . . . , n − 3, n − 1} in Lemma 2.2, above, is a special case of pattern P2 with k = 1. In fact, we show that a general pattern P2 may always be reduced to this special case. A further observation is that this specific pattern is nearly of the form P1 . Indeed, if the diagonal n − 2 were specified, the pattern would be of the form P1 . In fact, for any pattern of the form P2 , if the diagonal (r − 1)k were specified, the pattern would be of the form P1 . We now state a useful lemma for proving that Theorem 1.1, Item 2 implies Theorem 1.1, Item 1, when P is of the form P1 or P2 . Lemma 2.3. Let S be a partial n × n positive definite completable symmetric matrix and Q a permutation matrix of order n such that ⎡ ⎤ S1 ⎢ ⎥ S2 ⎢ ⎥ QT SQ = ⎢ ⎥, . .. ⎣ ⎦ S

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for some  ∈ N. Here each block Si is a partial symmetric matrix for i ∈ {1, . . . , }, and the elements outside of the blocks are all unspecified. Then the maximum determinant completion of Si , denoted Si , exists and is unique. Moreover, the unique maximum determinant completion of S is given by ⎡  ⎤ S1 0 · · · 0 ⎢ 0 S2 · · · 0 ⎥ ⎢ ⎥ T S = Q ⎢ . .. .. ⎥ Q . .. ⎣ .. . . . ⎦ 0 0 · · · S Proof. Since QT ·Q is an automorphism of the positive definite matrices, with inverse Q·QT , we have that QT SQ is positive definite completable and admits ˆ Moreover, under the map a unique maximum determinant completion, say S. Q · QT , every completion of QT SQ corresponds to a unique completion of S, with the same determinant, since the determinant is invariant under the ˆ T . Now we show that Sˆ transformation Q · QT . Therefore, we have S  = QSQ has the block diagonal form. Observe that Si is positive definite completable, take for instance the positive definite submatrices of Sˆ corresponding to the blocks Si . Thus Si is well defined, and by the determinant Fischer inequality, e.g., [19, Theorem 7.8.3], we have ⎡  ⎤ S1 0 · · · 0 ⎢ 0 S2 · · · 0 ⎥ ⎢ ⎥ ˆ S=⎢ . .. .. ⎥ , .. ⎣ .. . . . ⎦ 0 0 · · · S as desired.



In [20] it is shown that a partial Toeplitz matrix of the form P1 with rk = n − 1 can be permuted into a block diagonal matrix as in Lemma 2.3. We use this observation and extend it to all patterns of the form P1 , as well as patterns of the form P2 , in the following. Proposition 2.4. Let T be positive definite completable and of the form P1 or P2 . Then T  is Toeplitz. Proof. Let T be of the form P1 with data {t0 , tk , t2k , . . . , trk } and let p ≥ r be the largest integer so that pk ≤ n − 1. As in [20], there exists a permutation matrix Q of order n such that ⎡ ⎤ T0 ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ T 0 T ⎢ ⎥, Q TQ=⎢ ⎥ T 1 ⎢ ⎥ ⎢ ⎥ . .. ⎣ ⎦ T1

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where T0 is a (p+1)×(p+1) partial Toeplitz matrix occuring n−pk times and and T1 is a p×p partial Toeplitz. Moreover, T0 and T1 are both partially positive definite. Let us first consider the case p = r. Then T0 and T1 are actually fully specified, and the maximum determinant completion of QT T Q, as in Lemma 2.3, is obtained by fixing the elements outside of the blocks to 0. After permuting back to the original form, T  has zeros in every unspecified entry. Hence it is Toeplitz. Now suppose p > r. Then T0 is a partial Toeplitz matrix with pattern {1, 2, . . . , r} and data {t0 , tk , t2k , . . . , trk } and T1 is a partial Toeplitz matrix having the same pattern and data as T0 , but one dimension smaller. That is, T1 is a partial principal submatrix of T0 . By Theorem 1.2 both T0 and T1 are positive definite completable and their maximum determinant completions, T0 and T1 , are Toeplitz. Let {a(r+1)k , a(r+2)k , . . . , apk } be the data of T0 corresponding to the unspecified entries of T0 and let {b(r+1)k , b(r+2)k , . . . , b(p−1)k } be the data of T1 corresponding to the unspecified entries of T1 . By the permanence principle of [11], T1 is a principal submatrix of T0 and therefore bi = ai , for all i ∈ {(r +1)k, (r +2)k, . . . , (p−1)k}. By Lemma 2.3, the maximum determinant completion of QT T Q is obtained by completing T0 and T1 to T0 and T1 respectively, and setting the entries outside of the blocks to zero. After permuting back to the original form we get that T  is Toeplitz with data a(r+1)k , a(r+2)k , . . . , apk in the diagonals (r + 1)k, (r + 2)k, . . . , pk and zeros in all other unspecified diagonals. Now suppose that T is of the form P2 . By applying the same permutation as above, and by using the fact that n = (r + 1)k and each block T0 is of size r + 1, we see that the submatrix consisting only of blocks T0 is of size (n − rk)(r + 1) = ((r + 1)k − rk)(r + 1) = k(r + 1) = n. Hence,

⎡ ⎢ QT T Q = ⎣

⎤

T0 ..

.

⎥ ⎦,

T0 where T0 is a partial matrix with pattern {1, 2, . . . , r − 2, r} and data {t0 , tk , t2k , . . . , t(r−2)k , trk }. The unspecified elements of diagonal (r − 1)k of T are contained in the unspecified elements of diagonal r − 1 of the partial matrices T0 . By Lemma 2.2, the maximum determinant completion of T0 is Toeplitz with value t(r−1)k in the unspecified diagonal. As in the above, after completing QT T Q to its maximum determinant positive definite completion and permuting back to the original form, we obtain the maximum determinant Toeplitz completion of T with value t(r−1)k in the diagonal (r−1)k and zeros in every other unspecified diagonal, as desired.  We now turn our attention to patterns of the form P3 . Let J denote the n × n lower triangular Jordan block with eigenvalue 0. That is, J has ones on diagonal −1 and zeros everywhere else. We also let e1 , . . . , en ∈ Rn denote the columns of the identity matrix, i.e., the canonical unit vectors.

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n−1 T With this notation we have J = j=1 ej+1 ej . We state several technical results regarding J in the following lemma. Lemma 2.5. With J defined as above and k, l ∈ {0, 1, . . . , n}, the following hold. n−k 1. J k = j=1 ej+k eTj . 2. If l = n and k = 0, then J k (J T )l = 0, otherwise J k (J T )l has nonzero elements only in the diagonal l − k. 3. J k (J T )l − J n−l (J T )n−k = 0 if, and only if, l = n − k. Proof. For item 1 the result clearly holds when k ∈ {0, 1}. Now observe that for integers of suitable size (ek eTl )(ei eTj ) = 0 if, and only if, l = i in which case the product is ek eTj . Thus we have ⎛ ⎞⎛ ⎞ n−1 n−1 n−1 n−2     J2 = ⎝ ej+1 eTj ⎠ ⎝ ej+1 eTj ⎠ = (ej+1 eTj )(ej eTj−1 ) = ej+2 eTj . j=1

j=1

j=2

j=1

Applying an induction argument yields the desired expression for arbitrary k. For item 2, we use the result of item 1 to get ⎛ ⎞⎛ ⎞ n−k n−l   J k (J T )l = ⎝ ej+k eTj ⎠ ⎝ ej eTj+l ⎠ , j=1

j=1

n−max {k,l}

=



(ej+k eTj )(ej eTj+l ),

j=1 n−max {k,l}

=



ej+k eTj+l .

j=1

The nonzero elements of this matrix are contained in the diagonal j + l − (j + k) = l − k. Finally, for item 3 we have 

n−max{k,l}

J k (J T )l − J n−l (J T )n−k =

j=1



n−max{k,l}

ej+k eTj+l −

ej+n−l eTj+n−k .

j=1

This matrix is the zero matrix if, and only if, l = n − k.



We now state a special case of the Schur–Cohn Criterion using the matrix J. We let Sn denote the Euclidean space of symmetric matrices, Sn++ the cone of positive definite matrices, and SnT the subspace of symmetric Toeplitz matrices.

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Theorem 2.6 (Schur–Cohn Criterion, [22]). Let f (z) = a0 + a1 z + · · · + an z n be a polynomial with real coefficients. Let a := (a0 , . . . , an ) and A(a) :=

n−1  j=0

aj J j , B(a) :=

n−1 

 n−j aj J T .

j=1

Then every root of f (z) satisfies |z| > 1 if, and only if, Bez(a) := A(a)A(a)T − B(a)T B(a) ∈ Sn++ , where the matrix Bez(a) is the Toeplitz Bezoutian. Moreover, Bez(a)−1 is Toeplitz. The Schur–Cohn criterion is usually stated for the case where the roots are contained within the interior of the unit disk, but a simple reversal of the coefficients, as described in Chapter X of [25], leads to the above statement. For further information on the Toeplitz Bezoutian, and for a proof of the fact that Bez(a)−1 is Toeplitz, see e.g., [16]. We now present a result on the maximum determinant completion of partial Toeplitz matrices with pattern P3 . Proposition 2.7. Let T be positive definite completable and of the form P3 . Then T  is Toeplitz. Proof. Let T be as in the hypothesis with pattern P3 for some fixed integer k. Furthermore, let O ⊂ R++ × R2 consist of all triples (t0 , tk , tn−k ) so that the partial Toeplitz matrix with pattern P3 and data {t0 , tk , tn−k } is positive definite completable. Then it can be verified that O is an open convex set, and thus in particular connected. We let U ⊆ O consist of those triples (t0 , tk , tn−k ) for which the corresponding maximum determinant completion is Toeplitz and we claim that U = O. Clearly U =  ∅ as (tk , 0, 0) ∈ U for all tk > 0. We show that U is both open and closed in O, which together with the connectedness of O yields that U = O. First observe that the map F : O → Sn++ that takes (t0 , tk , tn−k ) to its corresponding positive definite maximum determinant completion is continuous. This follows from the classical maximum theorem of Berge, [4]. Next, the Toeplitz positive definite matrices, Sn++ ∩ SnT , form a closed subset of Sn++ since SnT is closed. Thus U = F −1 (Sn++ ∩ SnT ) is closed in O. To show that U is also open, we introduce the set P := {(p, q, r) ∈ R++ × R2 : p + qz k + rz n−k has all roots satisfy |z| > 1}. Since the region |z| > 1 is an open subset of the complex plane, P is an open set. We consider the map G : P → R3 defined as G(p, q, r) = ([Bez(p, q, r)−1 ]11 , [Bez(p, q, r)−1 ]k1 , [Bez(p, q, r)−1 ]n−k,1 ), where by abuse of notation Bez(p, q, r) is the Toeplitz Bezoutian of Theorem 2.6: Bez(p, q, r) = (pJ 0 + qJ k + rJ n−k )(pJ 0 + qJ k + rJ n−k )T − (rJ k + qJ n−k )(rJ k + qJ n−k )T .

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Then G is continuous and we show that its image is exactly U . By Theorem 2.6, for any (p, q, r) ∈ P we have Bez(p, q, r) ∈ Sn++ , Bez(p, q, r)−1 ∈ Sn++ ∩ SnT . Thus Bez(p, q, r)−1 is a completion of the partial matrix having pattern P3 and data {[Bez(p, q, r)−1 ]11 , [Bez(p, q, r)−1 ]k1 , [Bez(p, q, r)−1 ]n−k,1 }. It follows that G(P) ⊆ O. Moreover, expanding Bez(p, q, r) we obtain diagonal terms as well as terms of the form J 0 (J T )k and J 0 (J T )n−k , where the coefficients have been omitted. Applying item 2 of Lemma 2.5, Bez(p, q, r) has non-zero values only in entries of the diagonals 0, k, n − k. Note that the term J k (J T )n−k cancels out in the expansion. Thus by Theorem 2.1, Bez(p, q, r)−1 is a maximum determinant completion of the partial matrix with pattern P3 and data {[Bez(p, q, r)−1 ]11 , [Bez(p, q, r)−1 ]k1 , [Bez(p, q, r)−1 ]n−k,1 } and it follows that G(P) ⊆ U . To show equality, let (t0 , tk , tn−k ) ∈ U and let F (t0 , tk , tn−k ), as above, be the maximum determinant completion of the partial matrix with pattern P3 and data {t0 , tk , tn−k } which is Toeplitz. Let f0 , fk , and fn−k be the (1, 1), (k + 1, 1) and (n − k + 1, 1) elements of F (t0 , tk tn−k )−1 respectively. Then by the Gohberg–Semencul formula for the inversion of a symmetric Toeplitz matrix (see [12, 21]) we have 1 (f0 J 0 + fk J k + fn−k J n−k )(f0 J 0 + fk J k + fn−k J n−k )T f0 1 − (fn−k J k + fk J n−k )(fn−k J k + fk J n−k )T , f 0  = fk fn−k = Bez f0 , √ , √ . f0 f0 √  ∈ P and Since F (t0 , tk tn−k )−1 ∈ Sn++ , it follows that f0 , √ffk , f√n−k f0 0   = fk fn−k G f0 , √ , √ = (t0 , tk , tn−k ). f0 f0 F (t0 , tk tn−k )−1 =

Therefore G(P) = U . Moreover, from the above we have that   = fk fn−k −1 G (t0 , tk , tn−k ) = f0 , √ , √ , f0 f0 with f0 , fk , and fn−k defined above. Since G−1 is continuous, G−1 (U ) = P, and P is an open set, we conclude that U is open, as desired.  Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. The direction (2) =⇒ (1) follows from Proposition 2.4 and Proposition 2.7. For the direction (1) =⇒ (2), let s ∈ {1, 2, . . . , n − 1} and T be positive definite completable with pattern P = {k1 , . . . , ks }, k0 = 0, and data {t0 , t1 , . . . , ts }, where tj corresponds to diagonal kj , j ∈ {0, . . . , s}. Assume that T  is Toeplitz. Then by Theorem 2.1, (T  )−1 has nonzero entries only in the diagonals P ∪ {0} (and their symmetric counterparts). We denote by aj

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the value of the first column of (T  )−1 in the row kj + 1 for all j ∈ {0, . . . , s}, and define s s    n−kj A := aj J kj , B := aj J T . j=0

j=1

The Gohberg–Semencul formula gives us that (T  )−1 = a10 (AAT − B T B). Substituting in the expressions for A and B and expanding, we obtain that (T  )−1 is a linear combination of the following types of terms, along with their symmetric counterparts: J kj (J T )kj ,

J k0 (J T )kj ,

J kj (J T )kl − J n−kj (J T )n−kl , j = l.

By Lemma 2.5, the first type of term has nonzero entries only on the main diagonal, and the second type of term has nonzero entries only on the diagonals belonging to P . The third type of term has nonzero entries only on the diagonals ±|kj − kl |. As we have already observed in the proof of Proposition 2.7, the set of data for which T is positive definite completable is an open set. We may therefore perturb the data of T so that the entries a0 , . . . , aj of the inverse do not all lie on the same proper linear manifold. Then terms of the form J kj (J T )kl − J n−kj (J T )n−kl with j = l do not cancel each other out. We conclude that, for each pair j and l where j = l, we have |kj − kl | ∈ P or J kj (J T )kl − J n−kj (J T )n−kl = 0. By Lemma 2.5 the second alternative is equivalent to l = n − j. Using this observation we now proceed to show that P has one of the specified forms. Let 1 ≤ r ≤ s be the largest integer such that {k1 , . . . , kr } is of the form P1 , i.e., k2 = 2k1 , k3 = 3k1 , etc. If r = s, then we are done. Therefore we may assume s ≥ r + 1. Now we show that in fact s = r + 1. We have that kr+1 − k1 ∈ P or kr+1 = n − k1 . We show that the first case does not hold. Indeed if kr+1 − k1 ∈ P , then it follows that kr+1 − k1 ∈ {k1 , . . . , kr }. This implies that kr+1 ∈ {2k1 , . . . , rk1 , (r + 1)k1 } = {k2 , . . . , kr , (r + 1)k1 }. Clearly kr+1 ∈ / {k2 , . . . , kr }, and if kr+1 = (r + 1)k1 , then r is not maximal, a contradiction. Therefore kr+1 = n−k1 . To show that s = r+1, suppose to the contrary that s ≥ r + 2. Then kr+2 − k1 ∈ P or kr+2 = n − k1 . The latter does not hold since then kr+2 = kr+1 . Thus we have kr+2 −k1 ∈ {k1 , . . . , kr , kr+1 }, which implies that kr+2 ∈ {2k1 , . . . , rk1 , (r + 1)k1 , kr+1 + k1 } = {k2 , . . . , kr , kr + k1 , n}. Since kr+2 ∈ / {k2 , . . . , kr , n}, we have kr+2 = kr + k1 . Therefore, since kr < kr+1 < kr+2 , we have that 0 < kr+2 −kr+1 < k1 , and moreover, kr+2 −kr+1 ∈ / P . It follows that kr+2 = n − kr+1 = k1 , a contradiction. We have shown that P = {k1 , 2k1 , . . . , rk1 , ks } with ks = n − k1 . If r = 1, then P is of the form P3 . On the other hand if r ≥ 2, we observe that {ks − kr , . . . , ks − k2 } ⊆ P \ {k1 }, or equivalently, {ks − kr , . . . , ks − k2 } ⊆ {k2 , . . . , kr }.

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Since the above sets of distinct increasing elements have identical cardinality, we conclude that ks − k2 = kr . Rearranging, we obtain that ks = (r + 2)k1 and P is of the form P2 , as desired.  Remark 2.8. The results of this section have been stated for the symmetric real case for simplicity and for application to SDP in Section 3. With obvious modifications, our results extend to the Hermitian case.

3. Semidefinite Toeplitz Completions In this section we extend the results of Theorem 1.1 to positive semidefinite completions. In the case where all completions are singular, the maximum determinant is not useful for identifying a Toeplitz completion, however, a recent result of [29] allows us to extend our observations to the semidefinite case. Given a partial symmetric Toeplitz matrix, T , a positive semidefinite completion of T may be obtained by solving an SDP feasibility problem. Indeed, if T has pattern P and data {tk : k ∈ P ∪ {0}}, then the positive semidefinite completions of T are exactly the set F := {X ∈ Sn+ : A(X) = b},

(3.1)

where A is a linear map and b a real vector in the image space of A satisfying [A(X)]ik = Ei,i+k , X, bik = tk ,

i ∈ {1, 2, . . . , n − k}, k ∈ P ∪ {0}.

Here Ei,j is the symmetric matrix having a one in the entries (i, j) and (j, i) and zeros everywhere else and we use the trace inner product: X, Y  = tr(XY ). The maximum determinant is used extensively in SDP, for example, the central path of interior point methods is defined by solutions to the maximum determinant problem. If F is nonempty but does not contain a positive definite matrix, the maximum determinant may still be applied by perturbing F so that it does intersect the set of positive definite matrices. Consider the following parametric optimization problem X(α) := arg max {det(X) : X ∈ F(α)}, where F(α) := {X ∈ Sn+ : A(X) = b + αA(I)} and α > 0. For each α > 0, the solution X(α) is contained in the relative interior of F(α). It is somewhat intuitive that if the limit of these solutions is taken as α decreases to 0, we should obtain an element of the relative interior of F(0) = F. Indeed, the following result confirms this intuition. We denote by A∗ the adjoint of A. ¯ in Theorem 3.1. Let F =  ∅ and X(α) be as above. Then there exists X ¯ Moreover, Z¯ := the relative interior of F such that limα0 X(α) = X. ¯ Z¯ = 0 and Z¯ ∈ range(A∗ ). limα0 α(X(α))−1 exists and satisfies X Proof. See Section 3 of [29]. An immediate consequence of this result is the following.



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Corollary 3.2. Let T be an n × n partial symmetric Toeplitz matrix of the form P1 , P2 , or P3 . If T admits a positive semidefinite completion, then it admits a maximum rank positive semidefinite completion that is Toeplitz. Proof. Let T be as in the hypothesis with data {t0 , t1 , . . . , ts } and let F be the set of positive semidefinite completions, as above. If F ∩ Sn++ = ∅, then the maximum determinant completion is Toeplitz by Theorem 1.1 and is of maximum rank. Now suppose F ⊆ Sn+ \Sn++ and observe that for every α > 0, F(α) consists of solutions to to the completion problem having pattern P1 , P2 , or P3 with data {t0 + α, t1 , . . . , ts } and there exists a positive definite completion. Thus X(α) is Toeplitz for each α > 0 and since the Toeplitz ¯ of Theorem 3.1, is Toeplitz. The matrices are closed, the limit point X, relative interior of F corresponds to those matrices having maximum rank ¯ has maximum rank, as desired. over all of F, hence X  Remark 3.3. In Theorem 2.2 of [27] the author gives in the case of two prescribed diagonals (in the strict lower triangular part) necessary and sufficient conditions on the data for the existence of a Toeplitz positive semidefinite completion. In Theorem 10 of [15] the authors give in the case of pattern P3 necessary and sufficient conditions for the existence of a positive semidefinite completion. If one is able to verify that the conditions are the same, which will require some tenacity, then one would have an alternative proof that for the pattern P3 positive semidefinite completability implies the existence of a Toeplitz positive semidefinite completion. Their results are all stated for the real case, so one advantage of the approach here is that it readily generalizes to the complex Hermitian case. While Theorem 1.1 characterizes patterns for which the maximum determinant completion is automatically Toeplitz and Corollary 3.2 addresses the maximum rank completions, one may merely be interested in the existence of a Toeplitz completion when a positive semidefinite one exists. Obviously, the patterns in Theorem 1.1 fall in this category, but as we see in the following result, there are more. Theorem 3.4. Let n, k, r ∈ N. Define the patterns • P2 := {k, 2k, . . . , (r − 2)k, rk}, where n ≥ rk + 1, • P3 := {k, r}, where n ≥ k + r. If T is an n × n positive semidefinite completable partial Toeplitz matrix with a pattern in the set {P1 , P2 , P3 }, then T has a Toeplitz positive semidefinite completion. Proof. For pattern P1 this is a consequence of Corollary 3.2. Note that P2 and P3 are obtained from P2 and P3 , respectively, by relaxing the restriction on n. Using the results we already have for P2 and P3 we fill in some of the diagonals of T to obtain a new partial matrix of the form P1 . The details now follow. Suppose T has pattern P2 and consider the partial submatrix contained in rows and columns {1, k + 1, 2k + 1, . . . , rk + 1}. This partial matrix is

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Toeplitz, has a positive semidefinite completion and has pattern {1, 2, . . . , r − 2, r}. By Corollary 3.2, there exists a positive semidefinite Toeplitz completion for this submatrix, with the value a, say, in diagonal r−1. We can now assign the value a to diagonal (r − 1)k in the original partial matrix. The resulting partial matrix has now pattern {k, 2k, . . . , (r − 2)k, (r − 1)k, rk} (thus of type P1 ), and consequently has a Toeplitz positive semidefinite completion. When T has pattern P3 , we take the submatrix in rows and columns {1, 2, . . . , k + r}. This submatrix has pattern P3 , and thus by Corollary 3.2, there exists a positive semidefinite Toeplitz completion for this submatrix. We can use the data of this submatrix to fill in diagonals 1, 2, . . . , k + r − 1 of the original partial matrix. Now the partial matrix is of type P1 , and thus a positive semidefinite Toeplitz completion exists.  Whether or not Theorem 3.4 gives a full characterization of patterns for which the existence of a positive semidefinite completion implies that there is a Toeplitz positive semidefinite completion, is an open question.

4. The Singularity Degree of Some Toeplitz Cycles The Slater condition holds for the feasible set of an SDP if it contains a positive definite matrix. If the Slater condition does not hold for an SDP, there may be a duality gap and the dual may be unattained. Consequently, it may not be possible to verify whether a given matrix is optimal or not. One way to regularize an SDP that does not satisfy the Slater condition is by restricting the problem to the smallest face of Sn+ containing the feasible set. Since every face of Sn+ is a smaller dimensional positive semidefinite cone, every SDP may be transformed into an equivalent (possibly smaller dimensional) SDP for which the Slater condition holds. This transformation is referred to as facial reduction, see for instance [5, 6, 9]. The challenge, of course, is to obtain the smallest face. Most facial reduction algorithms look for exposing vectors, i.e., non-zero, positive semidefinite matrices that are orthogonal to the minimal face. Exposing vectors are guaranteed to exist by the following theorem of the alternative. Here we let F be the feasible set of an SDP that is defined by the affine equation A(X) = b, as in (3.1). Theorem 4.1 ( [7]). Suppose F = ∅. Then exactly one of the following holds. 1. F ∩ Sn++ = ∅. 2. There exists Z ∈ Sn+ ∩ range(A∗ ) such that ZX = 0 for all X ∈ F. This result guarantees the existence of exposing vectors when the Slater condition does not hold. By restricting the feasible set of an SDP to the kernel of an exposing vector, the dimension of the SDP is reduced. By repeatedly finding exposing vectors and reducing the size of the SDP, eventually the problem is reduced to the minimal face and the Slater condition holds. If the exposing vector obtained at each iteration is as in item 2 of Theorem 4.1 and of maximal rank over all such exposing vectors, then the number of times the original SDP needs to be reduced in order to obtain a regularized SDP is

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referred to as the singularity degree. This notion and the connection to error bounds for SDP is introduced in [30, Sect. 4]. For instance, if an SDP satisfies the Slater condition, then it has singularity degree 0 and the singularity degree is 1 if and only if there exists an exposing vector Z ∈ Sn+ ∩ range(A∗ ) such that rank(Z) + rank(X) = n for all X in the relative interior of F. In [31, Lemma 3.4] it is shown that for n ≥ 4, there exists a partial matrix (not Toeplitz) with all entries of the diagonals 0, 1, n − 1 specified so that the singularity degree of the corresponding SDP is at least 2. Here we apply the results of the previous sections to derive the singularity degree (or bounds for it) of a family of symmetric partial Toeplitz matrices with pattern P = {1, n − 1}. As in much of the matrix completion literature the partial matrix is viewed as arising from a graph and the pattern P corresponds to the graph of a cycle with loops. We record the following result mainly for reference as it is just an application of the Gohberg–Semencul formula for the inverse of a positive definite Toeplitz matrix. Proposition 4.2. Let T = (ti−j )ni,j=1 be a positive definite Toeplitz matrix, and suppose that (T −1 )k,1 = 0 for all k ∈ {3, . . . , n − 1}. Then T −1 has the form ⎤ ⎡ a c 0 d ⎥ ⎢ .. ⎥ ⎢c b . c ⎥ ⎢ ⎥ ⎢ .. (4.1) ⎥, ⎢0 c . b 0 ⎥ ⎢ ⎥ ⎢ .. .. .. ⎣ . c⎦ . . d 0 c a with b = a1 (a2 + c2 − d2 ).

 Proof. Let us denote the first column of T −1 by a c 0 the Gohberg–Semencul formula we have that 1 T −1 = (AAT − B T B), a where ⎡ ⎡ ⎤ 0 d 0 a 0 0 0 ⎢ ⎢ ⎥ . .. ⎢0 0 ⎢c a ⎥ d 0 ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ . . . 0 ⎥ , B = ⎢0 0 A = ⎢0 c 0 a ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ . . . . . . .. . . . ⎣ ⎣ . . . . 0⎦ 0 0 d 0 c a

···

0

c

..

.

..

.

..

. 0

d

T

. By

⎤

⎥ ⎥ ⎥ ⎥ . 0⎥ ⎥ ⎥ d⎦ 0



Example 4.3. Let n = 4 and consider the partial matrix with pattern P = {1, 3} and data {t0 , t1 , t3 } = {1 + α, cos(θ), cos(3θ)} for θ ∈ [0, π3 ] and α ≥ 0. Let F(α) denote the set of positive semidefinite completions for each α > 0 as in Section 3, let F = F(0), and let sd(F) denote the singularity degree of any SDP for which F is the feasible set. By Corollary 6 of [3] there exists a positive definite completion whenever α > 0 and there exists a positive

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semidefinite completion (but not a positive definite one) when α = 0. Then by Theorem 1.1 the maximum determinant completion is Toeplitz whenever α > 0 and there exists a maximum rank positive semidefinite completion that is Toeplitz when α = 0 by Corollary 3.2. Let X(α) denote the maximum determinant positive definite completion when α > 0. Then ⎛ ⎞ 1 + α cos(θ) x(α) cos(3θ) ⎜ cos(θ) 1 + α cos(θ) x(α) ⎟ ⎟. X(α) =: ⎜ ⎝ x(α) cos(θ) 1 + α cos(θ) ⎠ cos(3θ) x(α) cos(θ) 1 + α Here x(α) denotes the value of the unspecified entry. By considering det(X(α)) as a function of x(α) we derive, through the optimality condition, that  1 = α(α + 2) + (4 cos2 (θ) − 1)2 − (1 + α) . x(α) = 2 Indeed, the optimality condition is that the determinant of the matrix obtained by removing from X(α) row 1 and column 3 equals 0 (it subsequently helps to note that 1 + α is one of the roots of the so obtained equation). Taking the limit as α decreases to 0, we get ⎛ ⎞ 1 cos(θ) cos(2θ) cos(3θ) ⎜ 1 cos(θ) cos(2θ)⎟ ¯ := lim X(α) = ⎜ cos(θ) ⎟. X ⎝ cos(2θ) cos(θ) 1 cos(θ) ⎠ α0 cos(3θ) cos(2θ) cos(θ) 1 This matrix has maximum rank over all positive semidefinite completions ¯ has rank 2 whenever θ ∈ when α = 0 due to Corollary 3.2. Specifically, X π (0, 3 ] and rank 1 when θ = 0. To derive the singularity degree of F we need to find the maximal rank of an exposing vector having the properties of Theorem 4.1. To this end let Z(α) := αX(α)−1 and let Z¯ = limα0 Z(α). By Theorem 3.1, Z¯ exists and is an exposing vector for F (as long as it is not the zero matrix) as in Theorem 4.1. By Proposition 4.2 we have ⎛ ⎞ a(α) c(α) 0 d(α) ⎜ c(α) b(α) c(α) 0 ⎟ ⎟, (4.2) Z(α) =: ⎜ ⎝ 0 c(α) b(α) c(α) ⎠ d(α) 0 c(α) a(α) 1 where b(α) = a(α) (a(α)2 +c(α)2 −d(α)2 ). Let a, b, c, and d be the limit points of a(α), b(α), c(α), and d(α), respectively, as α decreases to 0. Then ⎛ ⎞ a c 0 d ⎜ c b c 0⎟ ⎟ Z¯ = ⎜ ⎝0 c b c ⎠ . d 0 c a

¯ ≥ 2 and if b = 0 then rank(Z) ¯ ≤ 1. We observe that if b = 0, then rank(Z) The first observation follows by considering the second and third columns. For

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the second observation, suppose Z¯ = 0 then since Z¯ is positive semidefinite we have a > 0 and c = 0. From the equation b = a1 (a2 + c2 − d2 ) we get 1 2 (a − d2 ), a ¯ = 1. which implies that a = ±d and rank(Z) Now we determine the value of b for θ ∈ [0, π3 ]. Since X(α)Z(α) = αI, we have ⎡ ⎤ ⎡ ⎤ 0 c(α) ⎢ 1⎥ ⎢b(α)⎥ ⎢ ⎥ ⎥ X(α) ⎢ ⎣c(α)⎦ = α ⎣0⎦ . 0 0 0=

Solving for b(α) we obtain the expression b(α) =

α(cos(θ)+cos(3θ)) (1+α)(cos(θ)+cos(3θ))−2x(α) cos(θ) , α(1+α+x(α)) (1+α)(1+α+x(α))−2 cos2 (3θ) ,

θ ∈ [0, π3 ], θ = θ=

π 4,

π 4.

Evaluating the limits we get that b = 0 if θ = π3 and b is non-zero for all ¯ ≥ 2 when θ ∈ (0, π ) and other values of θ in [0, π3 ]. It follows that rank(Z) 3 ¯ = 2 for these values of θ, we conclude that sd(F) = 1 when since rank(X) θ ∈ (0, π3 ). When θ = 0 it can be derived that a = b = 34 and c = d = − 38 . Then Z¯ is a rank 3 matrix and sd(F) = 1. For the case θ = π3 we have that ¯ ≤ 1 and now we show that every exposing vector for F that lies in rank(Z) Sn+ ∩ range(A∗ ) has rank at most 1. Indeed, for θ = π3 we have ⎛ ⎞ 1 1 − 12 −1 2 1 ⎜ 1 1 − 12 ⎟ 2 ¯ = ⎜ 21 ⎟ X 1 1 ⎠. ⎝− 1 2 2 2 1 −1 − 12 1 2 ¯ is formed by the vectors Now a basis for the kernel of X ⎡ ⎤ ⎡ ⎤ 1 1 ⎢−1⎥ ⎢ 0⎥ ⎢ ⎥ ⎥ v := ⎢ ⎣0⎦ , u := ⎣ 1 ⎦ . 0 1 Observe that range(A∗ ) consists of all the matrices with entries (1, 3) and ¯ = 0 and (2, 4) identically 0. Now if Z is an exposing vector for F, we have XZ T T T hence Z = λ(vv ) + μ(uu ) for some λ, μ ∈ R. But since uu ∈ / range(A∗ ) T ∗ and vv ∈ range(A ), it follows that μ = 0, rank(Z) ≤ 1 and sd(F) ≥ 2. ¯ = 2 and facial reduction guarantees at least a one In fact since rank(X) dimensional reduction at each iteration, sd(F) = 2. We conclude this example by summarizing our observations: sd(F) =

1, 2,

θ ∈ [0, π3 ), θ = π3 .

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Some of the observations of this example extend to general n ≥ 5. First we show that the partial matrix admits a unique positive semidefinite completion, which is Toeplitz. Proposition 4.4. Consider the partial symmetric n × n Toeplitz matrix with pattern P = {1, n − 1} and data {t0 , t1 , tn−1 } = {1, cos θ, cos((n − 1)θ)}, where θ ≤

π n−1 .

Then the unique positive semidefinite completion is n−1

(cos ((i − j)θ))i,j=0 = B T B, where

 B=

1 0

cos θ sin θ

cos(2θ) sin(2θ)

··· ···

 cos((n − 1)θ) . sin((n − 1)θ)

Proof. Let us denote the first column of a positive semidefinite completion  T by cos θ0 cos θ1 cos θ2 · · · cos θn−1 , where θ0 = 0, θ1 = θ, θn−1 = (n − 1)θ and θ2 , . . . , θn−2 ∈ [0, π]. If we look at the principal submatrix in rows and columns 1, n − 1 and n, we get the positive semidefinite matrix ⎛ ⎞ 1 cos(θn−2 ) cos((n − 1)θ) ⎠. ⎝ cos(θn−2 ) 1 cos(θ) cos((n − 1)θ) cos(3θ) 1 By [3, Proposition 2] we have that (n − 1)θ ≤ θn−2 + θ. Thus θn−2 ≥ (n − 2)θ.

(4.3)

Next, consider the (n − 1) × (n − 1) upper left corner with data {t0 , t1 , tn−2 } = {1, cos θ, cos θn−2 }. By [3, Corollary 2] we have that 2 max{θn−2 , θ} ≤ (n − 2)θ + θn−2 .

(4.4)

θn−2 ≤ (n − 2)θ.

(4.5)

This implies that Combining (4.5) with (4.3) we have θn−2 = (n − 2)θ. If instead we look at the principal submatrix in rows and columns 1,2, and n and combine it with the (n − 1) × (n − 1) lower right block, we obtain that the value in position (n, 2) is also cos((n − 2)θ). Thus along the (n − 2)th diagonal the value is cos((n − 2)θ). One can repeat this argument for other submatrices (or invoke induction) and obtain that the unique completion is constant along the kth diagonal with value cos(kθ) for k = 2, . . . , n − 2.  Now we extend the observations of Example 4.3 to general n ≥ 5, for π . the special case θ = n−1

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Example 4.5. For n ≥ 5 consider the n × n symmetric partial Toeplitz matrix π with pattern P = {1, n−1} and data {t0 , t1 , tn−1 } = {1+α, cos( n−1 ), cos(π)}, where α > 0. As in Example 4.3, we let F denote the set of positive semidefinite completions when α = 0 and we let X(α) denote the maximum determinant completion when α > 0. By Proposition 4.4, F consists of the rank 2 matrix   n−1 (i − j)π ¯ X := cos , n−1 i,j=0 ¯ = limα0 X(α). If Z(α) = αX(α)−1 and Z¯ is and by Theorem 3.1, X ¯ ≤ 1. Let the limit of Z(α) as α decreases to 0, we show that rank(Z) a(α), b(α), c(α), and d(α) be the elements of Z(α) as in Proposition 4.2 and (4.2) and let a, b, c, and d be the limit points of a(α), b(α), c(α), and d(α), respectively. We claim that if a = 0 then Z¯ = 0. Indeed, by the fact that Z¯ is positive semidefinite we have c = d = 0. Moreover, ¯ Z) ¯ = (n − 2)b, 0 = tr(X which implies that b = 0 and consequently Z¯ = 0. Thus we may assume a > 0 and the equation 1 b = (a2 + c2 − d2 ), a ¯ Z¯ = 0 and the above equation, we obtain holds. From X     π π 2 cos c + b = 0, a + cos c − d = 0. n−1 n−1 π )), d = a − 2b , and thus This gives c = −b/(2 cos( n−1  2 1 b2 1 b b=a+ − . a − π a 4 cos2 ( n−1 ) a 2

After rearranging, we obtain " # b2 1 1 − = 0. π a 4 4 cos2 ( n−1 ) ¯ ≤ 1 follows. We have thus shown that Consequently b = 0, and rank(Z) sd(F) ≥ 2 for all n ≥ 4. ¯ 11 = Numerical experiments suggest that Z¯ is the rank 1 matrix with (Z) ¯ 1n = (Z) ¯ n1 = n−1 and all other entries equal to 0. ¯ nn = (Z) (Z) 4

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[21] T. Kailath and J. Chun, Generalized Gohberg-Semencul formulas for matrix inversion, in: The Gohberg anniversary collection, Vol. I (Calgary, AB, 1988), vol. 40 of Oper. Theory Adv. Appl., pp. 231–246, Birkh¨ auser, Basel, 1989. [22] M.G. Kre˘ın and M.A. Na˘ımark, The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations, Linear and Multilinear Algebra 10 no. 4 (1981), 265–308. Translated from the Russian by O. Boshko and J.L. Howland. [23] N. Krislock and H. Wolkowicz, Explicit sensor network localization using semidefinite representations and facial reductions, SIAM Journal on Optimization 20 no. 5 (2010), 2679–2708. [24] M. Laurent, A connection between positive semidefinite and Euclidean distance matrix completion problems, Linear Algebra Appl. 273 (1998), 9–22. [25] M. Marden, Geometry of polynomials, 2nd ed., vol. 3 of Mathematical Surveys, American Mathematical Society, Providence, R.I., 1966. [26] B.N. Mukherjee and S.S. Maiti, On some properties of positive definite Toeplitz matrices and their possible applications, Linear Algebra Appl. 102 (1988), 211– 240. [27] G. Naevdal, On a generalization of the trigonometric moment problem, Linear Algebra Appl. 258 (1997), 1–18. [28] B. Recht, M. Fazel, and P. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev. 52 no. 3 (2010), 471–501. [29] S. Sremac, H.J. Woerdeman, and H. Wolkowicz, Complete facial reduction in one step for spectrahedra, Technical report, 34 p., University of Waterloo, Waterloo, Ontario, 2017; also available as arXiv:1710.07410. [30] J.F. Sturm, Error bounds for linear matrix inequalities, SIAM J. Optim. 10 no. 4 (2000), 1228–1248 (electronic). [31] S. Tanigawa, Singularity degree of the positive semidefinite matrix completion problem, Technical Report, Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan, 2016; also available as arXiv:1603.09586 Stefan Sremac and Henry Wolkowicz Department of Combinatorics and Optimization, Faculty of Mathematics University of Waterloo 200 University Ave. W, Waterloo, Ontario N2L 3G1 Canada e-mail: [email protected] [email protected] Hugo J. Woerdeman Department of Mathematics, Drexel University 3141 Chestnut Street, Philadelphia, PA 19104 USA e-mail: [email protected]

On commutative C ∗-algebras generated by Toeplitz operators with Tm-invariant symbols Nikolai Vasilevski Dedicated to Rien Kaashoek, a remarkable person and mathematician, on the occasion of his 80 th birthday.

Abstract. It is known that Toeplitz operators, whose symbols are invariant under the action a maximal Abelian subgroups of biholomorphisms of the unit ball Bn , generate the C ∗ -algebra being commutative in each standardly weighted Bergman space. In case of the unit disk (n = 1) this condition on generating symbols is also necessary in order that the corresponding C ∗ -algebra be commutative. In this paper, for n > 1, we describe a wide class of symbols that are not invariant under the action of any maximal Abelian subgroup of biholomorphisms of the unit ball, and which, nevertheless, generate via corresponding Toeplitz operators C ∗ -algebras being commutative in each standardly weighted Bergman space. These classes of symbols are certain proper subsets of functions that are invariant under the action of the group Tm , with m ≤ n, being a subgroup of the maximal Abelian subgroup Tn of biholomorphisms of Bn . Mathematics Subject Classification (2010). Primary 47B35; Secondary 47L80, 32A36. Keywords. Toeplitz operators, Bergman space, commutative C ∗ -algebras.

1. Introduction One of the fundamental tasks in the theory of Toeplitz operators is the characterization of the algebras generated by such operators with symbols possessing some specific properties. A first natural class of symbols, the continuous ones, was treated in early seventies of the past century by L. Coburn [3] and This work was partially supported by CONACYT Project 238630, M´ exico.

© Springer Nature Switzerland AG 2018 H. Bart et al. (eds.), Operator Theory, Analysis and the State Space Approach, Operator Theory: Advances and Applications 271, https://doi.org/10.1007/978-3-030-04269-1_18

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U. Venugopalkrishna [11]. It was shown that such Toeplitz operators commute modulo compact operators, and the structure of C ∗ -algebra generated by these Toeplitz operators as well as the corresponding commutative Calkin algebra (the quotient of the algebra by the ideal of compact operators) was completely characterized. Twenty-three years later a just commutative C ∗ -algebra generated by Toeplitz operators acting on the Bergman space over the unit disk was, in fact, described in [5]. B. Korenblum and K. Zhu showed there that Toeplitz operators with radial symbols are diagonal with respect to the standard monomial basis of the Bergman space and thus commute among each other. In a next few years it was unexpectedly discovered that, contrary to the case of Toeplitz operators acting on the Hardy space, there is a wide family of commutative C ∗ -algebras generated by Toeplitz operators with certain classes of symbols. Moreover, these algebras remain commutative in each standardly weighted Bergman space on the unit disk. Surprisingly the symbols of Toeplitz operators that generate commutative C ∗ -algebras are of a geometric nature: they are constant on cycles of pencils of hyperbolic geodesics, or, equivalently, they are invariant under the action of maximal Abelian subgroups of M¨obius transformations of the unit disk (see, for example, [8] for more details). Each such commutative C ∗ -algebra is diagonalizable in a sense that each its operator is unitary equivalent to the multiplication operator by the spectral sequence or spectral function of this operator. This result generated a vast literature devoted to the description of commutative algebras generated by Toeplitz operators, as well as, to the use of such algebras in different questions related to the Toeplitz operator theory. We mention the result of [4] giving a necessary and sufficient condition for symbols that generate via Toeplitz operators a C ∗ -algebra being commutative in each standardly weighted Bergman space on the unit disk. Under some technical condition the result says: a C ∗ -algebra generated by Toeplitz operators is commutative in each standardly weighted Bergman space if and only if the symbols of corresponding Toeplitz operators are of the above geometric nature. Papers [6, 7] develop ideas of [8] for the unit ball Bn case. For the n-dimensional ball Bn there are five different model maximal Abelian subgroups of biholomorphisms of the unit ball: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent, and quasi-nilpotent, giving in total n + 2 subgroups, as the last one depends on a parameter. Each other maximal Abelian subgroup is conjugate to one of the above model cases. As in the case of the unit disk [8], Toeplitz operators, whose symbols are invariant under the action of one of the above maximal Abelian subgroups of biholomorphisms of the unit ball, generate a C ∗ -algebra being commutative in each standardly weighted Bergman space A2λ (Bn ), λ ∈ (−1, ∞). Note that this condition on symbols is only a sufficient one. Further there is a unitary operator (different for different commutative C ∗ -algebras) that diagonalizes each operator from the algebra in question, that is, giving an unitary equivalence with the

On commutative C ∗ -algebras generated by Toeplitz operators

445

multiplication operator by its spectral function or spectral sequence. Note that this diagonalization gives us an immediate characterization of the majority of the main properties of the corresponding operators: boundedness (in case of unbounded symbols), compactness, spectrum, invertibility, common invariant subspaces, etc. Another surprising feature of the multidimensional case, not possible in one-dimensional case of the unit disk, is the existence of Banach (not C ∗ ) algebras generated by Toeplitz operators that are commutative in each standardly weighted Bergman space A2λ (Bn ). We will not discuss in detail this here, and just mention that they were firstly characterized in [9]. After that many papers were published extending the classes of symbols that generate via Toeplitz operators commutative Banach algebras. Note that Corollary 4.4 of this paper gives a further extension of such classes of symbols. The main concern of the paper is the C ∗ -algebras. As it was already mentioned, the results of [6, 7] give just a sufficient condition on symbols (invariance under the action of a maximal Abelian subgroup of biholomorphisms of the unit ball), so that the corresponding Toeplitz operators generate a commutative C ∗ -algebra. At the same time the result of [4] gives an if and only if condition. It turns out that the condition of [6, 7] is indeed only sufficient. And this is exactly what this paper is about: we describe a wide class of symbols that are not invariant under the action of any maximal Abelian subgroup of biholomorphisms of the unit ball, and which, at the same time, generate via corresponding Toeplitz operators C ∗ -algebras being commutative in each weighted Bergman space. These classes of symbols are certain proper subsets of functions that are invariant under the action of the group Tm , with m ≤ n, being a subgroup of the quasi-elliptic group Tn of biholomorphisms of the unit ball Bn . Note that the set of all Tm -invariant symbols produces a noncommutative C ∗ -algebra. What is important and interesting here is that these new commutative C ∗ -algebras possess the same diagonalization property: each operator from these algebras is diagonal with respect to some orthonormal basis of the weighted Bergman space A2λ (Bn ). But now this basis is not anymore the canonical monomial basis of A2λ (Bn ), in case of m < n it varies from algebra to algebra. If m = n, then Tm = Tn and we return back to the quasi-elliptic case of [6]. The paper is organized as follows. Section 2 contains preliminary material. In Section 3 we introduce our class of Tm -invariant symbols g of the form (3.2), show that each subspace Hκ in the orthogonal sum decomposition (3.1) of the Bergman space is invariant for Toeplitz operators Tg with the above symbols g, and calculate the scalar products Tg z α , z β  for elements z α , z β from the same invariant subspace. In Section 4 we characterize the action of Toeplitz operators Tg on each invariant subspace Hκ as well as the action of Toeplitz operators with symbols being the different multiples in the product

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decomposition (3.2) of g. Further we show (Theorem 4.3) that Toeplitz operators with these multiple-symbols commute among each other and give in their product the Toeplitz operator Tg . Corollary 4.4 describes then a new class of commutative Banach algebras generated by Toeplitz operators with our Tm -invariant symbols. Section 5 is devoted to the study of commutative C ∗ -algebras. Its main result, finalized in Theorem 5.6, describes a new noncanonical orthonormal basis in A2λ (Bn ), defined by the Toeplitz operators that generate the algebra in question, and shows that each element of this commutative C ∗ -algebra is diagonal with respect to the constructed basis. To do this we use a trick involving Toeplitz operators with Tm -invariant symbols acting on the classical Fock space. As a by-product, in the final Section 6 we characterize new commutative C ∗ -algebras generated by Toeplitz operators with quasi-radial and Tm -invariant symbols acting on the Fock space.

2. Preliminaries Let Bn be the unit ball in Cn , Bn = {z = (z1 , . . . , zn ) ∈ Cn : |z|2 = |z1 |2 + · · · + |zn |2 < 1}, and let S 2n−1 be the corresponding (real) unit sphere, the boundary of the unit ball Bn . In what follows we will use the notation τ (Bm ) for the base of the unit ball Bm , considered as a Reinhard domain, i.e., 2 τ (Bm ) = {(r1 , . . . , rm ) = (|z1 |, . . . , |zm |) : r2 = r12 + · · · + rm ∈ [0, 1)}.

We denote as well by Bm the real m-dimensional unit ball, Bm = {x = (x1 , . . . , xm ) ∈ Rm : x2 = x21 + · · · + x2m < 1}. m Then, of course, τ (Bm ) = Bm ∩ Rm + =: B+ . Given a multi-index α = (α1 , α2 , . . . , αn ) ∈ Zn+ , we will use the standard notation,

α! = α1 ! α2 ! · · · αn !, z α = z1α1 z2α2 · · · znαn , for p = (p1 , p2 , . . . , pn ) ∈ Zn we set |p| = p1 + p2 + · · · + pn . Denote by dV = dx1 dy1 . . . dxn dyn , where zl = xl + iyl , l = 1, 2, . . . , n, the standard Lebesgue measure in Cn ; and let dS be the corresponding surface measure on S 2n−1 . We introduce the standard one-parameter family of weighted measures, dvλ (z) =

Γ(n + λ + 1) (1 − |z|2 )λ dV (z), π n Γ(λ + 1)

λ > −1,

which are probability ones in Bn , the weighted space L2 (Bn , dvλ ), and its subspace, the weighted Bergman space A2λ (Bn ), which consists of all functions analytic in Bn .

On commutative C ∗ -algebras generated by Toeplitz operators

447

Recall (see, for example, [12, Section 1.3]) that the standard monomial orthonormal basis in A2λ (Bn ) is given by N Γ(n + |α| + λ + 1) α eα = eα (z) = z , α ∈ Zn+ . (2.1) α! Γ(n + λ + 1) The (orthogonal) Bergman projection Bλ of L2 (Bn , dvλ ) onto A2λ (Bn ) is given by  ϕ(ζ) dvλ (ζ) (Bλ ϕ)(z) = . n (1 − z · ζ)n+λ+1 B Finally, given a function a(z) ∈ L∞ (Bn ), the Toeplitz operator Ta with symbol a acts on A2λ (Bn ) as follows: Ta : ϕ ∈ A2λ (Bn ) −→ Bλ (aϕ) ∈ A2λ (Bn ). Recall as well (see, for example, [1]) that the Beta function of n variables is defined by ⎛ ⎞⎛ ⎞xm −1  m−1 m−1 M x −1  ⎝ B(x1 , . . . , xm ) = yj j ⎠ ⎝1 − yj ⎠ dy1 · · · dym−1 , Δm−1

j=1

j=1

:

; where Δm−1 = (y1 , . . . , ym−1 ) ∈ Rm−1 : 0 ≤ y1 + · · · + ym−1 < 1 is the + standard (m − 1)-dimensional simplex; recall as well that B(x1 , . . . , xm ) =

Γ(x1 ) · · · Γ(xm ) . Γ(x1 + · · · + xm )

In what follows we will frequently use the notation B[f ](x1 , . . . , xm ) = B[f (y1 , . . . , ym−1 )](x1 , . . . , xm ) ⎛ ⎞⎛ ⎞xm −1  m−1 m−1 M x −1  := f (y1 , . . . ym−1 ) ⎝ y j j ⎠ ⎝1 − yj ⎠ dy1 · · · dym−1 , Δm−1

j=1

j=1

so that B[1](x1 , . . . , xm ) = B(x1 , . . . , xm ).

3. Tm -invariant symbols Let k = (k1 , . . . , km ) be a tuple of positive integers whose sum is equal to n: k1 + · · · + km = n. The length of such a tuple may obviously vary from 1, for k = (n), to n, for k = 1 = (1, . . . , 1). Throughout the paper we fix a tuple k = (k1 , . . . , km ), with k1 ≤ k2 ≤ · · · ≤ km , and rearrange the n coordinates of z ∈ Bn in m groups, each one of which has kj , j = 1, . . . , m, entries. Let h ∈ Z+ be such that k1 = · · · = kh = 1 and kh+1 ≥ 2. We will use the notation z(1) = (z1,1 , . . . , z1,k1 ), z(2) = (z2,1 , . . . , z2,k2 ), . . . , z(m) = (zm,1 , . . . , zm,km )

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with z1,1 = z1 , z1,2 = z2 , . . . , z1,k1 = zk1 , z2,1 = zk1 +1 , . . . , z2,k2 = zk1 +k2 . Note that for h > 0, z(1) = z1 , . . . , z(h) = zh . We will also use an alternative representation of a point z = (z1 , . . . , zn ) ∈ Bn : where z(j) ∈ Bkj , j = 1, . . . , m.

z = (z(1) , . . . , z(m) ),

In general, given any n-tuple u, we will also use two its alternative representations u = (u1 , . . . , un ) = (u(1) , . . . , u(m) ), where u1,1 = u1 , u1,2 = u2 , . . . , u1,k1 = uk1 , u2,1 = uk1 +1 , . . . , u2,k2 = uk1 +k2 , . . . , um,km = un . For each κ = (κ1 , . . . , κm ) ∈ Zm + we introduce the finite-dimensional subspace Hκ of the Bergman space A2λ (Bn ): : ; Hκ := span eα : |α(j) | = κ(j), j = 1, . . . , m ; of course, if h > 0, then |α(1) | = α1 = κ1 , . . . , |α(h) | = αh = κh . Further we have that ∞  A2λ (Bn ) = Hκ , (3.1) |κ|=0

Introduce the action of T T

m

m

on B as follows: n

/ η = (η1 , . . . , ηm ) : z = (z(1) , . . . , z(m) ) −→ ηz = (η1 z(1) , . . . , ηm z(m) ).

Lemma 3.1. Let g ∈ L∞ (Bn ) be invariant under the action of the group Tm on Bn , i.e., g(ηz) = g(z) for all η ∈ Tm and z ∈ Bn . Then the Toeplitz operator Tg leaves invariant all spaces Hκ . Proof. Given α, β ∈ Zn+ , we calculate  Tg eα , eβ  = geα , eβ  = g(z)eα (z)eβ (z) dvλ (z) Bn  = g(η1 z(1) , . . . , ηm z(m) )eα (z)eβ (z) dvλ (z) Bn

=

=

m M

j=1 m M

|βj) |−|α(j) |

ηj

|βj) |−|α(j) |

ηj

 Bn

g(w(1) , . . . , w(m) )eα (w)eβ (w) dvλ (w)

Tg eα , eβ ;

j=1

we changed above the variables: w(j) = ηj z(j) for all j = 1, . . . , m. Thus Tg eα , eβ  = 0 if there is j = 1, . . . , m such that |α(j) | =  |β(j) |, and the result follows. 

On commutative C ∗ -algebras generated by Toeplitz operators

449

Note that if h > 0, then each Tm -invariant function g has the form g(z) = g(|z1 |, . . . , |zh |, z(h+1) , . . . , z(m) ). with g(|z1 |, . . . , |zh |, ηh+1 z(h+1) , . . . , ηm z(m) ) = g(|z1 |, . . . , |zh |, z(h+1) , . . . , z(m) ). We give now an alternative representation of points z ∈ Bn . We represent first each coordinate of z (which is the same as each coordinate of z(j) , j = 1, . . . , m) in the form zi = |zi |ti

or

zj, = |zj, |tj, ,

where ti and tj, belong to T. For each portion z(j) , j = 1, . . . , m, of a point z we introduce its “common” radius O rj = |zj,1 |2 + · · · + |zj,kj |2 , and represent the coordinates of z(j) in the form zj, = rj sj, tj, , where sj, =

|zj,1 | rj ,

k −1

 = 1, . . . , kj ,

so that s(j) = (sj,1 , . . . , sj,kj ) ∈ S+j := S kj −1 ∩ R+j . For h > 0 we have that rj = |zj | and sj,1 = 1, or zj = rj tj , for all j = 1, . . . , h. k

In what follows we restrict the class of Tm -invariant functions to the collection of functions of the form m M g(z) = a(r1 , . . . , rm ) bj (s(j) ) c(t(j) ), (3.2) j=h+1

where a = a(r1 , . . . , rm ) ∈ L∞ (τ (Bm )) is a k-quasi-radial function, bj = k −1 bj (s(j) ) ∈ L∞ (S+j ), and cj = c(t(j) ) ∈ L∞ (Tkj ), j = h + 1, . . . , m. Note that in parallel with the functions bj (s(j) ), where s(j) = (sj,1 , . . . , sj,kj ) ∈ k −1 S+j , we will consider the functions bj ( s(j) ), with s(j) = (sj,1 , . . . , sj,kj −1 ) ∈ kj −1 B+ , connected by   O bj ( s(j) ) = bj sj,1 , . . . , sj,kj −1 , 1 − (s2j,1 + · · · s2j,kj −1 ) , and defining thus the same objects. Remark 3.2. Although we may consider a wider class of Tm -invariant functions following the lines of this paper, we restrict ourselves to functions of the form (3.2) by at least two reasons. First, it is already a wide extension of the classes of quasi-radial quasi-homogeneous and quasi-radial pseudohomogeneous functions, considered in [9] and [10], respectively. And, second, this class already permits us to demonstrate new features the paper is devoted to, at the same time, leading to more simple tractable formulas.

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Take now a function g of the form (3.2), two multi-indices α, β ∈ Zn+ with |α(j) | = |β(j) | for all j = 1, . . . , m, so that z α and z β belong to the same subspace Hκ with κ = (|α(1) |, . . . , |α(m) |), and calculate:  Γ(n + λ + 1) n α β I(B ) := Tg z , z  = n g(z) z α z β (1 − |z|2 )λ dV (z). π Γ(λ + 1) Bn Changing the variables: zj = |zj | tj we have I(Bn )

 m n M M Γ(n + λ + 1) = n a bj |zq |αq +βq +1 (1 − |z|2 )λ d|z1 | · · · d|zn | π Γ(λ + 1) τ (Bn ) q=1 j=h+1  m n M M q −βq dtq × cj tα q itq Tn j=h+1

=:

q=1

Γ(n + λ + 1) I(τ (Bn )) · I(Tn ). π n Γ(λ + 1)

Then, I(Tn ) =

h  M j=1

T

m  M

dtj itj

j=h+1 m M

= (2π)n

T kj

cj (t(j) )

n M

α

tj,qj,q

−βj,q dtj,q itj,q

q=1

cj eα(j) , eβ(j) L2 (Tkj ) ,

j=h+1

√ α where eα(j) = ( 2π)−kj t(j)(j) is an element of the standard orthonormal basis of L2 (Tkj ). In the next calculation we change the variables: |zj,l | = rj sj,l , so that k −1 for each j = h + 1, . . . , m, d|zj,1 | · · · d|zj,kj | = rj j drj dSj , where dSj is the k −1

standard Euclidean volume element on S+j . We have  m n M M I(τ (Bn )) = a bj |zq |αq +βq +1 (1 − |z|2 )λ d|z1 | · · · |d|zn | τ (Bn )

 =

j=h+1 m M

a(r1 , . . . , rm )

τ (Bm )

×

m M j=h+1

q=1 |α(j) |+|β(j) |+2kj −1

rj

2 (1 − (r12 + · · · + rm ))λ dr1 · · · drm

j=1

 k S+j

=: I(τ (Bm )) ·

bj (s(j) ) −1

kj M

α

sj,lj,l

l=1 m M

j=h+1

k −1

I(S+j

).

+βj,l +1

dSj

On commutative C ∗ -algebras generated by Toeplitz operators

451

Recall that under our assumptions |α(j) | = |β(j) | for all j = 1, . . . , m. Thus substituting rj2 by rj and 2rj drj by drj , we have  m M √ √ |α |+kj −1 I(τ (Bm ) = 2−m a( r1 , . . . , rm ) rj (j) Δm

j=1

× (1 − (r1 + · · · + rm )) dr1 · · · drm λ

= 2−m B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1), √ √ where  a(r) :=  a(r1 , . . . , rm ) := a( r1 , . . . , rm ). k −1 In the next calculation we, first, use the parametrization of each S+j by its first kj − 1 coordinates sj1 , . . . , sj,kj −1 , so that O sj,kj = 1 − (s2j1 + · · · + s2j,kj −1 ) and dSj = O

dsj1 · · · dsj,kj −1 1 − (s2j1 + · · · + s2j,kj −1 )

:= O

d s(j) 1 − (s2j1 + · · · + s2j,kj −1 )

,

and, second, substitute s2j,l by sj,l and 2sj,l dsj,l by dsj,l . We have then k −1 I(S+j )

 =

k B+j

 =

k −1 S+j

bj ( s(j) ) −1

bj (s(j) )

kj M

α

sj,lj,l

+βj,l +1

dSj

l=1

kj −1

M

α

sj,lj,l

+βj,l +1

(1 − (s2j,1 + · · · + s2j,kj −1 )1/2(αj,kj +βj,kj ) d s(j)

l=1

= 2−(kj −1)



kj −1   M 1/2(α +β ) bj √sj,1 , . . . , √sj,k −1 sj,l j,l j,l j Δkj −1

× (1 − (sj,1 + · · · + sj,kj −1 )) −(kj −1)

=2

l=1 1/2(αj,kj +βj,kj )

dsj1 · · · dsj,kj −1

B[bj (sj,1 , . . . , sj,kj −1 )]( 12 (αj,1 +βj,1 )

+ 1, . . . , 12 (αj,kj +βj,kj ) + 1)

= 2−(kj −1) B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1), √ √ where bj ( s(j) ) := bj ( sj,1 , . . . , sj,kj −1 ) := bj ( sj,1 , . . . , sj,kj −1 ) and 1 = (1, . . . , 1). Gathering all the results we have: Tg eα , eβ  = ×

m M

Γ(n + λ + 1) −m 2 B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) π n Γ(λ + 1)

2−(kj −1) B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1)

j=h+1

× (2π)n

m M j=h+1

cj eα(j) , eβ(j) L2 (Tkj )

452 =

N. Vasilevski Γ(n + λ + 1) B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) Γ(λ + 1) m ? @ M × B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) . j=h+1

We sum up the result of the above calculations in the following proposition. Proposition 3.3. Let m M

g(z) = a(r1 , . . . , rm )

bj (s(j) ) cj (t(j) ),

j=h+1 k −1

where a = a(r1 , . . . , rm ) ∈ L∞ (τ (Bm )), bj = bj (s(j) ) ∈ L∞ (S+j ), cj = cj (t(j) ) ∈ L∞ (Tkj ), and cj (ηt(j) ) = cj (t(j) ) for all η ∈ T, j = h + 1, . . . , m. Then the Toeplitz operator Tg leaves invariant all spaces Hκ from the Bergman space decomposition (3.1), and for each z α , z β ∈ Hκ we have Γ(n + λ + 1) B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) Γ(λ + 1) m ? @ M B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) . (3.3)

Tg z α , z β  = ×

j=h+1

4. Toeplitz operator action and algebras We describe first the action of the Toeplitz operator Tg on basis elements eα of its invariant subspace Hκ . Lemma 4.1. Under the assumptions of Proposition 3.3 we have  Tg e α = Tg eα , eβ  eβ , eβ ∈Hκ

where Γ(n + |α| + λ + 1) √ B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) α! β! Γ(λ + 1) m ? @ M B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) . (4.1)

Tg eα , eβ  = ×

j=h+1

Proof. Using (3.3) and |α(j) | = |β(j) |, we have Tg eα , eβ  = =

1 z α  z β 

Tg z α , z β 

Γ(n + |α| + λ + 1) Γ(n + λ + 1) =√ α! β! Γ(n + λ + 1) Γ(λ + 1) × B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1)

On commutative C ∗ -algebras generated by Toeplitz operators ×

m ? M

B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj )

453

@

j=h+1

=

Γ(n + |α| + λ + 1) √ B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) α! β! Γ(λ + 1) m ? @ M × B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) .



j=h+1

Consider now particular cases of symbol g of the form (3.2). We start from the case of a k-quasi-radial symbol g(z) = a(r1 , . . . , rm ) ∈ L∞ (τ (Bm )), i.e., bj ≡ 1 and cj ≡ 1 for all j = h + 1, . . . , m. Then, for each j = h + 1, . . . , m, cj eα(j) , eβ(j) L2 (Tkj ) =

1, 0,

if α(j) = β(j) , if α(j) = β(j) .

Thus with |α(j) | = αj , kj = 1, for j = 1, . . . , h, and for α(j) = β(j) and bj ≡ 1, ,j = h + 1, . . . , m, we have Tg eα = Tg eα , eα  eα =

Γ(n + |α| + λ + 1) α! Γ(λ + 1)

× B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) =

⎧ m ⎨ M ⎩

B(α(j) + 1)

j=h+1

⎫ ⎬ ⎭

eα

Γ(n + |α| + λ + 1) B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) α! Γ(λ + 1) ×

kj m M M Γ(αj,l + 1) eα Γ(|α(j) | + kj )

j=h+1 l=1

Γ(n + |α| + λ + 1) B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) eα j=1 Γ(|α(j) | + 1)Γ(λ + 1)

= Jm

= γa,k,λ (α) eα = γa,k,λ (|α(1) |, . . . , |α(m) |) eα . This is exactly the result of [9, Lemma 3.1]. Let now cj ≡ 1 for all j = h + 1, . . . , m, which implies, in particular, α = β, then m M g(z) = a(r1 , . . . , rm ) bj (s(j) ) j=h+1

is a (particular case of) separately-radial symbol, i.e., a function which depends on |z1 |, . . . , |zn | only. Then the Toeplitz operator Tg is diagonalizable with respect to the standard monomial basis (2.1), Tg eα = γg (α) eα ,

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with (after a simple calculation) γg (α) = γa,k,λ (α)

m M j=h+1

9   Γ(kj + |α(j)|)  B[bj ( s(j) ] α(j) + 1 . J kj l=1 Γ(αj,l + 1)

Let finally g(z) = bj (s(j) ) cj (t(j) ) for some j ∈ {h + 1, . . . , m}, i.e., a ≡ 1 and both bl and cl are identically equal to 1 for all l = j. In this case Tg eα , eβ  does not equal to 0 if and only if eα and eβ are from the same Hκ and have the form α = (α(1) , . . . , α(j−1) , α(j) , α(j+1) , . . . , α(m) ), β = (α(1) , . . . , α(j−1) , β(j) , α(j+1) , . . . , α(m) ). Using (4.1) we have Γ(n + |α| + λ + 1) 1 = Tg eα , eβ  = J α ! Γ(λ + 1) α (j) !β(j) ! l=j (l) ⎧ ⎨ × B(|α(1) | + k1 , . . . , |α(m) | + km , λ + 1) ⎩

m M

B(α(l) + 1)

j=l=h+1

⎫ ⎬ ⎭

× B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) Jm Γ(n + |α| + λ + 1) 1 l=1 Γ(|α(l) | + kl ) Γ(λ + 1) = =J α ! Γ(λ + 1) Γ(|α| + n + λ + 1) α(j) !β(j) ! l=j (l) J kl m M q=1 Γ(αl,q + 1) × B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) Γ(|α(l) | + kl ) j=l=h+1

× cj eα(j) , eβ(j) L2 (Tkj ) Γ(|α(j) | + kj )  = = B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) . α(j) !β(j) ! To characterize the action of the operator Tbj cj on the space Hκ we introduce some notation: α(j)  := (α(1) , . . . , α(j−1) , α(j+1) , . . . , α(m) ), being the tuple α with the part α(j) omitted, and α(j)  α(j) := α, being the tuple   α restored by its parts α(j)  and α(j) . Given α ∈ Hκ , let Hκ (α(j)  β(j) : |β(j) | = κj }  ) := span {eβ = α(j)   be the α(j)  -level of Hκ . Then the above calculation implies that  T bj c j e α = Tbj cj eα , eβ  eβ . eβ ∈Hκ (α(j) )

That is, the operator Tbj cj leaves invariant each α(j)  -level Hκ (α(j)  ) ⊂ Hκ . Remark 4.2. The action of Toeplitz operators Tbj cj , j ∈ {h + 1, . . . , m}, does not depend on a weight parameter λ, while the values of γa,k,λ (α) do.

On commutative C ∗ -algebras generated by Toeplitz operators

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The above discussion leads to the following strengthening of the results of Proposition 3.3. Theorem 4.3. Let g(z) = a(r1 , . . . , rm )

m M

bj (s(j) ) cj (t(j) ),

j=h+1 k −1

where a = a(r1 , . . . , rm ) ∈ L∞ (τ (Bm )), bj = bj (s(j) ) ∈ L∞ (S+j ), cj = cj (t(j) ) ∈ L∞ (Tkj ), and cj (ηt(j) ) = cj (t(j) ) for all η ∈ T, j = h + 1, . . . , m. The operators Ta and Tbj cj , for j = h + 1, . . . , m, mutually commute and Tg = Ta Tbh+1 ch+1 · · · Tbm cm . The restriction of Ta on Hκ is a multiplication by γa,k,λ (α) operator γa,k,λ I, while the action of Tbj cj on basis elements of Hκ is as follows:  Γ(|α(j) | + kj ) = T bj c j e α = α(j) !β(j) ! e ∈H (α ) β

κ

 (j)

× B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) eβ . The following corollary widens essentially known classes of Banach algebras generated by Toeplitz operators that are commutative in each weighted Bergman space A2λ (Bn ), with λ ∈ (−1, ∞). Corollary 4.4. The Banach algebra, generated by Toeplitz operators Ta and Tbj cj , for j = h + 1, . . . , m, of the previous theorem, is commutative in each weighted Bergman space A2λ (Bn ), for λ ∈ (−1, ∞). Each such algebra is uniquely determined by a tuple k = (k1 , . . . , km ), a number h ∈ {0, 1, . . . , m}, and the set of generating symbols: a ∈ L∞ (τ (Bm )), k −1 bj cj , where bj (s(j) ) ∈ L∞ (S+j ), cj (t(j) ) ∈ L∞ (Tkj ) with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T, j = h + 1, . . . , m. Given a tuple k = (k1 , . . . , km ), we group all these data in one set dk = {L∞ (τ (Bm )); bh+1 ch+1 , . . . , bm cm }, and denote the corresponding algebra by Tλ (dk ), in case of operators acting on A2λ (Bn ).

5. C ∗ -algebras In this section we will be interested in the case when Tλ (dk ) is a C ∗ -algebra. For this we need either to have all bj cj , real-valued, so that the Toeplitz operators Tbj cj are self-adjoint, j = h + 1, . . . , m, or h = m. In the last case k = 1 = (1, . . . , 1) and d1 = {L∞ (τ (Bn ))}. That is, the C ∗ -algebra Tλ (d1 ) is generated by all Toeplitz operators with separatelyradial symbols a = a(|z1 |, . . . , |zn |) = a(r1 , . . . , rn ) ∈ L∞ (τ (Bn )). It is well known [6, Theorem 10.1] that each operator from this algebra is diagonal with respect to the standard monomial basis (2.1), and the mapping Ta −→ {γaλ (α)}α∈Zn+ ,

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where B[ a(r1 , . . . , rn )](α1 , . . . , αn , λ + 1) , B(α1 , . . . , αn , λ + 1) that associates each Toeplitz operator Ta to its eigenvalue sequence, is an isometric isomorphism (and even an unitary equivalence) from the algebra Tλ (d1 ) onto a certain subalgebra Γ (d1 ) of bounded sequences. This isomorphism easily leads to understanding of the majority of important properties of operators from Tλ (d1 ): Gelfand theory, compactness, invertibility, spectral properties, invariant subspaces, etc. We note that the C ∗ -algebra Tλ (d1 ) is one of the (n + 2) model commutative C ∗ -algebras generated by Toeplitz operators on the n-dimensional unit ball. The symbols of generating Toeplitz operators in all these cases are geometrically defined, in a sense that for each this case they are invariant under a maximal Abelian subgroup of biholomorphisms of the unit ball, and all such maximal Abelian subgroups (up to a conjugacy) are covered by these (n + 2) model commutative C ∗ -algebras. All these algebras possess a common important property: their operators are diagonalizable in a sense that they are unitary equivalent to multiplication operators by their spectral function or sequence. See [6] for more information and details. Surprisingly all C ∗ -algebras Tλ (dk ) essentially share the properties of Tλ (d1 ). All their operators are diagonalizable with respect to some orthonormal basis in A2λ (Bn ), and the association of an operator to its eigenvalue sequence gives an isometric isomorphism of the C ∗ -algebras Tλ (dk ) onto a certain subalgebra Γ (dk ) of bounded sequences (or functions defined on a countable set). But now the basis is not the standard one (2.1). It depends essentially on the set of generating symbols {bj cj : j = h + 1, . . . , m} and varies for different algebras. At the same time (Remark 4.2), for a given set {bj cj : j = h + 1, . . . , m}, the basis does not depend on a weight parameter λ. Let us study the properties of C ∗ -algebras Tλ (dk ) and their generating operators in more detail. We fix a tuple k = (k1 , . . . , km ) with 0 ≤ h < m, and the set of realk −1 valued functions bj cj , where bj (s(j) ) ∈ L∞ (S+j ), cj (t(j) ) ∈ L∞ (Tkj ) with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T, j = h+1, . . . , m. Then the commutative C ∗ algebra Tλ (dk ), with dk = {L∞ (τ (Bm )); bh+1 ch+1 , . . . , bm cm }, is generated by the operators γaλ (α) =

Ta ,

with a ∈ L∞ (τ (Bm )),

Tbh+1 ch+1 , . . . , Tbm cm ,

acting on the weighted Bergman space A2λ (Bn ). Or the algebra Tλ (dk ) is generated by two mutually commuting unital C ∗ -algebras: Tλ (L∞ (τ (Bm )), which is generated by all Toeplitz operators Ta with a ∈ L∞ (τ (Bm )), and T (bh+1 ch+1 , . . . , bm cm ), which is generated by Toeplitz operators Tbh+1 ch+1 , . . . , Tbm cm . All generators of the algebras Tλ (L∞ (τ (Bm )) and T (bh+1 ch+1 , . . . , bm cm )

On commutative C ∗ -algebras generated by Toeplitz operators

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leave invariant the spaces Hκ , κ ∈ Zm + , of the orthogonal sum decomposition 2 n (3.1) of the Bergman space Aλ (B ), and act on the basis elements eα ∈ Hκ as follows: Ta eα = γa,k,λ (|α(1) |, . . . , |α(m) |) eα , (5.1) where Γ(n + |α| + λ + 1) γa,k,λ (α) = Jm j=1 Γ(|α(j) | + 1)Γ(λ + 1) × B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1), and, for each j = h + 1, . . . , m,  Γ(|α(j) | + kj ) = Tbj cj e α = α(j) !β(j) ! e ∈H (α ) β

κ

 (j)

× B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) eβ .

(5.2)

We will use now a trick of [2, Subsection 4.2] involving operators acting on the Fock (Siegal-Bargmann) space. We start with an abstract Hilbert space Hκ whose orthonormal basis {eα } is enumerated by α = (α(1) , . . . , α(m) ) ∈ Z+ k1 × · · · Z+ km with |α(1) | = κ1 , . . . , |α(m) | = κm . Let then Tj (κ), j = h + 1, . . . , m, be the operators on Hκ which act on its basis elements according to (5.2). Then the unital finite dimensional C ∗ -algebra Tκ , generated by the operators Tj (κ), j = h + 1, . . . , m, is isomorphic and isometric to the algebra generated by the restriction of the operators Tbh+1 ch+1 , . . . , Tbm cm onto their common invariant subspace Hκ . Consider now a particular realization of the above abstract setting based on the Fock space and Toeplitz operators acting on it. Recall that the classical Fock space F 2 (Cn ) consists of all functions f analytic on Cn for which  2 f 2n = |f (z)|2 dμn (z) < ∞, where dμn (z) = π −n e−|z| dv(z). Cn

The standard orthogonal basis in F 2 (Cn ) consists of the normalized monomials 1 eα = √ z α , α ∈ Zn+ . α! The advantage of the Fock space, compared to the Bergman space A2λ (Bn ), is that it has an additional tensor product structure: for our tuple k = (k1 , . . . , km ) with k1 + · · · + km = n, F 2 (Cn ) = F 2 (Ck1 ) ⊗ · · · ⊗ F 2 (Ckm ). Similarly to (3.1), we have the splitting F 2 (Cn ) =



Hκ ,

(5.3)

κ∈Zm +

where Hκ = span {eα : |α(j) | = κj , j = 1, . . . , m}, but now Hκ = Hκ1 ⊗ · · · ⊗ Hκm ,

(5.4)

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where each Hκj ⊂ F 2 (Ckj ) is spanned by all eα(j) = √ 1

α(j) !

α

z(j)(j) with |α(j) | =

κj , j = 1, . . . , m. Note that the first h subspaces Hκj are one-dimensional. Given a function φ ∈ L∞ (Cn ), the Toeplitz operator Tφ , acting on the Fock space, is defined in a standard way: Tφ f = P (φf ), where P is the orthogonal Bargmann projection of L2 (Cn , dμn ) onto the Fock space F 2 (Cn ). As a basic step of further calculations we need the following auxiliary lemma. = Lemma 5.1. Let z = (rs1 t1 , . . . , rsk tk ) ∈ Ck , where r = |z1 |2 + · · · + |zm |2 , |z | k−1 sj = rj , so that s = (s1 , . . . , sm ) ∈ S+ , and tj ∈ T, j = 1, . . . , k. Let k−1 further b(s) ∈ L∞ (S+ ) and c(t) ∈ L∞ (Tk ) with c(ηt) = c(t) for all η ∈ T and t ∈ Tk . Then for each eα eβ ∈ F 2 (Ck ) we have Tbc eα , eβ  =

0, Γ(|α|+k) √ B[b( s)]( 12 (α α!β!

+ β) + 1) c eα , eβ L2 (Tk) ,

if |α| =  |β| . if |α| = |β|

Proof. Repeating literally the arguments of Lemma 3.1 we have that Tbc eα , eβ  = 0 for all |α| =  |β|. With |α| = |β|, we calculate  2 Tbc eα , eβ  = π −k b(s)c(t)eα eβ e−|z| dv(z) −k

2 =√ α!β!

Ck

 r

|α|+|β|+2k−1 −r 2

e

R+

 dr

k−1 S+

b(s)

k M

l +β1 +1 sα dS c eα , eβ L2 (Tk ) l

l=1

Γ(|α| + k)  = √ B[b( s)]( 12 (α + β) + 1) c eα , eβ L2 (Tk) . α!β!



Take now a tuple k = (k1 , . . . , km ) with 0 ≤ h < m, and the set of k −1 real-valued functions bj cj , where bj (s(j) ) ∈ L∞ (S+j ), cj (t(j) ) ∈ L∞ (Tkj ) with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T, j = h + 1, . . . , m, used for Toeplitz operators Tbh+1 ch+1 , . . . , Tbm cm , acting on the Bergman space A2λ (Bn ), and consider the Toeplitz operators Tbh+1 ch+1 , . . . , Tbm cm , acting on the Fock space F 2 (Cn ). Then the previous Lemma implies the following result. Theorem 5.2. The operators Tbj cj , for j = h + 1, . . . , m, mutually commute and leave invariant all spaces Hκ of the direct sum decomposition (5.3). The operators Tbj cj , j = h + 1, . . . , m, act on basis elements of Hκ according to (5.2), i.e,  Γ(|α(j) | + kj ) = Tbj cj eα = α(j) !β(j) ! e ∈H (α ) β

κ

 (j)

× B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) eβ . Furthermore, the restriction of each operator Tbj cj onto Hκ , represented in its tensor product decomposition:     (5.5) Hκ = Hκ1 ⊗ · · · ⊗ Hκj−1 ⊗ Hκj ⊗ Hκj+1 ⊗ · · · ⊗ Hκm

On commutative C ∗ -algebras generated by Toeplitz operators

459

has the form Tj (κ) := Tbj cj |Hκ = I ⊗ Tj,κj ⊗ I, where the matrix units of the operator Tj,κj on Hκj are given by Tj,κj eα(j) , eβ(j)  Γ(|α(j) | + kj )  = = B[b(sj )]( 12 (α(j) + β(j) ) + 1) c eα(j) , eβ(j) L2 (Tkj ) , α(j) !β(j) ! with eα(j) , eβ(j) ∈ Hκj . Let us introduce the unital C ∗ -algebra T(bh+1 ch+1 , . . . , bm cm ), which is generated by Toeplitz operators Tbh+1 ch+1 , . . . , Tbm cm acting on the Fock space F 2 (Cn ). And let Tκ (bh+1 ch+1 , . . . , bm cm ) and Tκ (bh+1 ch+1 , . . . , bm cm ), be the finite dimensional C ∗ -algebras which consist of the restrictions of all operators from T (bh+1 ch+1 , . . . , bm cm ) and T(bh+1 ch+1 , . . . , bm cm ) onto their invariant subspaces Hκ and Hκ , respectively. Corollary 5.3. The association Tbj cj −→ Tbj cj ,

j = h + 1, . . . , m,

generates an isometric isomorphism between the C ∗ -algebras T (bh+1 ch+1 , . . . , bm cm ) In particular, for each κ ∈

Zm +,

and

T(bh+1 ch+1 , . . . , bm cm ).

the association

Tbj cj |Hκ −→ Tbj cj |Hκ ,

j = h + 1, . . . , m,

generates an isometric isomorphism between the C ∗ -algebras Tκ (bh+1 ch+1 , . . . , bm cm )

and

Tκ (bh+1 ch+1 , . . . , bm cm ).

Denote by Ej,κj the chain being the lexicographically ordered standard monomial orthonormal basis in Hκj , which contains dj,κj := dim Hκj = (kj +κj −1)! (kj −1)! κ! elements. Let Mj,κj be the matrix of the operator Tj,κj formed by elements Tj,κj eβ(j) eα(j)  = Tj,κj eα(j) eβ(j)  Γ(|α(j) | + kj )  = = B[bj ( s(j) )]( 12 (α(j) + β(j) ) + 1) cj eα(j) , eβ(j) L2 (Tkj ) α(j) !β(j) ! in the ordered basis Ej,κj . Note that as Tj,κj is self-adjoint, the matrix Mj,κj is hermitian, moreover it is real and symmetric. Let then Uj,κj be the orthogonal matrix diagonalizing Mj,κj , i.e.,   T Dj,κj := Uj,κj Mj,κj Uj,κ = diag ξj,κj (1), . . . , ξj,κj (dj,κj ) . (5.6) j : ; Further, we denote by Fj,κj = fj,κj (p) p=1,...,dj,κ the (alternative) orthonormal basis in Hκj that consists of the elements fj,κj (p) = Uj,κj (p) Ej,κj ,

j

p = 1, . . . , dj,κj ,

where Uj,κj (p) is the pth row of the matrix Uj,κj .

(5.7)

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N. Vasilevski In this basis the operator Tj,κj is diagonal: Tj,κj fj,κj (p) = ξj,κj (p) fj,κj (p),

for all p = 1, . . . , dj,κj .

Consider now the orthonormal basis Fκ of the space Hκ which consists of all elements of the form fκ (p) := f1,κ1 (p1 ) ⊗ · · · ⊗ fm,κm (pm ), where p = (p1 , . . . , pm ), pj = 1, . . . , dj,κj , and fj,κj (pj ) ∈ Fj,κj , for all j = 1, . . . , m. Recall that first h spaces Hκj in (5.4) are one-dimensional, so that p1 = · · · = ph = 1 and fj,κj (pj ) = eκj , for j = 1, . . . , h. The dimension of Fκ is thus dκ := dh+1,κh+1 × · · · × dm,κm . Denote by dj,κj = {1, . . . , dj,κj } the set of indices that correspond to elements fj,κj (p) of the basis Fj,κj , and let dκ = d1,κ1 × · · · × dm,κm . We sum up the above in the following result. Lemma 5.4. The unital C ∗ -algebra Tκ (bh+1 ch+1 , . . . , bm cm ), which is generated by Tbj cj |Hκ , j = h + 1, . . . , m, is isomorphic and isometric to a certain unital C ∗ -algebra of functions γκ : dκ → C, and this isomorphism is generated by the following mapping Tb j c j | Hκ

−→ γκ,j ,

j = h + 1, . . . , m,

where γκ,j (p) = ξj,κj (pj ), with p = (p1 , . . . , pm ) ∈ dκ . Returning back to the Bergman space A2λ (Bn ) we consider its orthogonal (not orthonormal) basis formed by the monomials zα =

m M

α

z(j)(j) .

j=1

For each κ = (κ1 , . . . , κm ), the set {z α : |α(j) | = κj ,

j = 1, . . . , m}

is obviously an orthogonal basis in Hκ . For each j = h + 1, . . . , m, we collect α the monomials z(j)(j) , for all α(j) with |α(j) | = κj , and order them lexicographically into a chain Zj,κj (as we already did this for the basis in Hκj ). Introduce then a new orthonormal basis Fκ in Hκ normalizing the set of all polynomials: ? Fκ = fκ (p) = fκ (p)−1 fκ (p) : @ p = (p1 , . . . , pm ) ∈ dκ = d1,κ1 × · · · × dm,κm , with fκ (p) =

h M j=1

κ

zj j

m M j=h+1

Uj,κj (pj )Zj,κj ,

On commutative C ∗ -algebras generated by Toeplitz operators

461

where Uj,κj (pj ) is a the pth j row of the matrix Uj,κj that diagonalizes Mj,κj , see (5.6). Then : ; F := fκ (pκ ) : κ = (κ1 , . . . , κm ) ∈ Zm + , pκ = (pκ,1 , . . . , pκ,m ) ∈ dκ (5.8) is an orthogonal basis in A2λ (Bn ), whose construction is defined in essence by the action of the operators Tbh+1 ch+1 , . . . , Tbm cm . Introduce a countable set : ; D := (κ, pκ ) : κ = (κ1 , . . . , κm ) ∈ Zm pκ = (pκ,1 , . . . , pκ,m ) ∈ dκ . +, Then the results of this section imply the following Proposition 5.5. Each operator from the unital C ∗ -algebra T (bh+1 ch+1 , . . . , bm cm ) generated by Toeplitz operators Tbh+1 ch+1 , . . . , Tbm cm , where k −1

bj (s(j) ) ∈ L∞ (S+j

),

cj (t(j) ) ∈ L∞ (Tkj )

with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T and t(j) ∈ Tkj , j = h + 1, . . . , m, is diagonal with respect to the basis (5.8). For generating Toeplitz operators we have Tbj cj fκ (pκ ) = ξj,κj (pκ,j ) fκ (pκ ), (κ, pκ ) ∈ D. Furthermore, the unital C ∗ -algebra T (bh+1 ch+1 , . . . , bm cm ) is isomorphic and isometric to a certain C ∗ -algebra of functions (sequences, as being defined on a countable set) γ  : D → C, and this isomorphism is generated by the mapping Tbj cj −→ γ bj cj , j = h + 1, . . . , m, where γ bj cj (κ, pκ ) = ξj,κj (pκ,j ). To describe the commutative C ∗ -algebra Tλ (dk ) we need to add to the previous proposition the action of the Toeplitz operators Ta with a ∈ L∞ (τ (Bm )). Recall (5.1) that the operator Ta acts on the basis elements eα , or on monomials z α as follows: Ta z α = γa,k,λ (|α(1) |, . . . , |α(m) |) z α , where γa,k,λ (α) Γ(n + |α| + λ + 1) B[ a(r)](|α(1) | + k1 , . . . , |α(m) | + km , λ + 1). j=1 Γ(|α(j) | + 1)Γ(λ + 1)

= Jm

In particular, this implies that the action of Ta on the (alternative) basis elements fκ (pκ ) from Hκ is determined (apart of a function a, of course) only by the value of κ, and is given as follows: Ta fκ (pκ ) = γa,k,λ (κ1 , . . . , κm ) fκ (pκ ), for each pκ ∈ dκ . Thus we arrive at the following final result.

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Theorem 5.6. Each operator from the commutative C ∗ -algebra Tλ (dk ) generated by Toeplitz operators Ta with a ∈ L∞ (τ (Bm )) and Tbh+1 ch+1 , . . . , Tbm cm , k −1 where bj (s(j) ) ∈ L∞ (S+j ), cj (t(j) ) ∈ L∞ (Tkj ) with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T and t(j) ∈ Tkj , j = h + 1, . . . , m, is diagonal with respect to the basis (5.8). For generating Toeplitz operators we have Ta fκ (pκ ) = γa,k,λ (κ1 , . . . , κm ) fκ (pκ ), Tbj cj fκ (pκ ) = ξj,κj (pκ,j ) fκ (pκ ),

(κ, pκ ) ∈ D.

∗

Furthermore, the unital C -algebra Tλ (dk ) is isomorphic and isometric to a certain C ∗ -algebra of functions (sequences, as being defined on a countable set) γ k,λ : D → C, and this isomorphism is generated by the mapping Ta

−→

γ k,λ,a ,

Tbj cj

−→

γ k,λ,bj cj ,

j = h + 1, . . . , m,

where γ k,λ,a (κ, pκ ) = γa,k,λ (κ) and γ k,λj ,bj cj (κ, pκ ) = ξj,κj (pκ,j ).

6. C ∗ -algebras of Toeplitz operators on the Fock space F 2 (Cn ) The results of the previous section permits us to get practically for free the description of the commutative C ∗ -algebras generated by Toeplitz operators, with Tm -invariant symbols, acting on the Fock space F 2 (Cn ). We will sketch the principal results omitting the details. Again we start with a tuple k = (k1 , . . . , km ), a number h ∈ {0, 1, . . . , m}, and the set of generating symbols: a ∈ L∞ (Rm + ), bj cj , where bj (s(j) ) ∈ k −1 L∞ (S+j ), cj (t(j) ) ∈ L∞ (Tkj ) with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T, j = h + 1, . . . , m. We group then all these data in one set dk = {L∞ (Rm + ); bh+1 ch+1 , . . . , bm cm }, and denote the corresponding algebra by T(dk ). Multiplying the basis elements (5.7) of Hκj for all j and varying κ, we define an (alternative) orthonormal basis in F 2 (Cn ): ; : F := fκ (pκ ) : κ = (κ1 , . . . , κm ) ∈ Zm (6.1) + , pκ = (pκ,1 , . . . , pκ,m ) ∈ dκ with fκ (p) =

h M

m M

eκj j=1

Uj,κj (pj )Ej,κj ,

j=h+1

where again Uj,κj (pj ) is a the pth j row of the matrix Uj,κj that diagonalizes Mj,κj , see (5.6), and Ej,κj is the lexicographically ordered standard monomial orthonormal basis in Hκj . Then Lemma 5.4 implies the following analog of Proposition 5.5. Proposition 6.1. Each operator from the unital C ∗ -algebra T(bh+1 ch+1 , . . . , bm cm )

On commutative C ∗ -algebras generated by Toeplitz operators

463

generated by Toeplitz operators Tbh+1 ch+1 , . . . , Tbm cm , where k −1

bj (s(j) ) ∈ L∞ (S+j

),

cj (t(j) ) ∈ L∞ (Tkj )

with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T and t(j) ∈ Tkj , j = h + 1, . . . , m, is diagonal with respect to the basis (6.1). For generating Toeplitz operators we have Tbj cj fκ (pκ ) = ξj,κj (pκ,j ) fκ (pκ ), (κ, pκ ) ∈ D. Furthermore, the unital C ∗ -algebra T(bh+1 ch+1 , . . . , bm cm ) is isomorphic and isometric to a certain C ∗ -algebra of functions γ  : D → C, and this isomorphism is generated by the following mapping Tbj cj −→ γ bj cj ,

j = h + 1, . . . , m,

where γ bj cj (κ, pκ ) = ξj,κj (pκ,j ). Consider now Toeplitz operators Ta with k-quasi-radial symbols a = a(r1 , . . . , rm ) ∈ L∞ (Rm + ). Lemma 6.2. ToeplitzJoperators Ta with a = a(r1 , . . . , rm ) ∈ L∞ (Rm + ) acts on m basis elements eα = j=1 eα(j) as follows: Ta eα = γκ,a (|α(1) |, . . . , |α(m) |) eα , where 1 Γ(|α (j) | + kj ) j=1

γκ,a (|α(1) |, . . . , |α(m) |) = Jm ×

m M

 Rm +

|α(j) |+kj −1 −|r|2

rj

e

√ √ a( r1 , . . . , rm )

dr1 · · · drm .

j=1

Proof. Routine calculation using the coordinates z = (z(1) , . . . , z(m) ) with zj,l = rj sj,l tj,l , j = 1, . . . , m, l = 1, . . . , kj .  Note that γκ,a = γκ,a a(|α(1) |, . . . , |α(1) |) is constant of each Hκ as it depends on |α(1) | = κ1 ,. . . , |α(m) | = κm only. Finally we have Theorem 6.3. Each operator from the commutative C ∗ -algebra T(dk ) generated by Toeplitz operators Ta with a ∈ L∞ (Rm + ) and Tbh+1 ch+1 , . . . , Tbm cm , kj −1 kj where bj (s(j) ) ∈ L∞ (S+ ), cj (t(j) ) ∈ L∞ (T ) with cj (ηt(j) ) = cj (t(j) ) for all η ∈ T and t(j) ∈ Tkj , j = h + 1, . . . , m, is diagonal with respect to the basis (6.1). For generating Toeplitz operators we have Ta fκ (pκ ) = γa,k (κ1 , . . . , κm ) fκ (pκ ), Tbj cj fκ (pκ ) = ξj,κj (pκ,j ) fκ (pκ ),

(κ, pκ ) ∈ D.

∗

Furthermore, the unital C -algebra T(dk ) is isomorphic and isometric to a certain C ∗ -algebra of functions γ k : D → C, and this isomorphism is generated by the mapping Ta Tb j c j

−→ −→

γ k,a , γ k,bj cj ,

j = h + 1, . . . , m,

464

N. Vasilevski

where γ k,a (κ, pκ ) = γk,a (κ) and γ k,abj cj (κ, pκ ) = ξj,κj (pκ,j ).

References [1] H. Alzer, Inequalities for the Beta function of n variables, ANZIAM J. 44 (2003), 609–623. [2] W. Bauer and N. Vasilevki, On the structure of commutative Banach algebras generated by Toeplitz operators on the unit ball. Quasi-elliptic case. I: Generating subalgebras, J. Func. Analysis 265 (2013), 2956–2990. [3] L. Coburn, Singular Integral Operators and Toeplitz Operators on Odd Spheres, Indiana Univ. Math. J. 23 (1973), 433–439. [4] S. Grudsky, R. Quiroga-Barranco, and N. Vasilevski, Commutative C*-algebras of Toeplitz operators and quantization on the unit disk, J. Func. Analysis 234 (2006), 1–44. [5] B. Korenblum and K. Zhu, An application of Tauberian theorems to Toeplitz operators, J. Operator Theory 339 (1995), 353–361. [6] R. Quiroga-Barranco and N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball, I. Bargmann-type transforms and spectral representations of Toeplitz operators, Integr. Equat. Oper. Th. 59 no. 3 (2007), 379–419. [7] R. Quiroga-Barranco and N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball, II. Geometry of the level sets of symbols, Integr. Equat. Oper. Th. 60 no. 1 (2008), 89–132. [8] N.L. Vasilevski, Bergman Space Structure, Commutative Algebras of Toeplitz Operators and Hyperbolic Geometry, Integr. Equat. Oper. Th. 46 (2003), 235– 251. [9] N. Vasilevski, Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators, Integr. Equ. Oper. Theory 66 no. 1 (2010), 141–152. [10] N. Vasilevski, On Toeplitz operators with quasi-radial and pseudo-homogeneous symbols, in: Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory, vol. 2, pp. 401–417, Springer Verlag, Heidelberg, 2017. [11] U. Venugopalkrishna, Fredholm operators associated with strongly pseudoconvex domains in C n , J. Func. Analysis 9 (1972), 349–373. [12] Kehe Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer Verlag, Heidelberg, 2005. Nikolai Vasilevski Departamento de Matem´ aticas, CINVESTAV Apartado Postal 14-740 07000, M´exico, D.F. M´exico e-mail: [email protected]