Introduction to Model Spaces and their Operators (Cambridge Studies in Advanced Mathematics) [1 ed.] 1107108748, 9781107108745

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Introduction to Model Spaces and their Operators (Cambridge Studies in Advanced Mathematics) [1 ed.]
 1107108748, 9781107108745

Table of contents :
Contents
Preface
Notation
1 Preliminaries
1.1 Measure and integral
1.2 Poisson integrals
1.3 Hilbert spaces and their operators
1.4 Notes
2 Inner functions
2.1 Disk automorphisms
2.2 Bounded analytic functions
2.3 Inner functions
2.4 Unimodular boundary limits
2.5 Angular derivatives
2.6 Frostman’s Theorem
2.7 Notes
3 Hardy spaces
3.1 Three approaches to the Hardy space
3.2 The Riesz projection
3.3 Factorization
3.4 A growth estimate
3.5 Associated classes of functions
3.6 Notes
3.7 For further exploration
4 Operators on the Hardy space
4.1 The shift operator
4.2 Toeplitz operators
4.3 A characterization of Toeplitz operators
4.4 The commutant of the shift
4.5 The backward shift
4.6 Difference quotient operator
4.7 Notes
4.8 For further exploration
5 Model spaces
5.1 Model spaces as invariant subspaces
5.2 Stability under conjugate analytic Toeplitz operators
5.3 Containment and lattice operations
5.4 A decomposition for Ku
5.5 Reproducing kernels
5.6 The projection Pu
5.7 Finite-dimensional model spaces
5.8 Density results
5.9 Takenaka–Malmquist–Walsh bases
5.10 Notes
5.11 For further exploration
6 Operators between model spaces
6.1 Littlewood Subordination Principle
6.2 Composition operators on model spaces
6.3 Unitary maps between model spaces
6.4 Multipliers of Ku
6.5 Multipliers between two model spaces
6.6 Notes
6.7 For further exploration
7 Boundary behavior
7.1 Pseudocontinuation
7.2 Cyclicity via pseudocontinuation
7.3 Analytic continuation
7.4 Boundary limits
7.5 Notes
8 Conjugation
8.1 Abstract conjugations
8.2 Conjugation on Ku
8.3 Inner functions in Ku
8.4 Generators of Ku
8.5 Cartesian decomposition
8.6 2 × 2 inner functions
8.7 Notes
9 The compressed shift
9.1 What is a compression?
9.2 The compressed shift
9.3 Invariant subspaces and cyclic vectors
9.4 The Sz.-Nagy–Foias¸ model
9.5 Functional calculus for Su
9.6 The spectrum of Su
9.7 The C*-algebra generated by Su
9.8 Notes
9.9 For further exploration
10 The commutant lifting theorem
10.1 Minimal isometric dilations
10.2 Existence and uniqueness
10.3 Strong convergence
10.4 An associated partial isometry
10.5 The commutant lifting theorem
10.6 The characterization of {Su}′
10.7 Notes
11 Clark measures
11.1 The family of Clark measures
11.2 The Clark unitary operators
11.3 Spectral representation of the Clark operator
11.4 The Aleksandrov disintegration theorem
11.5 A connection to composition operators
11.6 Carleson measures
11.7 Isometric embeddings
11.8 Notes
11.9 For further exploration
12 Riesz bases
12.1 Minimal sequences
12.2 Uniformly minimal sequences
12.3 Uniformly separated sequences
12.4 The mappings Λ, V, and Γ
12.5 Abstract Riesz sequences
12.6 Riesz sequences in KB
12.7 Completeness problems
12.8 Notes
13 Truncated Toeplitz operators
13.1 The basics
13.2 A characterization
13.3 C-symmetric operators
13.4 The spectrum of Auϕ
13.5 An operator disintegration formula
13.6 Norm of a truncated Toeplitz operator
13.7 Notes
13.8 For further exploration
References
Index

Citation preview

C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 1 4 8 Editorial Board

B . B O L L O B Á S , W. F U LTO N , A . K ATO K , F. K I RWA N , P. S A R NA K , B . S I M O N , B . TOTA RO

INTRODUCTION TO MODEL SPACES AND THEIR OPERATORS The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory, and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further.

Stephan Ramon Garcia is an Associate Professor at Pomona College, California. He has earned multiple NSF research grants and five teaching awards from three different institutions. He has also authored over 60 research articles in operator theory, complex analysis, matrix analysis, and number theory. Javad Mashreghi is a Professor of Mathematics at Université Laval, Québec, where he has been selected Star Professor of the Year seven times for excellence in teaching. His main fields of interest are complex analysis, operator theory and harmonic analysis. He is the author of several mathematical textbooks, monographs, and research articles. He won the G. de B. Robinson Award, the publication prize of the Canadian Mathematical Society, in 2004. William T. Ross is the Roger Francis and Mary Saunders Richardson chair in mathematics at the University of Richmond, Virginia. He is the author of over 40 research papers in function theory and operator theory and also four books.

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS Editorial Board: B. Bollobás, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: www.cambridge.org/mathematics. Already published 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148

H. Geiges An introduction to contact topology J. Faraut Analysis on Lie groups: An introduction E. Park Complex topological K-theory D. W. Stroock Partial differential equations for probabilists A. Kirillov, Jr An introduction to Lie groups and Lie algebras F. Gesztesy et al. Soliton equations and their algebro-geometric solutions, II E. de Faria & W. de Melo Mathematical tools for one-dimensional dynamics D. Applebaum Lévy processes and stochastic calculus (2nd Edition) T. Szamuely Galois groups and fundamental groups G. W. Anderson, A. Guionnet & O. Zeitouni An introduction to random matrices C. Perez-Garcia & W. H. Schikhof Locally convex spaces over non-Archimedean valued fields P. K. Friz & N. B. Victoir Multidimensional stochastic processes as rough paths T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Representation theory of the symmetric groups S. Kalikow & R. McCutcheon An outline of ergodic theory G. F. Lawler & V. Limic Random walk: A modern introduction K. Lux & H. Pahlings Representations of groups K. S. Kedlaya p-adic differential equations R. Beals & R. Wong Special functions E. de Faria & W. de Melo Mathematical aspects of quantum field theory A. Terras Zeta functions of graphs D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, I D. Goldfeld & J. Hundley Automorphic representations and L-functions for the general linear group, II D. A. Craven The theory of fusion systems J. Väänänen Models and games G. Malle & D. Testerman Linear algebraic groups and finite groups of Lie type P. Li Geometric analysis F. Maggi Sets of finite perimeter and geometric variational problems M. Brodmann & R. Y. Sharp Local cohomology (2nd Edition) C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, I C. Muscalu & W. Schlag Classical and multilinear harmonic analysis, II B. Helffer Spectral theory and its applications R. Pemantle & M. C. Wilson Analytic combinatorics in several variables B. Branner & N. Fagella Quasiconformal surgery in holomorphic dynamics R. M. Dudley Uniform central limit theorems (2nd Edition) T. Leinster Basic category theory I. Arzhantsev, U. Derenthal, J. Hausen & A. Laface Cox rings M. Viana Lectures on Lyapunov exponents J.-H. Evertse & K. Györy Unit equations in Diophantine number theory A. Prasad Representation theory S. R. Garcia, J. Mashreghi & W. T. Ross Introduction to Model Spaces and their Operators

Introduction to Model Spaces and their Operators STEPHAN RAMON GARCIA Pomona College, California

JAVA D M A S H R E G H I Université Laval, Québec

W I L L I A M T. RO S S University of Richmond, Virginia

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107108745 c Stephan Ramon Garcia, Javad Mashreghi and William T. Ross 2016  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Garcia, Stephan Ramon. Introduction to model spaces and their operators / Stephan Ramon Garcia, Pomona College, California, Javad Mashreghi, Université Laval, Québec, William T. Ross, University of Richmond, Virginia. pages cm. – (Cambridge studies in advanced mathematics) Includes bibliographical references and index. ISBN 978-1-107-10874-5 1. Hardy spaces. 2. Operator theory. I. Mashreghi, Javad. II. Ross, William T., 1964– III. Title. QA320.G325 2015 515 .73–dc23 2015012064 ISBN 978-1-107-10874-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To our families: Gizem; Reyhan, and Altay Shahzad; Dorsa, Parisa, and Golsa Fiona

Contents

Preface Notation

page xi xiv

1

Preliminaries 1.1 Measure and integral 1.2 Poisson integrals 1.3 Hilbert spaces and their operators 1.4 Notes

1 1 8 22 31

2

Inner functions 2.1 Disk automorphisms 2.2 Bounded analytic functions 2.3 Inner functions 2.4 Unimodular boundary limits 2.5 Angular derivatives 2.6 Frostman’s Theorem 2.7 Notes

32 32 33 36 42 46 52 55

3

Hardy spaces 3.1 Three approaches to the Hardy space 3.2 The Riesz projection 3.3 Factorization 3.4 A growth estimate 3.5 Associated classes of functions 3.6 Notes 3.7 For further exploration

58 58 66 67 73 74 78 81

vii

viii

Contents

4

Operators on the Hardy space 4.1 The shift operator 4.2 Toeplitz operators 4.3 A characterization of Toeplitz operators 4.4 The commutant of the shift 4.5 The backward shift 4.6 Difference quotient operator 4.7 Notes 4.8 For further exploration

83 83 90 93 96 99 100 102 102

5

Model spaces 5.1 Model spaces as invariant subspaces 5.2 Stability under conjugate analytic Toeplitz operators 5.3 Containment and lattice operations 5.4 A decomposition for Ku 5.5 Reproducing kernels 5.6 The projection Pu 5.7 Finite-dimensional model spaces 5.8 Density results 5.9 Takenaka–Malmquist–Walsh bases 5.10 Notes 5.11 For further exploration

104 104 106 108 109 111 112 115 118 120 121 124

6

Operators between model spaces 6.1 Littlewood Subordination Principle 6.2 Composition operators on model spaces 6.3 Unitary maps between model spaces 6.4 Multipliers of Ku 6.5 Multipliers between two model spaces 6.6 Notes 6.7 For further exploration

126 126 129 134 137 139 141 142

7

Boundary behavior 7.1 Pseudocontinuation 7.2 Cyclicity via pseudocontinuation 7.3 Analytic continuation 7.4 Boundary limits 7.5 Notes

144 144 151 152 158 167

8

Conjugation 8.1 Abstract conjugations 8.2 Conjugation on Ku

170 170 173

Contents

8.3 8.4 8.5 8.6 8.7 9

Inner functions in Ku Generators of Ku Cartesian decomposition 2 × 2 inner functions Notes

The compressed shift 9.1 What is a compression? 9.2 The compressed shift 9.3 Invariant subspaces and cyclic vectors 9.4 The Sz.-Nagy–Foia¸s model 9.5 Functional calculus for Su 9.6 The spectrum of Su 9.7 The C ∗ -algebra generated by Su 9.8 Notes 9.9 For further exploration

ix

177 178 180 182 185 187 187 189 193 195 197 201 206 212 213

10 The commutant lifting theorem 10.1 Minimal isometric dilations 10.2 Existence and uniqueness 10.3 Strong convergence 10.4 An associated partial isometry 10.5 The commutant lifting theorem 10.6 The characterization of {Su } 10.7 Notes

215 216 217 222 223 224 229 230

11 Clark measures 11.1 The family of Clark measures 11.2 The Clark unitary operators 11.3 Spectral representation of the Clark operator 11.4 The Aleksandrov disintegration theorem 11.5 A connection to composition operators 11.6 Carleson measures 11.7 Isometric embeddings 11.8 Notes 11.9 For further exploration

231 231 235 239 245 247 250 251 256 258

12 Riesz bases 12.1 Minimal sequences 12.2 Uniformly minimal sequences 12.3 Uniformly separated sequences 12.4 The mappings Λ, V, and Γ

260 260 263 265 268

x

Contents

12.5 12.6 12.7 12.8

Abstract Riesz sequences Riesz sequences in KB Completeness problems Notes

13 Truncated Toeplitz operators 13.1 The basics 13.2 A characterization 13.3 C-symmetric operators 13.4 The spectrum of Auϕ 13.5 An operator disintegration formula 13.6 Norm of a truncated Toeplitz operator 13.7 Notes 13.8 For further exploration References Index

271 276 277 278 282 282 287 291 292 299 300 301 305 307 318

Preface

This is an introductory text on model spaces that is aimed towards both graduate students and active researchers who wish to enter this evolving subject at a comfortable and digestible pace. Model spaces have been studied, in one form or another, for the past 40 years, making connections to many areas of complex analysis (boundary behavior of analytic functions, analytic continuation, zero sets), operator theory (spectral properties, cyclic vectors, invariant subspaces, model operators for contractions, commutant lifting theorems, Hankel operators), engineering (the Darlington synthesis problem, control theory), and, more recently, to mathematical physics (completeness problems for Schrödinger and Sturm– Liouville operators). The purpose of this book is to present some of the basics of the subject in order to whet the appetite and to provide the newcomer with a solid foundation. Many of the topics in this book were inspired by several series of lectures given by us at workshops in Montréal, Helsinki, Lens, Rennes, and Kashan where, during the course of these lectures, we became convinced of the need for students and researchers from adjacent fields to have a friendly introduction to model spaces and their operators. This book is largely self-contained, although the reader is expected to be familiar with the basics of complex analysis, measure theory, and functional analysis. We briefly review these topics to establish our notation and, if necessary, to re-heat some possibly forgotten foundational topics. We develop and prove almost everything else, including a thorough treatment of the fundamentals of Hardy space theory, which is essential to the study of model spaces. Since the list of topics we plan to cover is readily available in the table of contents, we would like to spend a few moments making the case for model spaces. Why should you keep on reading? Historically, model spaces began xi

xii

Preface

with the desire to characterize the cyclic vectors and invariant subspaces of the backward shift operator S ∗ on the Hardy space H 2 . Beurling’s 1949 theorem completely characterized the non-trivial invariant subspaces of the unilateral shift S f = z f on H 2 as uH 2 , where u is an inner function on the open unit disk D. By taking orthogonal complements, we see that the proper invariant subspaces of S ∗ are the so-called model spaces Ku = (uH 2 )⊥ . The elements of uH 2 are easy to describe (multiples of the inner function u) while the elements of Ku are hidden behind annihilators and hence are more difficult to characterize. Indeed, which functions actually belong to Ku ? In 1970, Douglas, Shapiro, and Shields linked membership to a model space with certain continuation (analytic and pseudocontinuation) properties of these functions. Around the same time, Ahern and Clark explored the close relationship between the boundary behavior of functions in Ku and the existence of angular derivatives, building upon earlier work of Carathéodory, Frostman, and Riesz. Some of the most important theorems in operator theory involve modeling a class of abstract operators by concrete operators on familiar spaces. For example, there is the spectral theorem which models normal operators as multiplication operators on Lebesgue spaces. There are other representation theorems for subnormal operators and n-isometries in terms of multiplication operators on certain Hilbert spaces of analytic functions. Pushing this even further, there are the theorems of Sz.-Nagy and Foia¸s which model certain types of contractions as the compression of the unilateral shift to a model space. This program was highly successful and gave reasons to broaden the study of model spaces from the scalar-valued case discussed above to the vector-valued case. The discussion was broadened even further with the discovery of the close cousins of the model spaces, the de Branges–Rovnyak spaces. Sarason, in 1967, identified the commutant of the compressed shift. This result was greatly generalized by Sz.-Nagy and Foia¸s to a wider class of operators and became known as the commutant lifting theorem – now regarded as one of the crowning gems of operator theory. In 1972, Clark discovered a fascinating family of unitary operators whose associated spectral measures are ubiquitous in operator theory, complex analysis, and mathematical physics. These ideas were investigated further by Aleksandrov. Aleksandrov–Clark measures, as they have come to be known, have been used to study composition operators and are proving relevant in harmonic analysis. They also make connections to completeness problems for solutions to Schrödinger and Sturm–Liouville operators. A seminal article of Sarason from 2008 initiated the study of truncated Toeplitz operators, close relatives of Toeplitz operators whose domains are

Preface

xiii

model spaces. We discuss some of the foundational results of this evolving field, providing detailed proofs of the key results. The topics mentioned above, as well as some others, are covered in this book along with all the necessary background material and historical references. Since this is an introduction to model spaces, we cannot cover everything. Although certain topics are missing, the topics that we do cover, we cover in great detail. We do not skimp on the explanations or examples. First and foremost, this book is meant to help the reader learn about model spaces and to become fluent with the fundamental ideas. Finally, writing good books depends on valuable feedback from your colleagues. It this regard, we would like to thank John B. Conway, John E. McCarthy, Dan Timotin, and Dragan Vukotic for their comments on an earlier draft of this book. We also would like to thank Zachary Glassman for the wonderful drawings and Elizabeth Sarapata for the careful editing.

Notation

C  C = C ∪ {∞} C+ C− N = {1, 2, · · · } D = {z ∈ C : |z| < 1} De = {z ∈ C : |z| > 1} ∪ {∞} T = {z ∈ C : |z| = 1} A− M(T) M+ (T) m Dμ L2 = L2 (T, m)  f (n) 2 (Z), 2 (N) Pz (ζ) Pμ  μ(n) ∠ limz→ζ f (z) B(H) σ(A) σp (A) x⊗y σe (A) τζ,a H∞ H2

complex numbers Riemann sphere upper half plane lower half plane natural numbers open unit disk extended exterior disk unit circle the closure of A finite Borel measures on T (p. 1) positive finite Borel measures on T (p. 1) normalized Lebesgue measure on T (p. 1) symmetric derivative of a measure (p. 2) standard Lebesuge space (p. 7) Fourier coefficient of f (p. 8) square summable sequences (p. 8) Poisson kernel (p. 9) Poisson integral of a measure μ ∈ M(T) (p. 9) Fourier coefficient of a measure (p. 10) non-tangential limit of f at ζ (p. 14) bounded operators on a Hilbert space H (p. 24) spectrum of an operator A (p. 25) point spectrum of an operator A (p. 25) a rank one operator (p. 27) essential spectrum of an operator A (p. 31) disk automorphism (p. 32) bounded analytic functions on D (p. 33) Hardy space (p. 58) xiv

Notation cλ (z) = (1 − λz)−1 P S Tϕ S∗ Ku = (uH 2 )⊥ Qλ kλ (z) = 1−u(λ)u(z) 1−λz Pu O(D) Cϕ σ(u) Su κλ (z) σα Uα Auϕ Tu

xv

reproducing kernel for H 2 (p. 59) Riesz projection onto H 2 (p. 66) unilateral shift on H 2 (p. 83) Toeplitz operator on H 2 (p. 90) backward shift on H 2 (p. 99) model space (p. 104) difference quotient operator (p. 101) reproducing kernel for Ku (p. 111) projection onto Ku (p. 111) analytic functions on D (p. 126) composition operator (p. 126) spectrum of an inner u (p. 152) compressed shift (p. 189) normalized reproducing kernel for Ku (p. 208) Clark measure (p. 232) Clark unitary operator (p. 236) truncated Toeplitz operator (p. 282) space of truncated Toeplitz operators on Ku (p. 283)

1 Preliminaries

1.1 Measure and integral 1.1.1 Borel sets and measures Most of the “measuring” in this book will take place on the unit circle T = {z ∈ C : |z| = 1}. Since we assume that the reader has a background in graduate analysis, we quickly review the standard definitions without much fanfare. We let m := dθ/2π denote Lebesgue measure on T, normalized so that m(T) = 1. A subset of T is called a Borel set if it is contained in the Borel σ-algebra, the smallest σ-algebra of subsets of T that contains all of the open arcs of T. A Borel measure on T is a countably additive function that assigns a complex number to each Borel subset of T. Unless otherwise stated, our measures will always be finite. A Borel measure is positive if it assigns a nonnegative number to each Borel set. We let M(T) denote the set of all complex Borel measures on T and we let M+ (T) denote the set of all positive Borel measures on T. A function f : T →  C (where  C denotes the Riemann sphere C ∪ {∞}) satisfying the condition that f −1 (U) is a Borel set for any open set U ⊂ C is called a Borel function. We often need to distinguish between the “support” and a “carrier” of a measure. For μ ∈ M+ (T), consider the union U of all the open subsets U ⊂ T for which μ(U) = 0. The complement T \ U is called the support of μ. On the other hand, a Borel set E ⊂ T for which μ(E ∩ A) = μ(A)

(1.1)

for all Borel subsets A ⊂ T is called a carrier of μ. The support of μ is certainly a carrier, but a carrier need not be the support. Indeed, a carrier of a measure might not even be closed. For example, if f  0 is continuous and dμ = f dm, then a carrier of μ is T \ f −1 ({0}) (which is open) while the support of μ is the closure of this set. The support of a measure is unique while a carrier is not. 1

2

Preliminaries

The Hahn–Jordan Decomposition Theorem says that each μ ∈ M(T) can be written uniquely as μ = (μ1 − μ2 ) + i(μ3 − μ4 ),

μj ∈ M+ (T),

(1.2)

in which μ1 , μ2 and μ3 , μ4 , respectively, are carried on disjoint sets. Since T is a compact Hausdorff space, each Borel measure μ on T is regular in the sense that each positive measure μj in the Hahn–Jordan Decomposition of μ satisfies inf{μj (U) : U ⊃ E, U open} = sup{μj (F) : F ⊂ E, F closed}

(1.3)

for each Borel set E ⊂ T [158, p. 48]. Moreover, the quantity above is equal to μj (E). Recall that μ ∈ M+ (T) is absolutely continuous with respect to m (written μ  m) if μ(A) = 0 whenever m(A) = 0. We say that μ is singular with respect to m (written μ ⊥ m) if there are disjoint Borel sets A and B such that T = A∪B and μ(A) = m(B) = 0. Also recall that the Radon–Nikodym Theorem says that μ  m if and only if dμ = f dm, where f is a Lebesgue integrable  function on T (that is, | f | dm < ∞). By this we mean that μ(A) = A f dm for each Borel set A ⊂ T. The function f is unique up to a set of Lebesgue measure zero and is called the Radon–Nikodym derivative of μ (with respect to m). It is denoted by dμ/dm. One can also obtain dμ/dm as a “derivative” as follows. Definition 1.1 For μ ∈ M(T), the symmetric derivative (Dμ)(w) of μ at w ∈ T is defined to be   μ (e−it w, eit w) (1.4) (Dμ)(w) := lim+  −it , t→0 m (e w, eit w) whenever this limit exists. Here (e−it w, eit w) denotes the arc of T subtended by the points e−it w and eit w. Theorem 1.2

For each μ ∈ M(T), we have:

(i) (Dμ)(w) exists for m-almost every w ∈ T and Dμ =

dμ dm

m-almost everywhere; (ii) μ ⊥ m if and only if Dμ = 0 m-almost everywhere; (iii) If μ ∈ M+ (T) and μ ⊥ m, then Dμ = ∞ μ-almost everywhere. Moreover, μ is carried by the set {ζ : (Dμ)(ζ) = ∞}.

1.1 Measure and integral

3

The Lebesgue Decomposition Theorem says that every μ ∈ M(T) can be decomposed uniquely as μ = μa + μ s ,

(1.5)

where μa  m and μ s ⊥ m. The measure μa is called the absolutely continuous part of μ while μ s is called the singular part of μ. Furthermore, the singular part μ s can be decomposed as μ s = νd + νc , where  νd = cn δζn n1

is a measure with distinct atoms at ζn ∈ T (that is to say, for each Borel set E ⊂ T, δζn (E) = 1 if ζn ∈ E and zero otherwise) and weights cn = νd ({ζn }), and where νc is a singular measure with no atoms (that is, νc ({ζ}) = 0 for all ζ ∈ T). The measure νd is called the discrete part of μ s while νc is called the singular continuous part of μ s . See below for a more classical approach to measures using functions of bounded variation. We now review the weak-∗ topology on M(T). Let C(T) denote the algebra of complex-valued continuous functions on T endowed with the sup-norm  f ∞ := sup | f (ζ)|. ζ∈T

Note that C(T) is complete with respect to this norm and hence a Banach space. For each μ ∈ M(T), the linear functional  μ : C(T) → C, μ ( f ) := f dμ T

is bounded. The norm of μ is defined by  μ  := sup |μ ( f )| : f ∈ C(T) :  f ∞  1 and is equal to μ := |μ|(T), (1.6)

where |μ|(T) is the supremum of n1 |μ(En )| as {En }n1 runs over all finite partitions of T into disjoint Borel subsets. The quantity μ is called the total variation norm of μ. In terms of the Hahn decomposition (1.2) of μ, it satisfies  1  μj (T)  μ  μj (T). √ 2 1 j4 1 j4 Theorem 1.3 (Riesz Representation Theorem) If  is a bounded linear functional on C(T), then  = μ for some unique μ ∈ M(T).

4

Preliminaries

This allows us to define the weak-∗ topology on M(T). A sequence {μn }n1 ⊂ M(T) converges weak-∗ to μ if   f dμn = f dμ ∀ f ∈ C(T). (1.7) lim n→∞

T

T

The following tells us that closed balls in M(T) are weak-∗ compact. Theorem 1.4 (Banach–Alaoglu)

If {μn }n1 is a sequence in M(T) for which

sup μn  < ∞, n1

then there is a measure μ ∈ M(T) and a subsequence μnk that converges to μ in the weak-∗ topology. We will also need a version of the Hahn–Banach Theorem for C(T) and M(T). Theorem 1.5 Suppose that M is a linear manifold in C(T) whose annihilator  f dμ = 0 ∀ f ∈ M μ ∈ M(T) : T

is zero. Then M is dense in C(T). Furthermore, suppose that N is a linear manifold in M(T) whose pre-annihilator  f dμ = 0 ∀μ ∈ N f ∈ C(T) : T

is zero. Then N is weak-∗ dense in M(T).

1.1.2 Classical approach to measures The following classical approach to measure theory requires a discussion of functions of bounded variation and the Lebesgue–Stieltjes integral. We cover this material not only for students to reconnect with the classical roots of analysis, but to also help out with several proofs and examples later on. Definition 1.6

A function F : [0, 2π] → C is of bounded variation if  |F(x j+1 ) − F(x j )| < ∞, FBV := sup

(1.8)

P 0 jn −1 P

where the supremum is taken over all partitions P = {x0 , x1 , . . . , xnP } of [0, 2π], where 0 = x0 < x1 < x2 · · · < xnP = 2π.

1.1 Measure and integral

5

The expression (1.8) defines a semi-norm FBV , the total variation (semi)norm, on the set BV of all functions of bounded variation. Notice that FBV is not a true norm since FBV = 0 if F is a constant function. We gather together some important facts about BV. Proposition 1.7

If F ∈ BV, then:

(i) F  (x) exists for m-almost every x ∈ [0, 2π]; (ii) The one-sided limits F(x+ ) := lim+ F(t), t→x

F(x− ) := lim− F(t) t→x

+

exist for every x ∈ (0, 2π). Moreover, F(0 ) and F(2π− ) exist; (iii) F has at most a countable number of discontinuities; (iv) F = (F1 − F2 ) + i(F3 − F4 ), where each F j is increasing. For F ∈ BV and right continuous (that is, F(x) = lim+ F(t) for all x), define t→x

μF on the set of half-open intervals  [a, b) : a, b ∈ [0, 2π], a  b by

  μF [a, b) := F(b) − F(a).

By the Carathéodory Extension Theorem, μF extends to a unique Borel measure on [0, 2π]. Moreover μF  = FBV , that is to say, the total variation norm of μF defined in (1.6) equals the bounded variation norm of F defined in (1.8). The integral   f dF = f dμF , (1.9) [0,2π]

[0,2π]

defined for every f ∈ C[0, 2π] (continuous functions on [0, 2π]), is called the Lebesgue–Stieltjes integral of f with respect to F. In this classical setting, the Lebesgue Decomposition Theorem says that every F ∈ BV can be written as F = Fa + Fs , where Fa is absolutely continuous (that is, f is the anti-derivative of a Lebesgue integrable function) on [0, 2π] and Fs is singular (that is, Fs = 0 almost everywhere with respect to Lebesgue measure). Note that μF = μFa + μFs is the

6

Preliminaries

Figure 1.1 The Cantor devil’s staircase function

decomposition of the measure μF into its absolutely continuous and singular parts from (1.5). Furthermore, Fs can be decomposed as Fs = Fd + Fc , where Fd (x) =



F(y+ ) − F(y− )



yx

is a jump function and Fc is continuous with Fc = 0 almost everywhere with respect to Lebesgue measure on [0, 2π]. This gives us the decomposition μFs = μFd + μFc of μFs into its discrete part μFd and its continuous part μFc . For example, to produce a singular measure with no atoms one could take F to be the Cantor “devil’s staircase” function (Figure 1.1). Note that F is continuous and F  = 0 almost everywhere with respect to Lebesgue measure on [0, 2π]. Thus F = Fc . The desired singular continuous measure is then μF . The Riesz Representation Theorem tells us that every continuous linear functional on C[0, 2π] takes the form  f dF f → [0,2π]

for some unique (up to an additive constant) F ∈ BV. Moreover, the norm of this linear functional is FBV .

1.1 Measure and integral

7

The Banach–Alaoglu Theorem (Theorem 1.4) now takes the form of the Helly Selection Theorem: if {Fn }n1 ⊂ BV and supn1 Fn BV < ∞, then there is an F ∈ BV and a subsequence Fnk such that   f dFnk = f dF ∀ f ∈ C[0, 2π]. lim k→∞ [0,2π]

[0,2π]

1.1.3 Lebesgue spaces 2

2

Let L := L (T, m) denote the space of m-measurable (that is, Lebesgue measurable) functions f : T →  C such that  12

 2 | f | dm < ∞. (1.10)  f  := T

Retaining tradition, we equate two measurable functions which are equal almost everywhere. With this norm, L2 is a Hilbert space endowed with the inner product  f g dm.  f, g := T

From time to time we will need the spaces L p (0 < p < ∞) of measurable functions for which

  1p p | f | dm < ∞.  f  p := T

We also require the space L∞ := L∞ (T) of all essentially bounded measurable functions on T, equipped with the essential supremum norm  f ∞ := ess-supζ∈T | f (ζ)|. Here,  ess-supζ∈T | f (ζ)| := sup a  0 : m({ζ ∈ T : | f (ζ)| > a}) > 0 is the essential supremum of | f |. For μ ∈ M+ (T), we will also need the corresponding L p (μ) (0 < p < ∞) spaces of Borel measurable functions f on T such that

  1p p | f | dμ < ∞.  f L p (μ) := T

The identity 

 ζ dm(ζ) = n

T

e [0,2π]

⎧ ⎪ ⎪ ⎨1 if n = 0, =⎪ ⎩0 otherwise, 2π ⎪

inθ dθ

(1.11)

8

Preliminaries

shows that the family of functions {ζ → ζ n : n ∈ Z} is an orthonormal set in L2 . The coefficients  n  f (ζ)ζ dm(ζ) f (n) :=  f, ζ n  = T

of an f ∈ L2 with respect to this orthonormal set are called the (complex) Fourier coefficients of f . For each f ∈ L2 ,   f 2 = | f (n)|2 .

Theorem 1.8 (Parseval’s Theorem)

n∈Z

Furthermore,

    f (n)ζ n  = 0. lim  f − N→∞ −NnN

The previous theorem tells us several things. First, {ζ n : n ∈ Z} is an orthonormal basis for L2 . Second, the Fourier series   f (n)ζ n n∈Z 2

converges to f in the norm of L . In general, the Fourier series of an L2 function need not converge pointwise. However, a deep theorem of Carleson says that it converges pointwise m-a.e. to f [17]. Although we will not use this fact, the reader should be aware that such delicate matters exist. Finally, Theorem 1.8 tells us that the L2 norm of f coincides with the norm of the sequence { f (n) : n ∈ Z} of Fourier coefficients in the Hilbert space   12 |an |2 < ∞ 2 (Z) := a = {an }n∈Z : a2 (Z) := n∈Z

of all square-summable sequences of complex numbers, endowed with the inner product  an bn . a, b = n∈Z

We therefore identify the Hilbert spaces L2 and 2 (Z) via the correspondence f ↔ { f (n) : n ∈ Z}.

1.2 Poisson integrals The function Pz (ζ) :=

1 − |z|2 , |ζ − z|2

ζ ∈ T, z ∈ D,

1.2 Poisson integrals

9

is called the Poisson kernel of the unit disk D = {z : |z| < 1}. Note that Pz (ζ) > 0. A computation verifies that

ζ+z Pz (ζ) = Re ζ−z

 (1.12)

and a computation with geometric series yields  n Prw (ζ) = r|n| wn ζ , w ∈ T, r ∈ (0, 1).

(1.13)

n∈Z

With w = eiθ and ζ = eit , one can also establish the formula Preiθ (eit ) =

1 − r2 . 1 − 2r cos(θ − t) + r2

(1.14)

For fixed ζ ∈ T, Pz (ζ) is the real part of the analytic function z →

ζ+z , ζ−z

which makes z → Pz (ζ) a harmonic function on D. Integrating the series in (1.13) term by term and using the orthogonality relations (1.11), we see that  Pz (ζ) dm(ζ) = 1, z ∈ D. (1.15) T

An important property of the Poisson kernel is that for fixed δ > 0,   lim− sup Pr (eit ) = 0. r→1

δ|t|π

(1.16)

This is illustrated in Figure 1.2. One can also see this from the estimate Pr (eit ) 

1 − r2 , 1 − 2 cos δ + r2

δ  |t|  π.

For μ ∈ M(T), define the Poisson integral of μ by  P(μ)(z) := Pz (ζ)dμ(ζ), z ∈ D. T

By differentiating under the integral sign, we see that P(μ) is harmonic on D. Furthermore,   μ(n)r|n| wn , w ∈ T, r ∈ (0, 1), (1.17) P(μ)(rw) = n∈Z

where

  μ(n) :=

n

T

ζ dμ(ζ),

n ∈ Z,

10

Preliminaries

Pr (eit ) 10 8 6 4 2

–π

0

π

t

Figure 1.2 The graphs of Pr (eit ) for r = 0.2, 0.5, 0.8. Notice that Pr (eit ) > 0. Furthermore, notice how Pr peaks higher and higher near t = 0, for increasing values of r, while decaying rapidly away from the origin.

are the Fourier coefficients of the measure μ. We often write P( f ) in place of the more cumbersome P( f dm) for f ∈ L1 . For f ∈ C(T), we have the Poisson Integral Formula for the solution of the Dirichlet problem on D− . The classical Dirichlet problem for a planar domain Ω is: given a continuous function f on the boundary ∂Ω of Ω, find a function u which is continuous on Ω− that is harmonic on Ω and agrees with f on ∂Ω. Theorem 1.9 (Poisson Integral Formula) If f ∈ C(T), then P( f ) is harmonic on D and extends continuously to D− . Furthermore, lim P( f )(z) = f (w)

z→w

for every w ∈ T. Proof Let u := P( f ) and recall from our earlier discussion that u is harmonic on D. To complete the proof, we will show that for every fixed w ∈ T, lim u(z) = f (w).

z→w

(1.18)

Let ε > 0 be given. Use the continuity of f at w to produce a δ > 0 so that whenever ζ ∈ T and |w − ζ| < δ we have | f (w) − f (ζ)| < ε. From here we get      |u(z) − f (w)| =  f (ζ)Pz (ζ)dm(ζ) − f (w) Pz (ζ)dm(ζ) (by (1.15)) T T | f (ζ) − f (w)|Pz (ζ)dm(ζ)  T

1.2 Poisson integrals 

11



=

+ T∩{|ζ−w|δ

|t|δ

1 (F(t) − Dt) ∂t∂ (Pr (eit )). each having the integrand − 2π To estimate these two integrals, recall from (1.14) that

Pr (eit ) =

1 − r2 1 − 2r cos t + r2

and so ∂ 2r(1 − r2 ) sin t Pr (eit ) = − . ∂t (1 − 2r cos t + r2 )2 This means that if |t| > δ then   2r(1 − r2 )  ∂ P (eit )  2 r  ∂t  (1 − 2r cos δ + r2 )2  c(1 − r ), for some c > 0 independent of t and r, and so  ∂ (F(t) − Dt) (Pr (eit ))dt → 0 as r → 1− . ∂t |t|>δ

Thus the first integral in (1.20) tends to zero as r → 1− .

(1.21)

1.2 Poisson integrals

Now write the second integral 

in (1.20) as

|t|δ

∂ (Pr (eit )) dt ∂t |t|δ    F(t) − F(−t) ∂ 1 it − D t − Pr (e ) dt. = π 2t ∂t

=− |t|δ





13

1 2π

(F(t) − Dt)

0tδ

Given ε > 0, choose δ ∈ (0, π) so that    F(t) − F(−t) − D < ε,   2t

0 < t < δ.

Now use the fact, from (1.21), that −t ∂t∂ Pr (eit )  0 for 0 < t < δ to get           F(t) − F(−t) − D t − ∂ P (eit ) dt   1  r    π  2t ∂t  |t|δ  0tδ   ∂ ε t − Pr (eit ) dt  π ∂t 0tδ   ∂ ε t − Pr (eit ) dt  π ∂t [−π,π]  t=π  ε it  = + Pr (eit ) dt − tPr (e ) t=−π π [−π,π]

ε = (−2πPr (−1) + 2π) π  ε 1 − r2 = + 2π . − 2π 2 π 1 − 2r cos(−1) + r It follows that

     1 lim−  Pr (eit ) dF(t) − D < η r→1  2π   [−π,π]

for every η > 0, from which the result follows.



Remark 1.11 With a little more work (see [158, Thm. 11.10]), one can also show that for μ ∈ M+ (T) and w ∈ T lim P(μ)(rw)  (Dμ)(w)  (Dμ)(w)  lim− P(μ)(rw),

r→1−

r→1

14

Preliminaries

(a)

(b)

Figure 1.3 Two Stolz domains Γα (ζ) (shaded) anchored at ζ = 1. The one on the left is Γ3/2 (1) while the one on the right is Γ5/2 (1).

where

  μ (e−it w, eit w) (Dμ)(w) = lim  −it , it t→0+ m (e w, e w)

  μ (e−it w, eit w) (Dμ)(w) = lim+  −it  t→0 m (e w, eit w)

are the lower and upper symmetric derivatives of μ at w. Fatou’s theorem (Theorem 1.10) can be extended in the following way. For m-almost every ζ ∈ T, the Poisson integral P(μ)(z) approaches the finite limit (Dμ)(ζ), whenever this exists, as z → ζ in any Stolz domain  Γα (ζ) := z ∈ D : |z − ζ| < α(1 − |z|) , α > 1 anchored at ζ; see Figure 1.3. One also has Γα (ζ) ⊂ Γβ (ζ),

α < β.

Definition 1.12 For a function f : D → C and ζ ∈ T, we say that f (z) approaches L ∈ C non-tangentially, denoted L = ∠ lim f (z), z→ζ

(1.22)

if f (z) → L holds whenever z → ζ in every fixed Stolz domain. Observe that the Poisson Integral Formula (Theorem 1.9) follows from Fatou’s Theorem since, in this case, dμ = f dm. Thus for every w ∈ T we have  1 Dμ(w) = lim+ f (ζ) dm(ζ) = f (w) t→0 m(e−it w, eit w) [e−it w,eit w]

by the Fundamental Theorem of Calculus.

1.2 Poisson integrals

15

Remark 1.13 Theorem 1.10 is known as Fatou’s Theorem. We will also use Fatou’s Lemma: if { fn }n1 is a sequence of non-negative Borel functions on T, then   lim fn dμ  lim

T n→∞

n→∞

T

fn dμ.

We need several other facts about Poisson integrals and harmonic functions. Theorem 1.14 (Mean Value Property) If u is harmonic on D, then  u(rζ)dm(ζ) = u(0) T

for each r ∈ (0, 1). Proof Note that u = Re f for some analytic function f on D. By the Cauchy Integral Formula,  1 f (z) f (0) = dz. 2πi rT z Writing z = reiθ for 0  θ  2π, we write the integral above as  2π dθ f (reiθ ) . f (0) = 2π 0 

Now take real parts of both sides of the previous equation.

Theorem 1.15 (Maximum Modulus Theorem for Poisson integrals) If f ∈ L∞ , then |P( f )(z)|   f ∞ , Proof

z ∈ D.

For each z ∈ D we have   |P( f )(z)| =  Pz (ζ) f (ζ) dm(ζ) T Pz (ζ)| f (ζ)| dm(ζ)  T  Pz (ζ) dm(ζ)   f ∞

(Pz (ζ)  0)

T

=  f ∞ .

(by 1.15)



Here are two gems that often come in handy. Proposition 1.16 (Uniqueness of Fourier Coefficients)  μ(n) = 0 for all n ∈ Z, then μ is the zero measure.

If μ ∈ M(T) and

16

Proof

Preliminaries If μ ∈ M(T), the dual of C(T), and  n  ζ dμ(ζ) = 0, μ(n) = T

then by linearity,

n ∈ Z,

 T

p dμ = 0

for every trigonometric polynomial  p(ζ) = an ζ n ,

ζ ∈ T.

−NnN

By the Stone–Weierstrass Theorem, such polynomials are dense in C(T) and so  f dμ = 0 T

for all f ∈ C(T). By Theorem 1.3, μ is the zero measure.



For each fixed z ∈ D, the function ζ → Pz (ζ) is continuous on T. In fact. linear combinations of these functions form a dense subset of C(T). Proposition 1.17 Proof

The span of {Pz : z ∈ D} is dense in C(T).

If μ ∈ M(T) and  T

Pz (ζ)dμ(ζ) = 0

∀z ∈ D,

(that is, μ annihilates the span of {Pz : z ∈ D}), then the Poisson integral P(μ) is identically zero on D. Using the series expansion of P(μ)(rζ) from (1.17) we see that  k μ(k), k ∈ Z. (1.23) ζ P(μ)(rζ) dμ(ζ) = r|k| 0= T

Thus μ is a measure whose Fourier coefficients vanish. By the previous proposition, μ is the zero measure. By Theorem 1.5, the span of {Pz : z ∈ D} is dense in C(T). 

1.2.2 Herglotz’s theorem Since Pz (ζ) > 0 for all z ∈ D and ζ ∈ T, we see that P(μ) is a positive harmonic function on D whenever μ ∈ M+ (T). The following theorem of Herglotz says that the converse is true. We follow [63].

1.2 Poisson integrals

17

Theorem 1.18 (Herglotz) If u is a positive harmonic function on D, then there is a unique μ ∈ M+ (T) such that u = P(μ). Proof

The set U := {ur dm : 0 < r < 1}, where ur (ζ) = u(rζ),

ζ ∈ T,

is a collection of positive measures satisfying  ur dm = u(0) T

(Theorem 1.14). Since U is uniformly bounded in total variation norm, we can apply the Banach–Alaoglu Theorem (Theorem 1.4) to conclude that U has a weak-∗ accumulation point μ. Thus there is a sequence rn ↑ 1 such that the measures urn dm converge weak-∗ to μ. It follows from the Poisson Integral Formula (Theorem 1.9) and the definition (1.7) of weak-∗ convergence that μ ∈ M+ (T) and  P(μ)(z) = Pz (ζ)dμ(ζ)  = lim− Pz (ζ)u(rn ζ)dm(ζ) rn →1

T

= lim− u(rn z) rn →1

= u(z) for each z ∈ D. To prove that μ is unique, suppose that u = P(μ1 ) = P(μ2 ) for some μ1 , μ2 ∈ M+ (T). Then the measure ν = μ1 − μ2 satisfies P(ν)(rζ) = 0 for every 0 < r < 1 and ζ ∈ T. However,   k ν(n) =  ν(0) + ν(k) + ζ  ν(−k)). r|n| ζ n rk (ζ k 0 = P(ν)(rζ) = n∈Z

k1

Since the last expression is a power series in the variable r we must have (by the uniqueness of power series coefficients)  ν(0) = 0, and, for each fixed k  1, k

ζ k ν(k) + ζ  ν(−k) = 0,

∀ζ ∈ T.

By the uniqueness of the coefficients of a trigonometric polynomial (ultimately coming from the orthogonality relations from (1.11)), we see that  ν(k) =  ν(−k) = 0. It now follows that

18

Preliminaries  ν(n) = 0

∀n ∈ Z,

and so, by the uniqueness of Fourier coefficients (Proposition 1.16), we see that ν is the zero measure.  A slight variation of the preceding proof says a bit more: Corollary 1.19

If u is harmonic on D and  sup |u(rζ)| dm(ζ) < ∞, T

0 Nα . The estimate in (7.27) also gives us  1 − |an |  1 − rn   " = θn xn  θ1 xn < ∞ |1 − an | n1 θn n1 n1 n1 and so the Frostman condition in Theorem 2.17 is satisfied. This ensures that if B is a Blaschke product whose zeros are {an }n1 , then ∠ lim B(z) = η ∈ T. z→1

In a similar way,

 1 − |an |  " xn < ∞. |1 − an |2 n1 n1

In light of Theorem 2.21, B has a finite angular derivative at 1, even though 1 ∈ {an }−n1 = σ(B). Suppose that μ is the discrete measure  dμ = αn δeiθn ,

Example 7.26



n1

where αn > 0, n1 αn < ∞ (so μ is a finite measure), and θn ↓ 0. To determine whether or not the associated singular inner function sμ has a finite angular derivative at ζ = 1, we need to determine whether or not   1 1 dμ(ξ) = αn < ∞. 2 |1 − ξ| |1 − eiθn |2 T n1 Observe that |1 − eiθn |2 = 2(1 − cos θn ) " θn2 and so   αn 1 αn " . |1 − eiθn |2 n1 θn2 n1 For example, setting αn = 1/n4 and θn = 1/n produces a measure μ so that sμ has a finite angular derivative at ζ = 1 ∈ {αn }−n1 = supp μ = σ(sμ ).

7.5 Notes 7.5.1 Cyclic vectors for S ∗ At least in principle, the cyclic vector problem for S ∗ is solved ( f is non-cyclic for S ∗ if and only if f is PCBT – Proposition 7.15). However, this solution is not as explicit as the solution to the cyclic vector problem for S ( f is cyclic

168

Boundary behavior

for S if and only if f is outer – Corollary 4.5). Outer functions are, in a sense, readily identifiable via Proposition 3.22. On the other hand, PCBT functions are not so easily recognized. We refer the reader to some papers which partially characterize the non-cyclic vectors for S ∗ by means of such conditions as the growth of its Taylor coefficients [119, 171], the modulus of the function [61, 119], and gaps in its Taylor coefficients [1, 11, 61]. In the next chapter, we will parameterize the non-cyclic vectors in terms of a conjugation operator on a model space.

7.5.2 Pseudocontinuation and cyclic vectors The non-cyclic vectors for the backward shift on H 2 have pseudocontinuations of bounded type (Proposition 7.15). This phenomenon is not isolated to H 2 and appears in other Hilbert spaces of analytic functions. For example, in the Bergman space La2 (see (3.28)), and other closely related spaces, the non-cyclic vectors for the backward shift also have pseudocontinuations of bounded type, though this is not the full description of the non-cyclic vectors. For the Dirichlet space D (see (3.29)), the situation is much different in that there are non-cyclic vectors for the backward shift which do not have pseudocontinuations across any arc of T. See [14, 15, 154] for more details

7.5.3 Generalized analytic continuation There are a number of meaningful ways that one can consider two meromorphic functions, one on D and another on De , to be “extensions” of one another.



For example, if f = n0 an zn and g = n0 bn zn are analytic on D while

F = n0 cznn is analytic on De , then, under very mild technical conditions, one can consider F to be an extension of f /g if f (z) = g(z)F(z) by formal multiplication of power series. A Fourier series argument will show how this occurs with f /u and Fu as discussed in (7.1). We refer the reader to [154, 156] for other examples of generalized analytic continuation and how they appear in many unexpected areas of analysis.

7.5.4 Model spaces and tangential limits One can also discuss tangential limits of functions from model spaces [28, 36]. In fact, there is an alternate proof of Theorem 7.24 in [104] which can be used to prove a bit more. Indeed, suppose that u is inner, ζ ∈ T, and Ωζ is an approach region with vertex at ζ. By the term “approach region,” we mean an open subset of D whose closure intersects T only at ζ. For example,

7.5 Notes

169

an approach region might be the interior of a triangle whose vertex is at ζ (a Stolz approach region), or the interior of an internally tangent circle with contact point ζ (an oricyclic approach region), or perhaps one might want to consider other, even more tangential, approach regions. Whatever the approach region Ωζ , the theorem here is that every f ∈ Ku has a limit as z → ζ (with z ∈ Ωζ ) if and only if sup{kz  : z ∈ Ωζ } < ∞. Is there a condition, similar to Frostman’s Theorem (Theorem 2.21), that one could test to determine whether or not sup{kz  : z ∈ Ωζ } < ∞?

7.5.5 When Ahern–Clark fails There are some results from [103] which control the radial growth of functions from Ku when the conditions of Theorem 7.24 fail.

8 Conjugation

Each model space Ku comes equipped with a conjugation, a certain conjugatelinear operator on Ku that generalizes complex conjugation z → z on C. Not only does this conjugation cast a new light on pseudocontinuations, it also interacts with a number of important linear operators that act on model spaces.

8.1 Abstract conjugations We begin by introducing the following simple, but surprisingly fruitful, concept. Definition 8.1 A conjugation on a complex separable Hilbert space H is a mapping C : H → H that is: (i) conjugate linear: C(αx + βy) = αCx + βCy for all x, y in H and α, β ∈ C; (ii) involutive: C 2 = I; (iii) isometric: Cx = x for all x in H. In the literature, some authors use the term anti-linear instead of conjugatelinear. From this perspective, a mapping that satisfies the first and third conditions listed above is called an anti-unitary operator. Thus a conjugation is simply an anti-unitary operator that is involutive. Let us also remark that in light of the well-known polarization identity 4x, y = x + y2 − x − y2 + ix + iy2 − ix − iy2 ,

x, y ∈ H,

the isometric condition is equivalent to the seemingly stronger assertion that Cx, Cy = y, x, 170

x, y ∈ H.

8.1 Abstract conjugations

171

Notice how this condition, along with the fact that C 2 = I, implies the useful identity Cx, y = Cy, x,

x, y ∈ H.

(8.1)

Although our interest in conjugations stems primarily from a single, specific example in the context of model spaces, we first consider a number of apparently unrelated examples that will turn out to be relevant later on. Example 8.2 For a measure space (X, μ), the canonical conjugation on L2 (X, μ) is pointwise complex conjugation: [C f ](x) = f (x). Particular instances include the canonical conjugation C(z1 , z2 , . . . , zn ) = (z1 , z2 , . . . , zn )

(8.2)

on Cn (which can be regarded as the L2 space corresponding to a measure consisting of n distinct unit point masses) and the canonical conjugation C(z1 , z2 , z3 , . . .) = (z1 , z2 , z3 , . . .)

(8.3)

on the space 2 (N) of all square-summable sequences of complex numbers. Example 8.3

The Toeplitz conjugation on Cn is defined by C(z1 , z2 , . . . , zn ) = (zn , zn−1 , . . . , z1 ).

(8.4)

As its name suggests, the Toeplitz conjugation is intimately related to the study of Toeplitz matrices. In light of its appearance in the Szeg˝o recurrence from the theory of orthogonal polynomials on the unit circle [179, eq. 1.1.7], one might also choose to refer to (8.4) as the Szeg˝o conjugation. Example 8.4 Along similar lines, if one has a measure space (X, μ) that possesses a certain amount of symmetry, one can sometimes form a corresponding conjugation. For instance, the conjugation [C f ](x) = f (1 − x) on L2 [0, 1] arises in the study of the Volterra integration operator. Example 8.5 Another example along the lines of the previous one is the conjugation C on L2 (T, m) defined by (C f )(ζ) = f (ζ).

172

Conjugation

Note that C is conjugate linear and involutive. The fact that C is isometric comes from Parseval’s Theorem. Using (8.1), this conjugation yields the useful integral identity   f (ζ)g(ζ) dm(ζ) = f (ζ)g(ζ) dm(ζ), f, g ∈ L2 , (8.5) T

T

which will be used later on. Although varied in appearance, it turns out that the structure of conjugations is remarkably simple. The following observation is from [80, Lem. 1], although no doubt it must have been discovered numerous times before. Lemma 8.6 If C is a conjugation on H, then there exists an orthonormal basis {en }n1 of H such that Cen = en for all n. Proof Consider the set K = (I + C)H, noting that each vector in K is fixed by C and that K is a closed subset of H that is invariant under addition and multiplication by real scalars. Since x, y = Cy, Cx = y, x = x, y,

x, y ∈ K,

we see that K is a real Hilbert space (that is, a Hilbert space whose field of scalars is R) when endowed with the inner product  · , · . Let {en }n1 be an orthonormal basis for K. In light of the fact that   2x = (x + Cx) − i ix + C(ix) holds for all x in H, it follows that {en }n1 is also an orthonormal basis for the complex Hilbert space H = K + iK as well.  A basis of the type described in Lemma 8.6 is called a C-real orthonormal basis of H. Such bases will be important to us when we continue our study of various classes of operators defined on model spaces. At present, however, we require an additional result about general conjugations before arriving at the specific conjugation relevant to model spaces. Lemma 8.7 Every conjugation is unitarily equivalent to the canonical conjugation on an 2 -space of the appropriate dimension. Proof

If {en }n1 is a C-real orthonormal basis for H, then    αn en = αn en , C n1

n1

8.2 Conjugation on Ku

173

since C is conjugate linear and fixes each en . Now observe that the coordinate map U : H → 2 defined by Ux = {x, en }n1 is unitary (being an invertible isometry) and satisfies JU = UC where J denotes the canonical conjugation  on the appropriate 2 -space (see (8.3)).

8.2 Conjugation on Ku Being a complex Hilbert space, each model space Ku possesses many conjugations. Indeed, given any orthonormal basis {en }n1 of Ku , we may define a conjugation C on Ku by setting Cen = en and extending to the rest of Ku by conjugate-linearity. Of course, this observation is of limited usefulness since it makes no particular use of the function theoretic structure of Ku . It turns out, however, that Ku comes pre-equipped with a natural conjugation that respects, to a broad degree, the function theoretic properties of Ku . In order to understand this conjugation, we first remind the reader that Ku = H 2 ∩ uzH 2 when regarded as a space of functions on T via non-tangential boundary values (Proposition 5.4). Proposition 8.8 The conjugate-linear operator C : Ku → Ku , defined in terms of boundary functions on T by C f = f zu,

(8.6)

is a conjugation. In particular, | f | = |C f | almost everywhere on T so that f and C f share the same outer factor. Proof Since uu = 1 a.e. on T, it follows that C is conjugate-linear, isometric, and involutive. We need only prove that C maps Ku into Ku . If f ∈ H 2 is orthogonal to uH 2 , it follows that C f, zh =  f zu, zh = uzh, f z = uh, f  = 0

∀h ∈ H 2 .

In other words, C f ∈ H 2 . Similarly, C f, uh =  f zu, uh =  f z, h = 0 ∀h ∈ H 2 , from which we obtain C f ∈ Ku .



Remark 8.9 The conjugation C is defined by the inner function u. However, Ku = Kξu for any unimodular constant ξ. This equality will result in a

174

Conjugation

unimodular multiple of C when defining the conjugation on Kξu . Normally this unimodular constant is unimportant. However, it does come into play when one wants to normalize the inner function in a certain way (see Section 8.5). The conjugation on Ku defined by (8.6) is closely related to the notion of pseudocontinuation introduced in Section 7.1. We now make this connection more explicit. Letting g# (z) = g(z) denote the function obtained by conjugating the Taylor coefficients of g, the equation f = gzu tells us that the functions f (z)/u(z) on D and 1z g# ( 1z ) on De (the extended exterior disk) have matching non-tangential limits almost everywhere on T. Thus the difference between conjugation and pseudocontinuation is mostly one of interpretation. Our present approach is to think of the complementary function g as C f , a function belonging to Ku , and hence with domain D, as opposed to a function defined on De . Example 8.10 If u = z n , then Ku = assumes the form

 {1, z, . . . , z n−1 } and the conjugation C

C(a0 + a1 z + · · · an−1 z n−1 ) = an−1 + an−2 z + · · · + a0 zn−1 .

(8.7)

Indeed, working on T, one observes that C(z j ) = z j zzn = zn−1− j for 0  j  n − 1. The conjugation (8.7) is thus unitarily equivalent to the Toeplitz conjugation (8.4) defined on Cn . Example 8.11

More generally, suppose that  z − λi u= 1in 1 − λi z

is a finite Blaschke product having zeros λ1 , λ2 , . . . , λn , repeated according to multiplicity. Corollary 5.18 says that every f ∈ Ku takes the form f =

a0 + a1 z + · · · + an−1 z n−1 .  1in (1 − λi z)

Using this representation for f ∈ Ku , and thinking of the computations below on T, the conjugation becomes C f = zfu =z·

a0 + a1 z + · · · + an−1 zn−1  z − λi  · 1in (1 − λ j z) 1in 1 − λi z

8.2 Conjugation on Ku

=z· =

175

zn−1 (an−1 + an−2 z + · · · + a0 zn−1 )  z − λi ·  zn 1in (z − λ j ) 1in 1 − λi z

an−1 + an−2 z + · · · + a0 zn−1 .  1in (1 − λi z)

In other words, C

 a0 + a1 z + · · · + an−1 zn−1  an−1 + an−2 z + · · · + a0 zn−1 . =   1in (1 − λi z) 1in (1 − λi z)



Example 8.12 Suppose that f = n0 an zn and g = n0 bn zn are conjugate to each other in Ku (that is, g = C f and vice-versa). From Proposition 5.15 and its proof, it follows that   bn S ∗(n+1) u, g= an S ∗(n+1) u, f = n0

where  f 2 =

n0

 n0

|an |2 =



|bn |2 = g2 .

n0

Example 8.13 A straightforward computation reveals that the conjugation operator on Ku sends reproducing kernels to difference quotients: ⎛ ⎞ ⎜⎜⎜ 1 − u(λ)u(z) ⎟⎟⎟ 1 − u(λ)u(z) u(z) ⎟⎠ · zu(z) = · (Ckλ )(z) = ⎝⎜ 1 − λz z 1 − λz u(z) − u(λ) = (Qλ u)(z), = z−λ where, as usual, we consider all of the functions involved as functions on T (so that zz = 1 and u(z)u(z) = 1 a.e. on T). For each λ, the kernel kλ is outer since it is the quotient of the two outer functions 1−u(λ)u(z) and 1−λz (Corollary 3.24). In light of Proposition 8.8, we expect that the difference quotient Ckλ is simply an inner multiple of kλ . This is indeed the case as the following computation shows: 1 − λz (Ckλ )(z) u(z) − u(λ) = · kλ (z) z−λ 1 − u(λ)u(z) 1 − λz 1 − u(λ)u(z) z − λ   bu(λ) u(z) , = bλ (z) =

u(z) − u(λ)

·

176

Conjugation

where bw (z) = (z − w)/(1 − wz). When λ = 0, we obtain a specific identity that arises frequently in the study of Clark unitary operators (see Chapter 11), namely C(1 − u(0)u) = S ∗ u.

(8.8)

In particular, if u(0) = 0, then C1 = u/z and vice-versa. Example 8.14 If u has a finite angular derivative in the sense of Carathéodory at ζ ∈ T, then u(ζ) exists and is unimodular. Moreover, the boundary kernel kζ =

1 − u(ζ)u 1 − ζz

belongs to Ku (Theorem 7.24). In this case, something interesting occurs. Since ζ and u(ζ) are of unit modulus, we obtain Ckζ =

u − u(ζ) u(ζ) 1 − u(ζ)u = ζu(ζ)kζ . = · z−ζ ζ 1 − ζz

For either branch of the square root, it follows that the function 1

(ζu(ζ)) 2 kζ belongs to Ku and is fixed by C. Under certain circumstances, it is possible to construct C-real (orthonormal) bases for Ku using boundary kernels. This is relatively straightforward for finite-dimensional model spaces, as the following example demonstrates. Example 8.15

Let u be a finite Blaschke product with n zeros λ1 , λ2 , . . . , λn ∈ D,

repeated according to multiplicity. For the sake of simplicity, suppose that u(0) = 0. Fix α ∈ T and observe that the equation u(ζ) = α has precisely n distinct solutions ζ1 , ζ2 , . . . , ζn ∈ T. To see this, first observe that the equation u(z) = α has at most n solutions in C since it can be rewritten as a polynomial equation of degree n. Since |u(z)| = 1 if and only if z ∈ T, we see that these solutions must all lie on T. Evaluate the identity  1 − |λj |2 u (z) = u(z) 1 jn (1 − λj z)(z − λj )

8.3 Inner functions in Ku

177

(see (2.21)) at z = ζ ∈ T to get u (ζ) = u(ζ)ζ

 1 − |λj |2 1 jn

|1 − λj ζ|2

so that |u (ζ)| > 0 for all ζ ∈ T. Thus every root of the equation u(z) − α = 0 must be simple. Putting this all together, it follows that u(ζ) = α has precisely n distinct solutions on T and hence the functions e 2 (arg α−arg ζj ) 1 − αu(z) ,  |u (ζj )| 1 − ζj z i

ej (z) =

1  j  n,

form a C-real basis of Ku . It will turn out (see Theorem 11.4) that vectors ej are the eigenvectors of a certain certain unitary operator on Ku and thus form an orthonormal basis for Ku .

8.3 Inner functions in Ku Although H 2 contains all of the inner functions, it is not clear whether or not Ku contains any. The following result answers this question and is another nice application of conjugation on Ku . Theorem 8.16 The model space Ku contains an inner function if and only if u(0) = 0. In this case, the inner functions belonging to Ku are precisely u/z and all of its inner divisors. Proof

If u(0) = 0, then u = S ∗ u ∈ Ku . z

By Proposition 5.6, Ku is closed under the operation of removing inner divisors and so it follows that all of the inner divisors of u/z belong to Ku . Conversely, if f ∈ Ku is inner, then C f = f zu is also inner. Hence u = z f C f and so u(0) = 0. Moreover, if f is any inner function that divides u/z we may write u = z f g where g is inner. Since f = gzu a.e. on T, we conclude that  f ∈ Ku (Proposition 5.4). Example 8.17 If u = zn , then Ku is the space of polynomials of degree at most n − 1 (Example 5.17). Among this collection, the only inner functions are ξz j , where 0  j  n − 1 and ξ is a unimodular constant (Example 2.15).

178

Conjugation

8.4 Generators of Ku Fix an inner function u and let f = I1 F denote the inner–outer factorization of an f ∈ Ku \ {0}. Proposition 8.8 ensures that g = C f has the same outer factor as f , whence we may write g = I2 F for some inner function I2 . Since g = f zu a.e. on T it follows that I2 F = I1 Fzu and so Fzu CF = . F F Following [86] we make the following definition. I1 I2 =

Definition 8.18

(8.9)

Let F ∈ Ku be outer. The inner function

Fzu CF = F F is called the associated inner function of F with respect to u. IF =

Before proceeding, we remark that some papers refer to the inner function u itself as the associated inner function of F [61, Rem. 3.1.6]. Example 8.19 As pointed out in Example 8.13, we know that all of the kernel functions kλ are outer. Setting F = kλ we get Ckλ = IF = kλ

u−u(λ) z−λ 1−u(λ)u 1−λz

=

u−u(λ) 1−u(λ)u z−λ 1−λz

,

which is a certain Frostman shift of u with its zero at z = λ divided out with a single Blaschke factor. When u(0) = 0, then k0 = 1 is outer and u (8.10) I1 = . z If v1 and v2 are inner functions such that v1 v2 = IF , then the definition of IF implies that v1 F = v2 Fzu almost everywhere on T. Hence the functions f = v1 F and g = v2 F satisfy f = gzu a.e. on T. By Proposition 5.4, f and g belong to Ku and satisfy C f = g. Furthermore, IF F. (8.11) C(v1 F) = v2 F = v1 Putting this all together, we obtain the following result from [86].

8.4 Generators of Ku

179

Proposition 8.20 Let F ∈ Ku be outer. The set of all functions in Ku having outer factor F is precisely {vF : v inner and v|IF }.

(8.12)

We define a partial order  on (8.12) by declaring that v1 F  v2 F if and only if v1 |v2 . With respect to this ordering, the functions F and IF F are, up to unimodular constants, the unique minimal and maximal elements, respectively. Moreover, C restricts to an order-reversing bijection from the set (8.12) onto itself. Example 8.21 If u is inner with u(0) = 0, then F = k0 = 1 is outer. Hence the functions in Ku with this outer factor are the inner functions in Ku . By the previous proposition, these are the inner divisors of I1 = u/z (see (8.10)). Notice how this reproduces Theorem 8.16. Recall that f ∈ Ku generates Ku if  {S ∗nf : n  0} = Ku .

(8.13)

The following is part of [61, Thm. 3.1.5]. Proposition 8.22 Suppose that f, g ∈ H 2 and u is an inner function such that f = gzu on T. If u and the inner factor of g are relatively prime, then f generates Ku . Proof

From Proposition 5.4, we see that f ∈ Ku and so  M= {S ∗n f : n  0} ⊂ Ku

since Ku is S ∗ -invariant. Because M is a non-trivial S ∗ -invariant subspace of H 2 , it must be of the form Kv for some inner function v. Moreover, since Kv ⊂ Ku we have v|u (Corollary 5.9). Using the fact that f ∈ Kv , we may write f = hzv a.e. on T and so hzv = gzu for some h ∈ Ku . This implies that |g| = |h| a.e. on T so that the outer factors of g and h are identical up to a unimodular constant multiple (which we absorb in to the inner factors). Let vg and vh denote the inner factors of g and h, respectively. It follows that vh v = vg u, from which we obtain the identity vg v = vh u. Since vg and u are relatively prime, this identity proves that u/v = vg /vh is a unimodular constant. Thus  M = Kv = Ku , as claimed. Recognizing that g = C f in this setting immediately yields the following [86, Prop. 4.3].

180

Conjugation

Proposition 8.23 If f ∈ Ku and the inner factor of C f is relatively prime to u, then f generates Ku . Example 8.24 If u is a singular inner function (that is, u is not divisible by any Blaschke product), then Frostman’s Theorem (Theorem 2.22) asserts that kλ generates Ku for all λ ∈ D that do not lie in some exceptional set. Indeed, by Example 8.13, the inner factor of Ckλ is precisely bu(λ) ◦ u , bλ which is a Blaschke product whenever u(λ) is non-zero and does not lie in the exceptional set for u. Corollary 8.25 If f ∈ Ku is outer, then C f generates Ku . In particular, any self-conjugate outer function in Ku generates Ku . Example 8.26

Each difference quotient

u − u(λ) = Ckλ z−λ generates Ku since its conjugate C(Qλ u) = kλ is outer (Corollary 3.24). Qλ u =

Example 8.27 Any boundary kernel kζ ∈ Ku generates Ku . Indeed, Example 8.14 shows that any such function is a constant multiple of a self-conjugate outer function. Example 8.28 since

Every outer function F ∈ Ku is the sum of two generators F = 12 (1 + IF )F + i 2i1 (1 − IF )F.

Moreover, by (8.11), CF = IF F and thus C( 12 (1 + IF )F) = 12 (1 + IF )F, C( 2i1 (1 − IF )F) =

1 2i (1

− IF )F.

8.5 Cartesian decomposition Each f ∈ Ku enjoys a Cartesian decomposition f = a + ib where the functions a = 12 ( f + C f ),

b=

1 2i ( f

− C f ),

8.5 Cartesian decomposition

181

are both fixed by C. With respect to this decomposition, the conjugation C on Ku assumes the form C f = a − ib. To explicitly describe the functions belonging to Ku , it suffices to describe those functions that are C-real (that is, C f = f ). If f = C f , then f = f zu. Since u has unit modulus a.e. on T, by replacing u with a suitable unimodular constant multiple of u (which does not change the model space Ku ), we may assume u(ζ) = ζ for some ζ ∈ T. We can define kζ by kζ (z) =

1 − u(ζ)u(z) 1 − ζz

=

1 − ζu(z) 1 − ζz

which resides in the Smirnov class N + (see Definition 3.26) since the numerator is a bounded function while the denominator is a bounded outer function (Corollary 3.24). If u has a finite angular derivative in the sense of Carathéodory at ζ then kζ will belong to Ku , but at this point, we do not assume this extra condition. A computation yields 1 − ζu 1 − ζ z 1 − ζu 1 − ζ z · = · 1 − ζz 1 − ζu 1 − ζu 1 − ζz ⎛ ⎞

 ⎜⎜⎜ ζz − 1 ⎟⎟⎟ ζu−1 ⎟⎠ = ζu · ζ z ⎜⎝ 1−ζu 1 − ζz

kζ /kζ =

= ζζ zu = zu, which, when substituted into the equation f = f zu, reveals f /kζ = f /kζ almost everywhere on T. Thus a function f ∈ Ku satisfies f = C f if and only if f /kζ is real almost everywhere on T. Definition 8.29 A function f ∈ N + is called a real Smirnov function if its boundary function is real-valued almost everywhere on T. Denote the set of all real Smirnov functions by R+ . The following elegant theorem of Helson [110] provides an explicit formula for real Smirnov functions.

182

Conjugation

Theorem 8.30 A function f ∈ N + belongs to R+ if and only if there are inner functions ψ1 , ψ2 such that ψ 1 + ψ2 (8.14) f =i ψ1 − ψ 2 and ψ1 − ψ2 is outer. Proof One direction follows from direct computation using the fact that ψj ψj = 1 almost everywhere on T. For the other direction, note that τ(z) = i

1+z 1−z

maps D onto the upper half plane. Thus τ−1 ◦ f is of bounded type (that is, a quotient of H ∞ functions) and is unimodular almost everywhere on T. Hence τ−1 ◦ f = ψ1 /ψ2 is a quotient of inner functions and so f has the desired form.  Helson’s Theorem and the representation f = (α + iβ)kζ ,

α, β ∈ R+ ,

permit us to write any function f ∈ Ku as a rational expression involving a finite number of inner functions. Since each inner function has a pseudocontinuation to De given explicitly by the Schwarz reflection formula (see Example 7.4), we conclude that the pseudocontinuability of f itself can be explained entirely through the mechanism of Schwarz reflection.

8.6 2 × 2 inner functions In at least one instance, it turns out that conjugations on model spaces can yield structural insights into the nature of higher order inner functions. Definition 8.31 A matrix-valued function Θ : D → M2 (C) is called an inner function if all four of its entries belong to H ∞ and Θ(ζ) is unitary for a.e. ζ ∈ T. In other words, Θ is inner if sup Θ(z) < ∞, z∈D

where  ·  denotes the operator norm of a 2×2 matrix, and Θ−1 = Θ∗ a.e. on T. For Θ : D → M2 (C) to be an inner function, it is necessary that u = det Θ is a scalar-valued inner function. Indeed, u is a bounded analytic function on D

8.6 2 × 2 inner functions

183

whose boundary values are of unit modulus a.e. on T since the determinant of a unitary matrix must have unit modulus (the eigenvalues of a unitary matrix are of unit modulus and the determinant of a matrix is the product of its eigenvalues). The following result can be found in [87] along with several other related results, although it is stated in [80] without proof. This representation was exploited in [39] to study the characteristic function of a complex symmetric contraction. Theorem 8.32 Let u be an inner function and let a, b, c, d ∈ H ∞ . The analytic matrix-valued function   a −b Θ= (8.15) c d is unitary almost everywhere on T and satisfies det Θ = u if and only if: (i) a, b, c, d belong to Kzu ; (ii) |a|2 + |b|2 = 1 almost everywhere on T; (iii) Ca = d and Cb = c. Here C : Kzu → Kzu is the conjugation C f = f u on Kzu . Proof If Θ is unitary a.e. on T, then the scalar-valued function u = ad + bc is inner. Comparing entries in the identity Θ = (Θ∗ )−1 tells us that a = du and b = cu a.e. on T. By Proposition 5.4, we conclude that a, b, c, d ∈ Kzu , Ca = d, and Cb = c, proving (i) and (iii). Expanding the identity ΘΘ∗ = I and looking at the (1, 1) entry yields (ii). Conversely, if (i), (ii), and (iii) hold, then a.e. on T we have      2 a −b a bu |a| + |b|2 abu − abu ∗ =I = ΘΘ = bu au −b au abu − abu |a|2 + |b|2 so that Θ is unitary a.e. on T. Moreover, det Θ = ad + bc = aau + bbu = (|a|2 + |b|2 )u = u, as claimed.



We remark that (8.15) is entirely analogous to the representation of quaternions of unit modulus using 2×2 complex matrices. Recall that the quaternions H are a four-dimensional division algebra over R whose elements are of the form a + bi + c j + dk where the symbols i, j, k satisfy i2 = j2 = k2 = −1 and i j = k = − ji. The norm √ a + bi + c j + dk = a2 + b2 + c2 + d2

184

Conjugation

on H is multiplicative, in light of Euler’s famous four-square identity. It is well known that, as a division algebra, H is isomorphic to the subset of M2 (C) consisting of all matrices of the form   z −w . w z In this representation, the quaternions of unit modulus are precisely those for which |z|2 + |w|2 = 1. See the text [69] for further details. We now turn our attention to a matrix completion problem that originated in systems theory. The simplest version of the Darlington synthesis problem asks whether, given a ∈ H ∞ , it is possible to find b, c, d ∈ H ∞ such that (8.15) is inner. Theorem 8.32 tells us that a∞  1 and a ∈ Kzu , where the inner function u must equal the determinant of Θ. If a∞  1, then the following result asserts that we can produce a second function b ∈ Kzu such that |a|2 + |b|2 = 1 a.e. T [86, 87]. This is the key to our treatment of the Darlington synthesis problem. Lemma 8.33 If a ∈ Kzu and a∞  1, then there exists b ∈ Kzu such that |a|2 + |b|2 = 1 a.e. on T. Proof If a is an inner function, then we may let b = 0. So suppose that a is not an inner function. Since Ca = au, it follows that u − aCa = u(1 − |a|2 ) a.e. on T. From here we obtain u − aCa ∈ H ∞ \ {0} (note a is not inner) and so by Corollary 3.21,   2 log(1 − |a| ) dm = log |u − aCa| dm > −∞. T

T

Thus there exists an outer function h ∈ H ∞ such that |h|2 = 1 − |a|2 a.e. on T. Since the function u − aCa = u|h|2 belongs to H ∞ and has the same modulus as h2 , there exists an inner function v such that vh2 = u|h|2 . Since log |h| is integrable, h cannot vanish on a set of positive measure. Thus vh = hu holds a.e. on T. By Proposition 5.4, the function b = vh belongs to Kzu and satisfies  |a|2 + |b|2 = 1 a.e. on T. We are now ready to prove the following result of Arov [19] and Douglas and Helton [60]: Theorem 8.34 The Darlington synthesis problem with data a ∈ H ∞ has a solution if and only if: (i) a∞  1; (ii) a is pseudocontinuable of bounded type (PCBT).

8.7 Notes

185

Proof If a is PCBT, then a belongs to a model space Ku . Since Ku ⊂ Kzu by Corollary 5.9, we conclude that a ∈ Kzu . By Lemma 8.33, there exists a function b ∈ Kzu such that |a|2 + |b|2 = 1. By Theorem 8.32, it follows that the matrix-valued function   a −b Θ= Cb Ca 

is inner.

8.7 Notes 8.7.1 Darlington synthesis The paper [87] contains a detailed discussion of the solutions set to the 2 × 2 Darlington synthesis problem for various classes of upper-left corners a. Among other things, an algorithm to produce all possible solutions is given in the case where a is a rational function whose poles lie in De .

8.7.2 Real outer functions The Helson representation from (8.14) can be difficult to compute. For instance, it is a somewhat messy calculation to find a Helson representation for the Köbe function z = z + 2z2 + 3z3 + · · · , k(z) = (1 − z)2 which maps D bijectively onto C\(−∞, − 14 ]. We therefore propose a more constructive description of the functions in R+ , which originates in [85, 94]. First note that if f = If F is the inner–outer factorization of a function R+ , then     −4If (1 − If )2 F If F = · , −4 (1 − If )2 where the first term is in R+ and has the same inner factor as f and where the second is outer (Corollary 3.24) and in R+ . Thus to describe functions in R+ , it suffices to describe real outer (RO) functions. An infinite product expansion for real outer functions, in terms of Cayley-like transforms of inner functions, is obtained in [94]. One of the original motivations for the study of real outer functions stems from interest in exposed points of the unit ball b(H 1 ) of H 1 and the associated problem of characterizing rigid functions in H 1 . A function F ∈ b(H 1 ) is called exposed if there exists a real linear functional that attains its maximum at F and nowhere else on b(H 1 ). A function F ∈ H 1 is called rigid if it is determined

186

Conjugation

by its argument on T, in the sense that if G ∈ H 1 and arg F = arg G a.e. on T, then F and G are positive scalar multiples of each other. It is known that F ∈ H 1 is an exposed point of the unit ball of H 1 if and only if F1 = 1 and F is rigid [193]. If u is an inner function, then u=

1 + u (1 + u)2 = 1+u |1 + u|2

a.e. on T so that u and (1 + u)2 have the same argument a.e. on T. Thus the exposed points of b(H 1 ) must be outer functions. Moreover, an outer function F ∈ b(H 1 ) is not exposed if there exists a non-constant inner function u such that F/(1 + u)2 belongs to H 1 . This occurs if and only if there is a non-constant real outer function that is non-negative a.e. on T and multiplies F into H 1 . Some important references for this line of research are [94, 110, 114, 128, 147, 164, 168].

9 The compressed shift

We now examine the compression of the unilateral shift S to a model space Ku , resulting in the so-called compressed shift. We will discuss many of the basic properties one wants to know when introduced to a new operator: its spectrum, invariant subspaces, cyclic vectors, the C ∗ -algebra generated by the operator, its commutant, etc.

9.1 What is a compression? We begin with a discussion that will place the compressed shift in a more general setting. Let H be a separable complex Hilbert space. Given T ∈ B(H) and a subspace M of H, define the operator R : M → M,

R = PM T |M ,

where PM is the orthogonal projection of H onto M. If one decomposes T according to the orthogonal decomposition H = M ⊕ M⊥ , then T has the matrix representation  R [T ] = ∗

 ∗ . ∗

In certain situations, we have the additional property that f (R) = PM f (T )|M for all analytic polynomials f . 187

(9.1)

188

The compressed shift

Definition 9.1 When (9.1) is satisfied for all analytic polynomials f , we say that R is a compression of T to the subspace M and that T is a dilation of R to H. Remark 9.2 The operator theory literature is not always in agreement as there are several definitions of the term “compression.” Some use the term compression to mean R = PM T |M without insisting that f (R) = PM f (T )|M for all analytic polynomials f . The following theorem is due to Sarason [167]. Theorem 9.3 Let T ∈ B(H), M be a subspace of H, and R = PM T |M . Then the following are equivalent: (i) R is a compression of T ; (ii) There is a subspace N of H such that N ⊥ M, T N ⊂ N and T (M⊕N) ⊂ M ⊕ N; (iii) There are subspaces N and K of H such that the three subspaces M, N and K are mutually orthogonal, T N ⊂ N, T (M ⊕ N) ⊂ M ⊕ N, and H = M ⊕ N ⊕ K. Proof

(i) =⇒ (ii): Let M =

 {T n M : n  0}.

Clearly, M ⊂ M and T M ⊂ M . Define N = M  M, which implies that M ⊥ N and that M ⊕ N = M is invariant under T . To finish, we need to verify that T N ⊂ N. Indeed, for each x ∈ M and n  0 we have PM T (T n x) = PM T n+1 x = Rn+1 x = RRn x = RPM T n x = RPM (T n x). According to the definition of M , we obtain the identity PM T y = RPM y,

y ∈ M .

In particular, choosing any y ∈ N, we get PM T y = RPM y = 0. This implies that T y ∈ M  M = N and thus T N ⊂ N. (ii) =⇒ (iii): Let K = (M ⊕ N)⊥ . The definition of K implies that the subspaces M, N, and K are mutually orthogonal and that H = M ⊕ N ⊕ K. (iii) =⇒ (i): With respect to the orthogonal decomposition H = M⊕N ⊕K, the operator T has the matrix representation ⎡ ⎤ ⎢⎢⎢R 0 ∗⎥⎥⎥ ⎢⎢⎢ ⎥ [T ] = ⎢⎢⎢ ∗ ∗ ∗⎥⎥⎥⎥⎥ . ⎣ ⎦ 0 0 ∗

9.2 The compressed shift

189

Notice how the assumption T N ⊂ N yields the second column, and the assumption T (M ⊕ N) ⊂ (M ⊕ N) yields the third row. The R in the upper left corner comes from definition of R = PM T |M . The matrix representation above shows that for each n  1, the operator T n has the matrix representation ⎡ n ⎤ ⎢⎢⎢R 0 ∗⎥⎥⎥ ⎢ ⎥ [T n ] = ⎢⎢⎢⎢⎢ ∗ ∗ ∗⎥⎥⎥⎥⎥ ⎣ ⎦ 0 0 ∗ with respect to the orthogonal decomposition H = M ⊕ N ⊕ K. Thus the operators T and R fulfill the property PM T n |M = Rn for all n  0. By linearity we see that f (R) = PM f (T )|M for any analytic polynomial f . Thus R is the compression of T to the subspace M.  A particularly useful case of Theorem 9.3 is the following. Corollary 9.4 Let T ∈ B(H) and let M be a subspace of H such that T M⊥ ⊂ M⊥ . Then R = PM T |M is a compression of T to M. Proof

Apply Theorem 9.3 to N = M⊥ and K = {0}.



Note that R∗ = PM T ∗ |M ,

(9.2)

which will be useful later on.

9.2 The compressed shift Recall from Proposition 5.13 that the orthogonal projection Pu from L2 onto the model space Ku is given by the formula (Pu f )(λ) =  f, kλ , where kλ is the reproducing kernel for Ku . Definition 9.5

For an inner function u, the operator Su : Ku → Ku ,

Su f = Pu S f

is called the compressed shift operator. When we study the Sz.-Nagy–Foia¸s model in Section 9.4, we will see that a contraction that satisfies several simple conditions is unitarily equivalent to a corresponding compressed shift. Let us first prove several facts about Su . First and foremost is (see Definition 9.1): Theorem 9.6

Su is a compression of S .

190

Proof

The compressed shift In Corollary 9.4, let M = Ku and note that S Ku⊥ = S (uH 2 ) ⊂ uH 2 = Ku⊥ .



Since Su is a compression of S , we have the identity Sun f = Pu S n f,

f ∈ Ku , n  0.

Let us now prove some other useful facts about Su . Proposition 9.7 Proof

Su∗ f = Pu (ζ f ) = S ∗ f for all f ∈ Ku .

By (9.2) and the fact that Ku is S ∗ -invariant, we have Su∗ = Pu S ∗ f = S ∗ f

∀ f ∈ Ku .

To prove that Su∗ f = Pu (ζ f ), we first observe that ζu f ∈ (H 2 )⊥ for all f ∈ Ku . Indeed, for all n  0, ζu f, ζ n  = u f, ζ n+1  =  f, uζ n+1  = 0. The previous line says that P(ζu f ) = 0 and so, by (5.26), Pu (ζ f ) = P(ζ f ) − uP(ζu f ) = P(ζ f ) = Tz f = S ∗ f.



Next recall the conjugation C on Ku from (8.6) defined in terms of boundary values on T by C f = f zu. The following result says that the compressed shift Su is a complex symmetric operator [80, 81]. We will explore this topic in greater detail in Chapter 13. Proposition 9.8 Proof

Su = CSu∗C.

For f, g ∈ Ku , use the previous proposition and (8.1) to get CSu∗C f, g = Cg, Su∗C f  = Cg, S ∗C f  = S Cg, C f  = ζuζg, uζ f  = ζg, f  = ζ f, g = S f, g = S f, Pu g = Pu S f, g = Su f, g

which proves the desired identity.



The following operator identities, motivated by the identity I − S S ∗ = c0 ⊗ c0 on H 2 (where cλ is the Cauchy kernel), will appear several times in this book. Lemma 9.9

I − Su Su∗ = k0 ⊗ k0 and I − S u∗ S u = Ck0 ⊗ Ck0 = S ∗ u ⊗ S ∗ u.

Proof

9.2 The compressed shift

191

Pu 1 = k0 .

(9.3)

First note that

To see this, use the identity u = u(0) + zS ∗ u along with Proposition 5.14 to get   Pu 1 = 1 − uP(u) = 1 − uP u(0) + zS ∗ u = 1 − uu(0) + 0 = k0 . Next, for each f ∈ Ku we have (I − Su Su∗ ) f = f − Pu (S S ∗ f )

(Proposition 9.7)

= f − Pu ( f − f (0)) = f (0)Pu 1 =  f, k0 k0

(by (9.3))

= (k0 ⊗ k0 ) f. This proves the operator identity I − Su Su∗ = k0 ⊗ k0 .

(9.4)

To verify the second identity, we need the formula C( f ⊗ g)C = C f ⊗ Cg,

f, g ∈ Ku ,

(9.5)

which can be seen with the computation [C( f ⊗ g)C](h) = C[( f ⊗ g)(Ch)] = C[Ch, g f ] = Ch, gC f = h, CgC f

(by (8.1))

= (C f ⊗ Cg)(h). Next we note that C(I − Su Su∗ )C = CC − CSu Su∗C = I − CSuCCSu∗C = I − Su∗ Su

(Proposition 9.8)

and so I − Su∗ Su = C(I − Su Su∗ )C = C(k0 ⊗ k0 )C

(by (9.4))

= (Ck0 ) ⊗ (Ck0 ).

(by (9.5))

Finally, note from (8.8) that Ck0 = S ∗ u.



192

The compressed shift

From Theorem 6.9 recall the conjugate linear map J : Ku → Ku# defined by J f = f # , where f # (z) := f (z), and the unitary operator U : Ku → Ku# ,

U = JC.

(9.6)

The following facts can be verified by direct computation. Since we will be dealing with the two different model spaces Ku and Ku# , we will let C denote the conjugation on Ku and C # denote the conjugation on Ku# . Lemma 9.10

For an inner function u, we have the following:

(i) J −1 g = g# for all g ∈ Ku# ; (ii) JC = C # J; (iii) (JC)∗ = CJ −1 = J −1C # . Proposition 9.11

For an inner function u, the following identities hold:

(i) JSu J −1 = S u# ; (ii) JSu∗ J −1 = S u∗# ; (iii) USu U ∗ = S u∗# . First note that for all f, g ∈ Ku# we have

Proof

JSu J −1 f, g = J −1 g, S u J −1 f  = J −1 g, Pu S J −1 f  = J −1 g, ζ J −1 f   = g(ζ)ζ f (ζ) dm(ζ) T ζ f (ζ)g(ζ) dm(ζ) (by (8.5)) = T

= S f, g = S f, Pu# g = Pu# S f, g = S u# f, g, which implies that JSu J −1 = S u# . Now bring in the identities from Lemma 9.10, along with Proposition 9.8, to get USu U ∗ = (JC)Su (JC)∗ = C # JSu J −1C # = C # S u# C # = S u∗# . Finally, JSu∗ J −1 = JCSuCJ −1 = USu U ∗ = S u∗# .



We conclude this section with a proof that Sun and Su∗n both tend to zero in the strong operator topology (see Chapter 1) as n → ∞. This observation will be important when we consider the Sz.-Nagy–Foia¸s model theory (Section 9.4).

9.3 Invariant subspaces and cyclic vectors

Lemma 9.12

193

Let u be an inner function. Then for each f ∈ Ku we have lim Su∗n f  = lim Sun f  = 0.

n→∞

n→∞

Proof Since Su∗n f = S ∗n f , the first assertion follows from the fact that for each f ∈ Ku , we have  | f (k)|2 → 0 Su∗n f 2 = S ∗n f 2 = kn

as n → ∞ (Theorem 1.8). The second assertion follows from Proposition 9.11 because Sun is unitarily equivalent to S u∗n# . 

9.3 Invariant subspaces and cyclic vectors From Beurling’s Theorem (Theorem 4.3), the unilateral shift S is a cyclic operator (any outer function serves as a cyclic vector) and so is the backward shift S ∗ (any function that does not have a pseudocontinuation of bounded type serves as a cyclic vector (see Proposition 7.15)). We now focus our attention on the cyclic vectors and invariant subspaces of the compressed shift Su . Proposition 9.13 The compressed shift Su is a cyclic operator with cyclic vector k0 . That is to say,  {Sun k0 : n  0} = Ku . Proof

From Proposition 5.15 we know that  {S ∗n u : n  1} = Ku

and so   {Sun k0 : n  0} = {(CSu∗C)n k0 : n  0}  = {CSu∗nCk0 : n  0}  = C {S ∗(n+1) u : n  0}

(Proposition 9.8) (C 2 = I) (by (8.8))

= CKu = Ku . Proposition 9.14



If u and v are inner functions and v|u, then:

(i) vH 2 ∩ Ku is an Su -invariant subspace of Ku ; (ii) vH 2 ∩ Ku = vKu/v .

194

The compressed shift

Moreover, every Su -invariant subspace of Ku is of the form vH 2 ∩ Ku where v is an inner function that divides u. Proof If M is a subspace of Ku and Su M ⊂ M, then M⊥ = Ku M satisfies Su∗ M⊥ ⊂ M⊥ . Since Su∗ = S ∗ |Ku , Proposition 5.2 ensures that M⊥ = Kv for some inner function v. However, since Kv ⊂ Ku , we see that v|u (Corollary 5.9). Putting this all together we get M = Ku  Kv = vH 2 ∩ Ku . From Lemma 5.10 we see that Ku = Kv ⊕ vKu/v ,

(9.7)

which implies that vH 2 ∩ Ku = vKu/v . To see that vKu/v is Su -invariant, notice that Su∗ = S ∗ |Ku and Kv is S ∗ -invariant. From (9.7) it follows that orthogonal complement of Kv in Ku , that is, vKu/v ,  must be Su -invariant. Recall the definition of the greatest common divisor (gcd) of two inner functions from (4.5) and Corollary 4.9. Corollary 9.15

If f ∈ Ku and has inner factor ϑ and v = gcd(ϑ, u), then  {Sun f : n  0} = vH 2 ∩ Ku .

Thus f ∈ Ku is a cyclic vector for Su if and only if the inner factor of f is relatively prime to u. In particular, every outer function in Ku is cyclic for Su . Proof Set M = vH 2 ∩ Ku and observe that M is Su -invariant (Proposition 9.14). Note that ϑH 2 ⊂ vH 2 (Theorem 4.7) and so f ∈ M. Thus we have  {Sun f : n  0} ⊂ M. To show equality, we will use Theorem 1.27. Indeed, let g ∈ M and g ⊥ Sun f for all n  0. This implies 0 = g, Sun f  = g, Pu S n f  = Pu g, ζ n f  = g, ζ n f  and so g⊥

 {S n f : n  0} = ϑH 2 ⊂ vH 2 .

However, since g ∈ vH 2 , we conclude that g ≡ 0.



Corollary 9.16 The compressed shift Su is irreducible, meaning there are no proper non-trivial subspaces of Ku that are invariant for both Su and Su∗ .

9.4 The Sz.-Nagy–Foia¸s model

195

Proof If M  Ku is Su -invariant, then M = vH 2 ∩ Ku for some (nonconstant) inner function v dividing u. If M were also Su∗ -invariant, then M would be S ∗ -invariant. By Proposition 5.5, the outer factor of any function in M would also belong to M. However, vH 2 ∩ Ku contains no outer functions and so M = {0}. 

9.4 The Sz.-Nagy–Foia¸s model One of the main reasons that model spaces are worthy of study in their own right stems from the so-called model theory developed by Sz.-Nagy and Foia¸s, which shows that a wide range of Hilbert space operators can be realized concretely as restrictions of the backward shift operator to model spaces. These ideas have since been generalized in many directions (for example, de BrangesRovnyak spaces, vector-valued Hardy spaces, etc.) and we make no attempt to provide an encyclopedic account of the subject, referring the reader instead to the influential texts [27, 141, 143, 153, 186]. Instead, we present few a results to illustrate the connection between operator theory and model spaces. In the following, we let H denote a separable complex Hilbert space. If T ∈ B(H), then, by rescaling, we may assume that T is a contraction. As such, T enjoys a decomposition of the form T = K ⊕ U (see [186, p. 8] for more details) where U is a unitary operator and K is a completely nonunitary (CNU) contraction, by which we mean there does not exist a non-trivial reducing subspace for K (invariant for both K and K ∗ ) upon which K is unitary. Since the structure of unitary operators is described by the Spectral Theorem, the study of arbitrary bounded Hilbert space operators can be focused on CNU contractions. With a few additional hypotheses, one can obtain a concrete functional model for such operators [186]. The fact that the operator Su∗ = S ∗ |Ku (see Proposition 9.7) satisfies conditions (i) and (ii) of the following theorem comes from Lemma 9.12 and Lemma 9.9, respectively. Theorem 9.17 (Sz.-Nagy–Foia¸s)

If T ∈ B(H) is a contraction satisfying:

(i) T n x → 0 for all x ∈ H, that is, T n → 0 in the strong operator topology; (ii) rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1; then there exists an inner function u such that T is unitarily equivalent to S ∗ |Ku . Proof We let  denote the unitary equivalence of Hilbert spaces or their √ ∗ operators. Since the defect operator D = I − T T (which is well-defined since T is a contraction and so I − T ∗ T is a positive operator) has rank 1, we see that ran D (the range of D) is isomorphic to C so that 4 3 := ran D  H 2 . (9.8) H n1

196

The compressed shift

It follows that for each n  1 we have  )  1 1 * DT j x2 = (I − T ∗ T ) 2 T j x, (I − T ∗ T ) 2 T j x 0 jn

0 jn

=

 )

(I − T ∗ T )T j x, T j x

*

0 jn

=

   T j x, T j x − T ∗ T T j x, T j x

0 jn

=

 

T j x2 − T j+1 x2



0 jn

= x2 − T n+1 x2 . Since, by hypothesis, T n x → 0 for each x ∈ H, we conclude that  DT j x2 = x2 , x ∈ H, j0

and hence the operator Φ : H → H 2 defined by Φx = (Dx, DT x, DT 2 x, DT 3 x, . . .) is an isometric embedding of H into H 2 (here we have identified a function in H 2 with its sequence of Taylor coefficients). Since Φ is an isometry, its image is closed in H 2 and is clearly S ∗ -invariant. By Corollary 5.2, ran Φ = Ku for some u (the possibility that ran Φ = H 2 is ruled out because the argument we are about to give will show that T  S ∗ , violating the assumption that rank(I − T T ∗ ) = 1). Now observe that ΦT x = (DT x, DT 2 x, DT 3 x, . . .) = S ∗ Φx,

x ∈ H,

that is to say, the following diagram commutes: H

T H

Φ

Φ

? H2

? S∗ - 2 H

Letting U : H → Ku denote the unitary operator obtained from Φ by reducing its codomain from H 2 to Ku , it follows that UT = (S ∗ |Ku )U = Su∗ U,

9.5 Functional calculus for Su

197

that is to say, the following diagram commutes: H

T H

U

U

? Ku

S u∗ - ? Ku

Thus T is unitarily equivalent to the restriction of S ∗ to Ku .



Since Proposition 9.11 says that Su is unitarily equivalent to Su∗# , the Sz.Nagy-Foia¸s Theorem (Theorem 9.17) can be restated as follows: Theorem 9.18 (Sz.-Nagy–Foia¸s) which satisfies:

If T is a contraction on a Hilbert space

(i) T ∗n x → 0 for all x ∈ H, that is, T ∗n → 0 as n → ∞ in the strong operator topology; (ii) rank(I − T ∗ T ) = rank(I − T T ∗ ) = 1; then there exists an inner function u such that T is unitarily equivalent to Su .

9.5 Functional calculus for Su For any T ∈ B(H) one can define p(T ) ∈ B(H) for any analytic polynomial p(z) = a0 + a1 z + · · · + an zn by setting p(T ) = a0 I + a1 T + · · · + an T n . A much studied problem in operator theory is to determine how to define the operator f (T ) for other classes of functions besides polynomials. This line of inquiry falls under the broad heading of functional calculus. In particular, for an operator T (perhaps from a certain class of operators), we want to define a map f → f (T ) from a certain class of functions containing the analytic polynomials in such a way that this map is a continuous homomorphism (in some appropriate topology particular to the class of functions).

198

The compressed shift

There is the holomorphic functional calculus, which says that if f is analytic in an open neighborhood of σ(T ) then one can define f (T ) by the operatorvalued Cauchy integral formula  1 f (z)(zI − T )−1 dz, f (T ) = 2πi Γ where Γ is some appropriate system of curves surrounding σ(T ) and the integral is interpreted as an operator-valued Riemann integral. If T ∈ B(H) is self-adjoint, then there is the Borel functional calculus, where f (T ) can be meaningfully defined for any bounded Borel function on the real line. Another functional calculus comes from the Sz.-Nagy–Foia¸s theory which defines, for a contraction T , the operator f (T ) where f belongs to the disk algebra A (see Definition 5.23). For completely non-unitary contractions, one can meaningfully define f (T ) for f ∈ H ∞ . We now proceed to define the H ∞ -functional calculus for Su . For ϕ ∈ H ∞ , define the operator ϕ(Su ) := Pu T ϕ |Ku ,

(9.9)

where Tϕ is a Toeplitz operator on H 2 (Definition 4.11). More explicitly, ϕ(Su ) is given by the formula ϕ(Su )( f ) = Pu (ϕ f ),

f ∈ Ku .

(9.10)

Another widely used notation for ϕ(Su ) is in terms of the truncated Toeplitz operator Auϕ on Ku given by the formula Auϕ : Ku → Ku ,

Auϕ f = Pu (ϕ f ).

(9.11)

Clearly, ϕ(Su )  T ϕ  = ϕ∞ . The mapping Λ : H ∞ → B(Ku ),

ϕ → ϕ(Su )

is called the H ∞ -functional calculus for the operator Su . It was developed in [184, 186]. Theorem 9.19 Let u be an inner function. Then the mapping Λ is linear, multiplicative, and contractive. Furthermore, Λz = Auz = Su . Proof In the course of defining the functional calculus, we have already seen that Λ is linear and contractive. To prove that Λ is multiplicative, we proceed as follows. Let ϕ, ψ ∈ H ∞ . For g ∈ H 2 we use the formula

9.5 Functional calculus for Su

199

Pu g = g − uP(ug) from Proposition 5.14 to get, for f ∈ Ku , Auϕ Auψ f = Pu ϕ(Pu (ψ f ))   = Pu ϕ(ψ f − uP(uψ f )) = Pu (ϕψ f ) − Pu (uϕP(uψ f )) = Auϕψ f since uϕP(uψ f ) ∈ uH 2 and so Pu (uH 2 ) = 0. Thus we have Auϕ Auψ − Auϕψ ,

ϕ, ψ ∈ H ∞ ,

(9.12) 

which completes the proof. Here are some other properties of the functional calculus. Theorem 9.20

If u is inner and ϕ ∈ H ∞ , then the following assertions hold:

(i) ϕ(Su )∗ = Tϕ |Ku ;

ϕ(n)| < ∞, then (ii) If ϕ is in the Wiener algebra, that is, n0 |   ϕ(n)Sun , ϕ(Su ) = n0

where this series converges in the operator norm; (iii) ϕ(Su ) = 0 if and only if ϕ ∈ uH ∞ . Proof

To prove (i), use (9.9) to see that ϕ(Su )∗ = Pu T ϕ∗ |Ku = Pu Tϕ |Ku .

However, by Proposition 5.5, Ku is invariant under Tϕ and so ϕ(Su )∗ = Tϕ |Ku . To prove (ii), set   ϕ(n)zn , N  1. ϕN (z) = 

Then we have ϕN (Su ) =

0nN

 ϕ(n)Sun and, using Theorem 9.19, we get

0nN

     ϕ(n)Sun − ϕ(S u ) = ϕN (S u ) − ϕ(S u ) 0nN

= Λ(ϕN − ϕ)  ϕN − ϕ∞   | ϕ(n)| → 0 n>N

200

The compressed shift

as N → ∞. Notice in the above how we used the fact that for a function ψ in the Wiener algebra, the Fourier series of ψ converges uniformly on T. For the proof of (iii), clearly ϕ(Su ) = 0 if and only if ϕ(Su )∗ = 0. By (i), the latter is equivalent to Tϕ |Ku = 0. But, from Proposition 5.8, we know that ker Tϕ = Kv , where v is the inner factor of ϕ. Thus, ϕ(Su ) = 0 if and only if Ku ⊂ Kv , and this inclusion happens precisely when u|v (Corollary 5.9). Since  v is the inner factor of ϕ, we have u|v if and only if ϕ ∈ uH ∞ . Theorem 9.21 that

Let u be an inner function and ϕ ∈ H ∞ . If {ϕn }n1 ⊂ H ∞ such M = sup ϕn ∞ < ∞, n1

then the following assertions hold: (i) If lim ϕn (ζ) = ϕ(ζ),

n→∞

a.e. on T,

then ϕn (Su ) → ϕ(Su ) in the strong operator topology; (ii) If lim ϕn (λ) = ϕ(λ),

n→∞

λ ∈ D,

then ϕn (Su ) → ϕ(Su ) in the weak operator topology. Proof

If (i) holds, then, for each f ∈ Ku , ϕn (Su ) f − ϕ(Su ) f  = Pu (ϕn f ) − Pu (ϕ f )  ϕn f − ϕ f 

and

 ϕn f − ϕ f 2 =

T

|ϕn − ϕ|2 | f |2 dm.

Since, by assumption, ϕn → ϕ, a.e. on T and |ϕn (ζ) − ϕ(ζ)| | f (ζ)|  (ϕn ∞ + ϕ∞ )| f (ζ)|  2M| f (ζ)|, we can apply the Dominated Convergence Theorem to see that lim ϕn f − ϕ f  = 0.

n→∞

Thus ϕn (Su ) f → ϕ(Su ) f in norm for all f ∈ H 2 , or equivalently, ϕn (Su ) → ϕ(Su ) in the strong operator topology.

9.6 The spectrum of Su

201

To prove (ii), fix f, g ∈ Ku and observe that ϕn (Su ) f − ϕ(Su ) f, g = Pu (ϕn f ) − Pu (ϕ f ), g = ϕn f − ϕ f, Pu g = ϕn f − ϕ f, g.  = (ϕn − ϕ) f g dm. T

To finish the proof, we need a fact from [35, Prop. 2] which says that if ϕn → ϕ pointwise on D and ϕn is uniformly bounded in H ∞ -norm, then ϕn → ϕ in the weak-∗ topology of H ∞ , that is to say,   ϕn h dm → ϕh dm T

T

for every h ∈ L1 . Apply this result to h = f g ∈ L1 above to see that ϕn (Su ) →  ϕ(Su ) in the weak operator topology.

9.6 The spectrum of Su In Chapter 7 we defined the spectrum of the inner function u = Bsμ as the set σ(u) = λ ∈ D− : lim |u(z)| = 0 = (B−1 ({0}))− ∪ supp μ. z→λ

The main theorem of this section says that σ(u) = σ(Su ) [125, 140]. Theorem 9.22 (Livšic–Möller)

σ(Su ) = σ(u).

The proof of this theorem needs some preliminary discussion. Recall from Proposition 1.30 that Su is a contraction and so σ(Su ) ⊂ D− . Thus (I − λSu )−1 is a bounded operator for each λ ∈ D. Lemma 9.23

If u is an inner function and λ ∈ D, then kλ = (I − λSu )−1 k0 .

Proof From (4.15) we see that cλ = (I − λS )−1 c0 . Hence, for each f ∈ Ku , we can use (4.15) to get f (λ) =  f, cλ  =  f, (I − λS )−1 c0  = (I − λS ∗ )−1 f, c0 .

202

The compressed shift

We can now use the identity Pu c0 = k0 from Proposition 5.13 to get f (λ) = (I − λS u∗ )−1 f, c0  = Pu (I − λS u∗ )−1 f, c0  = (I − λS u∗ )−1 f, Pu c0  = (I − λS u∗ )−1 f, k0  =  f, (I − λSu )−1 k0 . This proves that kλ = (I − λSu )−1 k0 .



Proof of Theorem 9.22 We follow parts of [166]. As we have seen before, Su is a contraction and so σ(Su ) ⊂ D− (Proposition 1.30). If λ ∈ D− \ σ(u), then from Theorem 7.18 it follows that u − u(λ) ∈ H∞. u(λ)  0 and Qλ u = z−λ Furthermore, for any f ∈ Ku , −

1 u 1 u AQλ u (Su − λI) f = − A Au f u(λ) u(λ) Qλ u z−λ 1 u =− A f u(λ) Qλ u(z−λ) 1 u A f =− u(λ) u−u(λ) 1 u =− A f + Au1 f u(λ) u = 0 + f,

(Lemma 9.12)

since Auu = 0 and Au1 = I (Theorem 9.20). This means that (Su − λI) is invertible and hence λ  σ(Su ). Thus σ(Su ) ⊂ σ(u). To show σ(u) ⊂ σ(Su ), assume that Su − λI is invertible. We now argue that λ cannot be a point in D where u(λ) = 0. Indeed, by direct computation, we see that if λ ∈ D with u(λ) = 0, then kλ = cλ ∈ Ku and Su∗ cλ = S ∗ cλ = λcλ . This means that λ is an eigenvalue of Su∗ and thus λ ∈ σ(Su∗ ). Using the fact that σ(Su∗ ) = {λ : λ ∈ σ(Su )}, we see that λ ∈ σ(Su ), contradicting our assumption that Su − λI is invertible. Thus λ  σ(u) ∩ D. To show that λ  T ∩ σ(u) we proceed as follows. For η ∈ D, recall from Lemma 9.23 that kη = (I − ηSu )−1 k0 . Observe that the right-hand side of the above equation is a conjugate-analytic (vector-valued) function in a neighborhood of λ (remember that the complement of the spectrum of a bounded operator is an open set). Thus the function

9.6 The spectrum of Su

203

η → kη extends to a (vector-valued) conjugate-analytic function on an open neighborhood of λ. Finally, 1 − u(η)u(0) = kη , k0  is conjugate-analytic near λ and thus u must be analytic near λ. By Corollary 7.20, we conclude that λ  σ(u) ∩ T.  We can also determine the point spectrum σp (Su ) of Su . Corollary 9.24

For an inner function u, σp (Su ) = σ(u) ∩ D = {λ ∈ D : u(λ) = 0}.

Proof

By Lemma 9.12 we have lim Sun f  → 0,

n→∞

f ∈ Ku .

If λ ∈ σp (Su ) and Su f = λ f with  f  = 1, then |λ|n = λn f  = Sun f  → 0,

n → ∞.

Thus σp (Su ) ⊂ D. Since σp (Su ) ⊂ σ(Su ) and σ(Su ) = σ(u) (Theorem 9.22), we deduce that σp (Su ) ⊂ σ(u) ∩ D = {λ ∈ D : u(λ) = 0}. Now fix λ ∈ D such that u(λ) = 0 and set f = Qλ u (see (4.16)). Then by Theorem 9.19 and Theorem 9.20 (iii), along with the identity (z − λ) f (z) = u(z), we get (λI − Su ) f (Su ) = −u(Su ) = −Auu = 0.

(9.13)

Note that f = Qλ u = Ckλu ∈ Ku (Example 8.12). Furthermore, since f  0 and Ku ∩ uH ∞ = {0}, we conclude that f  uH ∞ and thus, by Theorem 9.20 (iii), f (Su )  0. Therefore, there is a g ∈ Ku such that h = f (Su )g ∈ Ku and h  0.  The identity (9.13) applied to g reveals that λ ∈ σp (Su ). By Corollary 9.24, we know that σp (Su ) = σ(u) ∩ D = {λ ∈ D : u(λ) = 0}

(9.14)

σp (Su∗ ) = σp (S u# ) = {λ ∈ D : u(λ) = 0}

(9.15)

and hence (Proposition 9.11). We can now identify the eigenspaces of Su∗ and Su .

204

The compressed shift

Corollary 9.25 Let u be an inner function and let λ ∈ D be such that u(λ) = 0. Then ker(Su∗ − λI) = Ckλ and ker(Su − λI) = CCkλ . Proof

Since u(λ) = 0 we have kλ = cλ ∈ Ku . Moreover, since Su∗ = S ∗ |Ku , ker(Su∗ − λI) = ker(S ∗ − λI) ∩ Ku .

But since ker(S ∗ − λI) = Ccλ , we see that ker(Su∗ − λI) = Ccλ . For the second equality, we recall the unitary operator U : Ku → Ku# from (9.6) defined by (U f )(ζ) = ζ f (ζ)u(ζ),

ζ ∈ T.

Observe that (U ∗ g)(ζ) = ζg(ζ)u(ζ).

(9.16)

We also recall Proposition 9.11, which says that USu U ∗ = Su#∗ . Since Su − λI = U ∗ (Su∗# − λI)U, we see that ker(Su − λI) = U ∗ ker(Su∗# − λI) = CU ∗ cλ . Using (9.16) and Example 8.12 we get (U ∗ cλ )(ζ) = ζcλ (ζ)u(ζ) = ζ

1 1 − ζλ

u(ζ) =

u(ζ) = Qλ u = Ckλ . ζ−λ

Finally, we compute the essential spectrum (see Chapter 1) σe (Su ) of Su . Proposition 9.26 Proof

σe (Su ) = σ(u) ∩ T.

Our proof follows [166]. We already know that σe (Su ) ⊂ σ(Su )

(Proposition 1.41)

= σ(u).

(Theorem 9.22)

If λ ∈ D, recall the difference quotient operator Qλ := S ∗ (I − λS ∗ )−1 on H 2 from (4.16). Also recall that Qλ f =

f − f (λ) , z−λ

f ∈ H2.

As we have seen before, S ∗ Ku ⊂ Ku implies Qλ Ku ⊂ Ku . Thus Rλ := Qλ |Ku = Su∗ (I − λSu∗ )−1



9.6 The spectrum of Su

205

defines a bounded operator on Ku . We also see that for f ∈ Ku ,

 f − f (λ) (Su − λI)Rλ f = Pu (z − λ) z−λ = f − f (λ)Pu 1 = f − f (λ)k0

(by (9.3))

= (I − k0 ⊗ kλ ) f. Thus (Su − λI)Rλ = I − k0 ⊗ kλ ,

(9.17)

which means that Su − λI is right invertible modulo a compact operator (recall that a finite-rank operator is compact). To show that Su − λI is left invertible (modulo a compact operator), we first note that CRλC = CSu∗ (I − λSu∗ )−1C = CSu∗CC(I − λSu∗ )−1C = Su (I − λSu )−1 = R∗λ , where C is the conjugation on Ku . A computation similar to the previous one shows that C(Su − λI)RλC = (Su∗ − λI)R∗λ .

(9.18)

From (9.17) we observe that C(Su − λI)RλC = C(I − k0 ⊗ kλ )C

(by (9.17))

= I − C(k0 ⊗ kλ )C = I − (Ck0 ) ⊗ (Ckλ ) ∗

= I − S u ⊗ Qλ u.

(by (9.5)) (by Example 8.12)

Combine this with (9.18) to conclude (Su∗ − λI)R∗λ = I − (Ck0 ) ⊗ (Ckλ ) = I − S ∗ u ⊗ Qλ u. Take adjoints of the previous line to get Rλ (Su − λI) = I − (Ckλ ) ⊗ (Ck0 ) = I − Qλ u ⊗ S ∗ u. Thus λ  σe (Su ) and so σe (Su ) ∩ D = ∅. We conclude that σe (Su ) ⊂ T. Since every λ ∈ T \ σ(u) is in the resolvent set for Su , and hence λ  σe (Su ), we will finish the proof by showing that every point of σ(u) ∩ T belongs to σe (Su ). Indeed, if λ ∈ σ(u), then Su − λI is not invertible (Theorem 9.22). However, since λ ∈ T, z − λ is an outer function (Corollary 3.24) and so S − λI has dense range as an operator on H 2 (Corollary 4.5). This implies that

206

The compressed shift

Su − λI also has dense range as an operator on Ku . By Corollary 9.24, Su − λI is injective. But since we are assuming that Su − λI is not invertible, it must be the case that Su − λI does not have closed range and thus can not be Fredholm.  Hence λ ∈ σe (Su ) (Corollary 1.40). We leave it to the reader to use the final part of the proof of the previous proposition, along with the proof of Proposition 4.1, to verify the following. Corollary 9.27

σe (S ) = T.

9.7 The C ∗ -algebra generated by Su For a family X of operators in B(H), let C ∗ (X ) denote the unital C ∗ -algebra generated by X . In other words, C ∗ (X ) is the closure, in the norm of B(H), of the unital algebra generated by the operators in X and their adjoints. Since we are frequently interested in the case where X = {A} is a singleton, we often write C ∗ (A) in place of C ∗ ({A}). The commutator ideal C (C ∗ (X )) of C ∗ (X ) is the smallest norm-closed two-sided ideal of B(H) that contains all of the commutators [A, B] := AB − BA, where A and B range over all elements of C ∗ (X ). Since the quotient algebra C ∗ (X )/C (C ∗ (X )) is an abelian C ∗ -algebra, it is isometrically ∗-isomorphic to C(Y), the set of all continuous functions on some compact Hausdorff space Y [52, Thm. 1.2.1]. We denote this relationship by C ∗ (X )  C(Y). C (C ∗ (X ))

(9.19)

The Toeplitz algebra C ∗ (S ), where S is the unilateral shift on H 2 , has been extensively studied since the seminal work of Coburn in the late 1960s [47, 48]. Indeed, the Toeplitz algebra is now one of the standard examples discussed in many well-known texts (for example, [20, Sect. 4.3; 56, Ch. V.1; 59 Ch. 7]). In this setting, we have C (C ∗ (S )) = K , the ideal of compact operators on H 2 , and Y = T, that is, C ∗ (S )/K  C(T). It also follows that C ∗ (S ) = {Tϕ + K : ϕ ∈ C(T), K ∈ K },

9.7 The C ∗ -algebra generated by Su

207

and, moreover, that each element of C ∗ (S ) enjoys a unique decomposition of the form Tϕ + K [20, Thm. 4.3.2]. We now prove the following analogue of Coburn’s work where C ∗ (S ) is replaced with C ∗ (Su ), the C ∗ -algebra generated by the compressed shift Su . For ϕ ∈ C(T) we extend our definition of a truncated Toeplitz operator (defined earlier for ϕ ∈ H ∞ when discussing the functional calculus for Su ) and set Auϕ : Ku → Ku ,

Auϕ f = Pu (ϕ f ).

We will see Auϕ when ϕ ∈ L∞ (and even sometimes ϕ ∈ L2 ) again when we discuss truncated Toeplitz operators more thoroughly in Chapter 13. Theorem 9.28

If u is an inner function, then we have the following:

(i) The commutator ideal C (C ∗ (Su )) of C ∗ (S u ) is equal to K u , the algebra of compact operators on Ku ; (ii) C ∗ (Su )/K u is isometrically ∗-isomorphic to C(σ(u) ∩ T); (iii) If ϕ ∈ C(T), then Auϕ is compact if and only if ϕ|σ(u)∩T = 0; (iv) C ∗ (Su ) = {Auϕ + K : ϕ ∈ C(T), K ∈ K u }; (v) If ϕ ∈ C(T), then σe (Auϕ ) = ϕ(σe (Auz )) = ϕ(σ(u) ∩ T); (vi) For ϕ ∈ C(T), Auϕ e = sup{|ϕ(ζ)| : ζ ∈ σ(u) ∩ T}. Remark 9.29 (i) If u is a finite Blaschke product then σ(u) ∩ T = ∅ (Proposition 7.19) and, since Ku is finite dimensional (Proposition 5.16), Auϕ is compact for every ϕ ∈ C(T). Hence in this case C ∗ (Su ) = K u = B(Ku ). Thus the only interesting cases occur when u is not a finite Blaschke product. (ii) It should also be noted that many of the statements in Theorem 9.28 can be obtained using the explicit triangularization theory developed by Ahern and Clark in [3] (also see the exposition in [141, Lec. V]). (iii) When we discuss truncated Toeplitz operators more thoroughly in Chapter 13, we will see that the symbol ϕ which defines Auϕ is not unique. In fact Auϕ = Auψ if and only if ψ − ψ ∈ uH 2 + uH 2 . So if one wanted to be more precise in statement (iv) of Theorem 9.28, one should write C ∗ (Su ) as the set of all A ∈ B(Ku ) such that A = Auϕ + K for some ϕ ∈ C(T) and some K ∈ K u . Towards a proof of Theorem 9.28, we start with two lemmas. Lemma 9.30

If ϕ ∈ C(T), then Auϕ is compact if and only if ϕ|σ(u)∩T ≡ 0.

208

The compressed shift

Proof (⇐) Suppose that ϕ|σ(u)∩T ≡ 0. Let  > 0 and pick ψ in C(T) such that ψ vanishes on an open set containing σ(u) ∩ T and ϕ − ψ∞ < . Since Auϕ − Auψ   ϕ − ψ∞ < , and the compact operators are norm closed in B(Ku ) (Proposition 1.38), it suffices to show that Auψ is compact. We will do this by proving that if { fn }n1 is a sequence in Ku that tends weakly to zero, then Auψ fn → 0 in norm (Proposition 1.37). To this end, let K denote the closure of ψ−1 (C\{0}) and observe that K ⊂ T\σ(u). By Proposition 7.21, each fn has an analytic continuation across K and so fn (ζ) =  fn , kζ  → 0 for each ζ ∈ K. Since u is analytic on a neighborhood of K we obtain | fn (ζ)| = | fn , kζ |   fn  kζ    fn  |u (ζ)| 2

1

(by Theorem 7.24) 

1 2

 sup  fn  sup |u (ξ)| < ∞ ξ∈K

n1

for each ζ in K. By the Dominated Convergence Theorem, it follows that  Auψ fn 2 = Pu (ψ fn )2  ψ fn 2 = |ψ|2 | fn |2 → 0 K

as n → ∞, whence tends to zero in norm, as desired. (⇒) Suppose that ϕ ∈ C(T) and Auϕ is compact. Let Auψ fn

κλ =

kλ kλ 

be the normalized reproducing kernels for Ku and define 2  1 − |λ|2  1 − u(λ)u(z)  2 Fλ (z) = |κλ (z)| =  .  1 − |u(λ)|2  1 − λz  Observe that Fλ (z) > 0 and 1 2π



π

−π

Fλ (eit ) dt = κλ  = 1.

Suppose ξ = e ∈ σ(u) ∩ T. By Proposition 7.19, there is a sequence {λn }n1 ⊂ D such that λn → ξ and |u(λn )| → 0. If |t − α|  δ > 0, then iα

Fλn (eit )  Cδ First we show that

1 − |λn |2 → 0. 1 − |u(λn )|2

 π   1 lim ϕ(ξ) − ϕ(eit )Fλn (eit ) dt = 0. n→∞ 2π −π

(9.20)

(9.21)

9.7 The C ∗ -algebra generated by Su

To do this, note that  π    ϕ(ξ) − 1   1 it it ϕ(e )F (e ) dt λn   2π 2π −π

209

|ϕ(ξ) − ϕ(eit )|Fλn (eit ) dt

|t−α|δ

1 + 2π

 |ϕ(ξ) − ϕ(eit )|Fλn (eit ) dt.

δ|t−α|π

The first integral can be made small by the continuity of ϕ (choosing an appropriate δ) and the fact that Fλn always integrates to one. Once δ > 0 is fixed, the second integral goes to zero by (9.20) and the Dominated Convergence Theorem. This verifies (9.21). Next we show that  ϕFλn dm = 0. (9.22) lim n→∞

T

Here is where the compactness of Auϕ becomes important. We need the fact that κλn → 0 weakly in Ku . To prove this, note that if f ∈ Ku ∩ H ∞ then G G 1 − |λn |2 1 − |λn |2 | f (λn )| = | f (λn )| | f, κλn | =   f  , ∞ kλn  1 − |u(λn )|2 1 − |u(λn )|2 which goes to zero since |λn | → 1 and |u(λn )| → 0. To see that  f, κλn  → 0 for a general f ∈ Ku , we let  > 0 be given and use the density of Ku ∩ H ∞ in Ku (Proposition 5.21) to produce a g ∈ Ku ∩ H ∞ with  f − g < . Then | f, κλn | = | f − g, κλ | + |g, κλn |   f − g κλn  + |g, κλn |   + |g, κλn | →  as n → ∞. It follows that κλn → 0 weakly in Ku . To verify (9.22), observe that    ϕF dm = |ϕκ , κ | = |ϕκ , P κ | λn λn λn λn u λn   T

= |Pu (ϕκλn ), κλn | = |Auϕ κλn , κλn |  Auϕ κλn  κλn  = Auϕ κλn . Now use the facts that Auϕ is compact and κλn → 0 weakly as n → ∞ to conclude that Auϕ κλn  → 0. This proves (9.22). Combining (9.21) with (9.22) shows that ϕ(ξ) = 0 and completes the proof of the lemma. 

210

The compressed shift

Lemma 9.31 For each ϕ, ψ ∈ C(T), the semicommutator Auϕ Auψ − Auϕψ is compact. In particular, the commutator [Auϕ , Auψ ] is compact. Proof Let p(z) = on T and note that

i

pi zi and q(z) =

Aup Auq − Aupq =



j

qj z j be trigonometric polynomials

pi qj (Auzi Auzj − Auzi+ j ).

i, j

We claim that the preceding operator is compact. Since the sums involved are finite, it suffices to prove that Auzi Auzj − Auzi+ j is compact for each pair of i, j ∈ Z. If i, j ∈ Z are of the same sign, then Auzi Auzj − Auzi+ j = 0 (9.12) is trivially compact. If i and j are of different signs, then upon relabeling and taking adjoints (if necessary), it suffices to show that if n  m  0, then the operator Auzn Auzm −Auzn−m is compact (the case n  m  0 is done by taking adjoints). In light of the fact that Auzn Auzm − Auzn−m = Auzn−m (Auzm Auzm − I), we need only show that Auzm Auzm − I is compact for each m  1. However, since Auz Auz − I has rank one (Lemma 9.9), this follows from the identity  Auz (Auz Auz − I)Auz . Auzm Auzm − I = 0m−1

Having shown that − is compact for every pair of trigonometric polynomials p and q, the desired result follows since we may uniformly approximate any given ϕ, ψ in C(T) by trigonometric polynomials (Stone– Weierstrass Theorem) and use the estimate Auϕ   ϕ∞ for any ϕ ∈ C(T).  Aup Auq

Aupq

Lemma 9.32 If u is an inner function that is not a single Blaschke factor, then k0 and Ck0 are linearly independent. Proof

Suppose Ck0 = ak0 for some a ∈ C. A little algebra will show that u=

u(0) + az 1 + azu(0)

which implies that u is a single Blaschke factor.



Proof of Theorem 9.28 Before proceeding further, let us remark that statement (iii) has already been shown (see Lemma 9.30). We first claim that   (9.23) C ∗ (Su ) = C ∗ {Auϕ : ϕ ∈ C(T)} ,

9.7 The C ∗ -algebra generated by Su

211

noting that the containment ⊂ in the preceding holds automatically. Since (Auz )∗ = Auz , it follows that Aup ∈ C ∗ (Auz ) for any trigonometric polynomial p. We then uniformly approximate any given ϕ ∈ C(T) by trigonometric polynomials to see that Auϕ ∈ C ∗ (Auz ). This establishes the containment ⊃ in (9.23). We next prove statement (i) of Theorem 9.28, that is, C (C ∗ (Su )) = K u .

(9.24)

The containment C (C ∗ (Su )) ⊂ K u follows from (9.23) and Lemma 9.31. On the other hand, Corollary 9.16 says that Su is irreducible, whence the algebra C ∗ (Su ) itself is irreducible (that is, contains no non-trivial projections). Lemma 9.9 tells us that [Su , Su∗ ] = Ck0 ⊗ Ck0 − k0 ⊗ k0 is compact. Furthermore, [Su , Su∗ ] is not the zero operator (Lemma 9.32). It follows that C ∗ (Su ) ∩ K u  {0}. We now use the general fact that any irreducible C ∗ -subalgebra of B(H) that contains a non-zero compact operator contains all of the compact operators [52, Cor. 3.16.8] to obtain K u ⊂ C (C ∗ (Su )), which establishes (9.24). We claim that C ∗ (Su ) = {Auϕ + K : ϕ ∈ C(T), K ∈ K u },

(9.25)

which is statement (iv) of Theorem 9.28 (see also Remark 9.29 (iii)). The containment ⊃ in the preceding follows because C ∗ (Su ) contains K u by (9.24) along with every element of the form Auϕ with ϕ in C(T) by (9.23). The containment ⊂ in (9.25) holds by another application of (9.23). The map γ : C(T) → C ∗ (Su )/K u defined by γ(ϕ) = Auϕ + K

u

is a homomorphism (Lemma 9.31) and hence γ(C(T)) is a dense subalgebra of C ∗ (Su )/K u by (9.23). In light of Lemma 9.30, we see that ker γ = {ϕ ∈ C(T) : ϕ|σ(u)∩T ≡ 0},

(9.26)

whence the map 3 γ : C(T)/ ker γ → C ∗ (Su )/K

u

(9.27)

defined by 3 γ(ϕ + ker γ) = Auϕ + K

u

is an injective ∗-homomorphism. By [56, Thm. I.5.5], it follows that 3 γ is an isometric ∗-isomorphism.

212

The compressed shift

Since C(T)/ ker γ  C(σ(u) ∩ T)

(9.28)

by (9.26), we get σe (Auϕ ) = σC(σ(u)∩T) (ϕ) = ϕ(σ(u) ∩ T) = ϕ(σe (Auz )), where σC(σ(u)∩T) (ϕ) denotes the spectrum of ϕ as an element of the Banach algebra C(σ(u) ∩ T). This yields statement (v). Putting (9.27) and (9.28) together shows that C ∗ (Auz )/K u is isometrically ∗-isomorphic to C(σ(u) ∩ T), which proves statement (ii). The fact that 3 γ is isometric also proves statement (vi). It remains to justify statement (iv). To this end, we will use a result of Clark to be shown in Chapter 11 (see Theorem 11.4) which asserts that for each α ∈ T, the operator Uα := Su +

α 1 − u(0)α

k0 ⊗ Ck0

(9.29)

on Ku is a cyclic unitary operator. Since Uα ≡ Su

(mod K u ),

we obtain ϕ(Uα ) ≡ Auϕ

(mod K u )

(9.30)

for every ϕ in C(T). This last fact follows because the norm on B(Ku ) dominates the quotient norm on B(Ku )/K u and since any ϕ ∈ C(T) can be uniformly approximated by trigonometric polynomials. Since K u ⊂ C ∗ (Su ), we now have C ∗ (Uα ) + K u = C ∗ (Su ), 

which yields the desired result.

9.8 Notes 9.8.1 Further references Some of the proofs in this chapter come from [95, 141, 166, 186]. More advanced ideas can be found in those references.

9.9 For further exploration

213

9.8.2 Vector-valued model spaces For n ∈ N, let HC2 n denote the vector-valued Hardy space, which consists of functions f : D → Cn that are analytic and for which  sup 0 k  1.

266

Riesz bases

This inequality shows that 

(1 − |λn |) 

n1



cn−1 (1 − |λ1 |) =

n1

1 − |λ1 | k  1,

and similarly 1 − |λk λn | = 1 − |λn | + |λn |(1 − |λk |)  (1 + cn−k ) (1 − |λk |), Since

and

n > k  1.

  w − z 2 (1 − |z|2 ) (1 − |w|2 )  = 1 −  1 − w z |1 − w z|2   |w| − |z| 2 (1 − |z|2 ) (1 − |w|2 )  = , 1 −  1 − |w| |z|  (1 − |w| |z|)2

along with 0  1 − |w| |z|  |1 − w z|, we deduce that   |w| − |z|   w − z    1 − wz   1 − |wz| . Hence,

   λk − λn    1 − λk λn 

   |λk | − |λn |  1 − |λk λn |



1 − c|n−k| 1 + c|n−k|

for all n, k  1, n  k, so that     λk − λn   1 − c j 2   > 0.  1 − λk λn  1 + cj j1 k1



kn

When the sequence {λn }n1 lies on a ray, we have the following. Proposition 12.12 Suppose {λn }n1 ⊂ D lies on a ray [0, eiθ ). Then {λn }n1 is uniformly separated if and only if it is exponential. Proof Without loss of generality, we assume that the ray is the interval [0, 1) and that 0  r1 < r2 < · · · < 1.

12.3 Uniformly separated sequences

267

One direction follows from Proposition 12.11. For the other direction, suppose {rn }n1 is uniformly separated, that is,   rk − rn    inf  1 − rk rn  > 0. n1 k1 kn

Certainly there is a δ > 0 such that rn+1 − rn  δ, 1 − rn+1 rn

n  1.

This inequality is equivalent to 1 − rn+1 

(1 − δ)(1 − rn ) , 1 + δrn

n  1.

Thus, 1 − rn+1  c(1 − rn ),

n1

with c = 1 − δ.



The next result is found in [97, p. 277–278] and connects uniformly separated sequences with Carleson measures (see Chapter 11 for more on Carleson measures). Proposition 12.13 Let {λn }n1 ⊂ D be a uniformly separated sequence with separation constant δ. Then  (1 − |λn |2 )| f (λn )|2  M f 2 , f ∈ H2, (12.10) n1

where M is a positive constant constant depending only on δ. Furthermore, if {λn }n1 ⊂ D such that (12.10) holds, then {λn }n1 is a finite union of uniformly separated sequences. Define the discrete measure μ on D by  μ= (1 − |λn |2 )δλn .

(12.11)

n1

Writing (12.10) as

 D

| f |2 dμ  M f 2 ,

f ∈ H2,

with μ as in (12.11), Proposition 12.13 says that μ is a Carleson measure for H 2 .

268

Riesz bases

12.4 The mappings Λ, V, and Γ Let C∞ = {(a1 , a2 , . . .) : aj ∈ C} be the set of all sequences of complex numbers. Fix a sequence {xn }n1 ⊂ H and define the mapping Λ : H → C∞ ,

Λx = {x, xn }n1 .

(12.12)

In other words, Λ maps x to its generalized Fourier coefficients with respect to {xn }n1 . This linear mapping has several interesting properties. For example, one can see that {xn }n1 is complete in H if and only if Λ is injective. By Proposition 12.2, we see that {xn }n1 is minimal if and only if the range of Λ contains all of the sequences en = {δkn }k1 ,

n  1.

The range of Λ is contained in C∞ . We now explore what happens when the range of Λ is contained in the smaller space 2 := 2 (N). Proposition 12.14 Let {xn }n1 ⊂ H be such that ran Λ ⊂ 2 . Then Λ is a bounded linear operator from H into 2 , and, for each x ∈ H, x, xn  = Λx, en 2 ,

n  1.

(12.13)

Proof Let {ym }m1 ⊂ H be such that ym → y for some y ∈ H and Λym → a for some a ∈ 2 . We want to show that Λy = a. Indeed, write a = {ak }k1 and fix n  1. Since  |ym , xn  − an |2  |ym , xk  − ak |2 = Λym − a22 , k1

we obtain lim ym , xn  = an .

m→∞

Because ym → y, we see that lim ym , xn  = y, xn 

m→∞

and so an = y, xn . Since a = Λy, the Closed Graph Theorem implies that Λ is a bounded operator from H into 2 . The formula in (12.13) is obtained by  taking the inner product of Λx and en .

12.4 The mappings Λ, V, and Γ

269

Note that under the hypotheses of Proposition 12.14, the operator Λ is surjective if and only if {xn }n1 is minimal. Furthermore, Λ is an isometric isomorphism between H and 2 if and only if {xn }n1 is complete and minimal. This next proposition helps us estimate the operator norm of Λ by controlling it on a dense set. Proposition 12.15 Let {xn }n1 ⊂ H. Suppose there exists a dense subset E ⊂ H and a constant M > 0 such that Λx2  Mx,

x ∈ E.

(12.14)

Then ran Λ ⊂ 2 and the preceding estimate holds for all x ∈ H. Proof Fix y ∈ H. Since E is dense in H, there is a sequence {ym }m1 ⊂ E such that ym → y. Since {ym }m1 is a Cauchy sequence in E, we see from (12.14) that {Λym }m1 is a Cauchy sequence in 2 . Therefore, Λym converges to some a ∈ 2 . The hypothesis of the proposition says that for each ym ∈ E we have Λym ∈ 2 . Thus, as seen in the proof of Proposition 12.14, we conclude that ym , xn  = Λym , en 2 ,

m, n  1.

Let m → ∞ to deduce y, xn  = a, en 2 ,

n  1.

This means that Λy = a, and thus we have Λym → Λy ∈ 2 . Finally, if we set x = ym in (12.14) and then let m → ∞, we see that (12.14) holds for all y ∈ H.  We denote the set of finitely (or compactly) supported sequences in C∞ by We introduce this class to temporarily avoid dealing with convergence issues in the definitions below. However, under some mild restrictions, all of the corresponding equations remain valid for infinite sequences. Fix {xn }n1 ⊂ H and define  Va = an xn , V : Cc∞ → H, Cc∞ .

n1

where a = {an }n1 ∈ Cc∞ . The operator V depends on the sequence {xn }n1 . We write  V for the mapping corresponding to the biorthogonal sequence { xn }n1 (if it exists). The same convention will apply for the operators U and Λ to be defined in a moment.

270

Riesz bases

If x is a finite linear combination of elements from {xn }n1 and a ∈ Cc∞ , one can check that the following identities hold:  x, xn xn , (12.15) x= n1

 a= a, en 2 en ,

(12.16)

n1

Λx =

 x, xn en ,

(12.17)

n1

 = Λx

 x, xn en ,

(12.18)

n1

Va =

 a, en 2 xn ,

(12.19)

n1

 Va =

 a, en 2 xn .

(12.20)

n1

We will systematically use the identities above in the rest of the chapter. Note that all of the summations in the identities above are actually finite sums. Therefore, under the same assumptions, and by using (12.16), (12.18), and (12.19), we have  =a ΛVa (12.21) and, by (12.15), (12.18), and (12.19),  = x. V Λx Moreover, by (12.15) and (12.19), Va, y =

(12.22)

' ( a, en 2 xn , y n1

=



a, en 2 xn , y

n1

'  ( = a, y, xn en 2 

n1

= a, Λy2 .

(12.23)

The Gram matrix of the sequence {xn }n1 ⊂ H is the infinite matrix Γ = [xm , xn ]m,n1 . Note that Γ is a self-adjoint matrix. We let  Γ denote the Gram matrix for the corresponding biorthogonal sequence { xn }n1 (if it exists). One of our goals is to explore conditions under which Γ can be interpreted as a bounded operator on 2 . For the time being, we note that for each a ∈ Cc∞ ,  ' (  Va2 = am xm , an xn = am an xm , xn , m1

n1

m1 n1

12.5 Abstract Riesz sequences

271

a ∈ Cc∞ .

(12.24)

whence Va2 = Γa, a2 , The identity (12.24), along with the identity Γ = {|Γa, a2 | : a ∈ Cc∞ , a2 = 1} (coming from the fact that Γ is self-adjoint), says that V extends to a bounded operator from 2 into H if and only if Γ is a bounded operator on 2 . Similarly, the identity (12.23) says that V extends to a bounded operator from 2 into H if and only if Λ is bounded from H into 2 . We summarize this observation as a proposition. Proposition 12.16

If any one of the following three operators

Λ : H → 2 ,

V : 2 → H,

Γ : 2 → 2 ,

is well-defined and bounded, then so are the other two. Under the hypotheses of Proposition 12.16, we see from (12.23) and (12.24) that V = Λ∗

and

Γ = V ∗ V.

(12.25)

Recall from Propositions 12.14 or 12.15 that if ΛH ⊂ 2 , we may conclude that Λ : H → 2 is a bounded operator.

12.5 Abstract Riesz sequences Definition 12.17 A sequence {xn }n1 ⊂ H is said to be a Riesz basis if it is the image of an orthonormal basis under an isomorphism, that is to say, a bounded invertible operator. As a consequence of the Riesz–Fisher theorem [158, p. 85], this is equivalent to saying that a sequence {xn }n1 is a Riesz basis if and only if there is an isomorphism U : H → 2 such that Uxn = en ,

n  1.

(12.26)

The operator U is called the orthogonalizer of {xn }n1 . Since {xn }n1 is complete, U is the unique invertible operator satisfying (12.26). A sequence {xn }n1 is called a Riesz sequence in H if it is a Riesz basis for its closed linear span. By (12.26), xn  = U −1 en   U −1 ,

n  1,

(12.27)

272

Riesz bases

and so a Riesz sequence must be bounded. Moreover, for each x ∈ H, we have Ux = {Ux, en 2 }n1 .

(12.28)

One has more flexibility with Riesz bases. For example, orthonormal sequences are not stable under small perturbations. However, if {yn }n1 is an orthonormal basis for H (and thus a Riesz basis) and the sequence {xn }n1 is such that  xn − yn 2 < 1, n1

then {xn }n1 is a Riesz basis for H [152, Sec. 86]). Proposition 12.18 Let {xn }n1 be a Riesz basis for H. Then {xn }n1 is uniformly minimal and the corresponding unique biorthogonal sequence { xn }n1 is given by  xn = U ∗ Uxn ,

n  1.

Moreover, the sequence { xn }n1 is also a Riesz basis for H with orthogonalizer  = U ∗−1 . U Proof

By the definition of U we have U ∗ Uxm , xn  = Uxm , Uxn  = em , en 2 = δmn ,

m, n  1.

Hence by Proposition 12.2, {xn }n1 is minimal and {U ∗ Uxn }n1 is its unique corresponding biorthogonal sequence. Use (12.27), the estimate  xn  = U ∗ Uxn  = U ∗ en   U ∗ ,

n  1,

and Corollary 12.7 to conclude that {xn }n1 is uniformly minimal. Finally, the bounded invertible operator U ∗−1 : H → 2 satisfies U ∗−1 xn = en ,

n  1.

Hence, by definition, the sequence { xn }n1 is a Riesz basis for H with ∗−1  orthogonalizer U . The next result establishes the connection between the operators Λ and U. Proposition 12.19 Let {xn }n1 be a Riesz basis for H and let U be its orthogonalizer. Then we have Λ = U ∗−1

and

 = U. Λ

 are both bounded invertible operators from H onto 2 . In particular, Λ and Λ

12.5 Abstract Riesz sequences

273

Proof By Proposition 12.18,  xn = U ∗ Uxn , n  1. By (12.28), we have, for each x ∈ H,  = {x, Λx xn }n1 = {x, U ∗ Uxn }n1 = {Ux, Uxn 2 }n1 = {Ux, en 2 }n1 = Ux.  = U. In particular, Λ  is a bounded invertible operator from H Thus, Λ onto 2 . Finally, by Proposition 12.18, { xn }n1 is also a Riesz basis for H with  = U ∗−1 . In the argument above, we can interchange the roles of {xn }n1 and U   { xn }n1 to conclude that Λ = U. We now provide several equivalent characterizations of Riesz bases. The following one is the most popular and is considered in several textbooks to be the definition of a Riesz basis. Proposition 12.20 Let {xn }n1 be a minimal and complete sequence in a Hilbert space H. Then {xn }n1 is a Riesz basis for H if and only if there is a positive constant M such that 2  1  2   |an |   an xn   M |an |2 (12.29) M n1 n1 n1 for all {an }n1 ∈ Cc∞ . Proof Assume that {xn }n1 is a Riesz basis and let U : H → 2 denote its orthogonalizer, that is to say, Uxn = en for n  1. For each sequence {an }n1 ∈ Cc∞ we have 2  2     2    an xn  = U −1 an en  "  an en  2 = |an |2  n1

n1

n1

n1

which proves (12.29). Now assume that (12.29) holds. This assumption can be rewritten as 1 a22  Va2  Ma22 , M 

a ∈ Cc∞ .

Hence V extends to a bounded invertible operator from 2 onto H. Since V −1 xn = en for n  1, we see that {xn }n1 is a Riesz basis for H with  orthogonalizer U = V −1 . We now provide several other characterizations of Riesz bases.

274

Riesz bases

Theorem 12.21 Let {xn }n1 be a minimal and complete sequence in H and let { xn }n1 be its unique associated biorthogonal sequence. Then the following statements are equivalent: (i) {xn }n1 is a Riesz basis for H; (ii) There are two positive constants M and M  such that   2   an xn   M |an |2 n1

and

(12.30)

n1

2     an |an |2 xn   M  n1

(12.31)

n1

for all {an }n1 ∈ Cc∞ ; (iii) { xn }n1 is complete and there are two positive constants c and c such that   2   an xn   c |an |2 (12.32) n1

and

n1

2     an |an |2 xn   c n1

(12.33)

n1

for all {an }n1 ∈ Cc∞ ;  ⊂ 2 ; (iv) ran Λ ⊂ 2 and ran Λ (v) The Gram matrices Γ and  Γ are bounded operators from 2 into itself; (vi) The Gram matrix Γ is a bounded invertible operator of 2 onto itself. Proof

The strategy of the proof is illustrated with the following diagram: ⇒ == == == == == == == ==

== == == == == == == == ⇒

(iii)

(i) =========⇒ (ii) =========⇒                    

(iv)                    

(vi) ⇐=========================== (v) (i) ⇒ (iii): By Proposition 12.18, the sequence {xn }n1 is a Riesz basis for H if and only if { xn }n1 is as well. Hence, by Proposition 12.20, we see that

12.5 Abstract Riesz sequences

275

(12.32) and (12.33) are precisely the left-hand side of the inequality (12.29) xn }n1 . for the sequences {xn }n1 and { (iii) ⇒ (iv): The inequality (12.32) can be rewritten as ca2  Va,

a ∈ Cc∞ .

This inequality and (12.22) imply that  = x  2  V Λx cΛx for all finite linear combinations of elements of the sequence {xn }n1 . There fore, by Proposition 12.15, we deduce that ΛH ⊂ 2 . This argument also applies to the biorthogonal sequence { xn }n1 and so ΛH ⊂ 2 . (i) ⇒ (ii): By Proposition 12.18, { xn }n1 is a Riesz basis for H. Thus by xn }n1 . In Proposition 12.20, the inequality (12.29) holds for both {xn }n1 and { particular, we obtain (12.30) and (12.31). (ii) ⇒ (iv): In the light of (12.19) and (12.20), the assumptions (12.30) and (12.31) can be rewritten as Va2  Ma22

and

 Va2  M  a22

for all a = {an }n1 ∈ Cc∞ . These inequalities respectively mean that V and  V ∞ 2 are bounded on Cc . Hence they extend to bounded operators from  into H.  are bounded operators from H into 2 . In By Proposition 12.16, Λ and Λ 2  ⊂ 2 . particular, we have ΛH ⊂  and ΛH  are (iv) ⇒ (v): According to Proposition 12.15, the operators Λ and Λ Γ extend bounded from H into 2 . Therefore, by Proposition 12.16, Γ and  to be bounded operators on 2 . (v) ⇒ (vi): Since Γ and  Γ are bounded operators on 2 , again Proposition  : H → 2 and V : 2 → H are bounded operators. 12.16 implies that Λ The identities (12.21) and (12.22) now show that the operator V is actually  Therefore, by a bounded invertible operator from 2 onto H with V −1 = Λ. ∗ 2 (12.25), Γ = V V is a bounded invertible operator on  . (vi) ⇒ (i): The relation (12.24) shows that Γ is a positive operator. Thus, by considering the positive invertible operator Γ1/2 , we see that there is a constant M > 0 such that 1 a22  Γa, a2  M 2 a22 , M2 

a ∈ 2 .

Hence, again by (12.24), we can say 1 a2  Va  Ma2 , M

a ∈ 2 .

276

Riesz bases

These estimates implicitly mean that V is a bounded invertible operator from 2 onto its range. Moreover, Ven = xn for n  1, and the completeness of {xn }n1 forces V2 = H. Therefore V is a bounded invertible operator from 2 onto H. At the same time, the relation Ven = xn implies that {xn }n1 is a Riesz  basis for H with orthogonalizer V −1 . This completes the proof.

12.6 Riesz sequences in KB Theorem 12.3 says that if a sequence {cλn }n1 of Cauchy kernels is minimal, then {λn }n1 must be a Blaschke sequence. Denoting the corresponding Blaschke product by B, we have cλn = kλn . In particular, since |λn | → 1 as n → ∞, we have 1 kλn  = → ∞. (1 − |λn |2 )1/2 By (12.27), this means (since a Riesz sequence must be bounded) that the sequence {kλn }n1 is never a Riesz sequence in KB . Instead, we can consider the normalized reproducing kernels, that is, (1 − |λ|2 )1/2 kλ , = kλ  1 − λz and then ask if {κλn }n1 forms a Riesz sequence. By Theorem 12.4, the unique biorthogonal sequence associated with {κλn }n1 in KB is (1 − |λn |2 )1/2 λn  Ckλn , κλn = kλn   kλn = (12.34) Bn (λn ) |λn | κλ =

where C denotes the conjugation (8.6) on KB . By (12.12), the operator Λ : KB → C∞ corresponding to {κλn }n1 is  Λ f =  f, κλn  n1  =  f, kλn /kλn  n1 = (1 − |λn |2 )1/2  f, kλn  n1 = (1 − |λn |2 )1/2 f (λn ) . (12.35) n1

 : KB → C∞ corresponding to { Similarly, by (12.34), the operator Λ κλn }n1 is given by  f =  f, Λ κλn  n1 ⎫ ⎧ ⎪ ⎪ ⎬ ⎨ (1 − |λn |2 )1/2 λn  f, Ckλn ⎪ =⎪ ⎭ ⎩ |λ | n Bn (λn ) n1

12.7 Completeness problems ⎧ ⎫ ⎪ ⎪ ⎨ (1 − |λn |2 )1/2 λn ⎬ =⎪ kλn , C f ⎪ ⎩ ⎭ |λ | n Bn (λn ) n1 ⎧ ⎫ 2 1/2 ⎪ ⎪ λn ⎨ (1 − |λn | ) ⎬ =⎪ C f, kλn ⎪ ⎩ ⎭ |λ | n Bn (λn ) n1 ⎧ ⎫ 2 1/2 ⎪ ⎪ λn ⎨ (1 − |λn | ) ⎬ =⎪ (C f )(λn )⎪ ⎩ ⎭ . |λ | n Bn (λn ) n1

277

(12.36)

Theorem 12.22 Let {λn }n1 be a Blaschke sequence of distinct points in D, let B be the corresponding Blaschke product, and let {κλn }n1 be the associated sequence of normalized reproducing kernels. Then the following statements are equivalent: (i) {κλn }n1 is a Riesz basis for KB ; (ii) {κλn }n1 is uniformly minimal; (iii) {λn }n1 is a uniformly separated sequence. Proof (i) =⇒ (ii): Theorem 12.18 says that any Riesz basis is uniformly minimal. (ii) =⇒ (iii): This is precisely the content of Theorem 12.9. (iii) =⇒ (i): By Proposition 12.13 and the fact that δ = inf |Bn (λn )| > 0, n1

we have



(1 − |λn |2 )| f (λn )|2  M f 2 < ∞

n1

and

 1 − |λn |2 M |(C f )(λn )|2  2  f 2 < ∞ ∀ f ∈ KB . 2 |Bn (λn )| δ n1

In light of (12.35) and (12.36), the preceding two inequalities imply that  map KB into 2 . Since {κλn }n1 is minimal and complete in KB , Λ and Λ  Theorem 12.21 now implies that it is a Riesz basis for KB .

12.7 Completeness problems For an inner function u and Λ a sequence of distinct points in D, when does  {kλ : λ ∈ Λ} = Ku ? KΛ := Such problems are known as completeness problems.

278

Riesz bases

By considering orthogonal complements and the reproducing property of kλ , we see that KΛ = Ku precisely when there are no functions from Ku \ {0} that vanish on Λ. We have already seen that KΛ = Ku if Λ is not a Blaschke sequence or if Λ has an accumulation point in D (Proposition 5.20). A little more thought shows that the same holds when Λ has an accumulation point in T \ σ(u) since otherwise there would exist a non-zero function in Ku whose analytic continuation vanishes on a set having a limit point in its domain (Proposition 7.21). For general Blaschke sequences, the problem becomes more delicate and we can rephrase the completeness problem in terms of kernels of Toeplitz operators on H 2 . Proposition 12.23 Suppose u is an inner function, Λ is a Blaschke sequence of distinct points in D, and B is the Blaschke product corresponding to Λ. Then KΛ  Ku

⇐⇒

ker TuB  {0}.

Proof Note that if KΛ  Ku , then there is an f ∈ Ku \ {0} that vanishes on Λ. This happens precisely when f = gB for some g ∈ H 2 \ {0}. Since f ∈ Ku , we know that u f ∈ zH 2 . This means that uBg ∈ zH 2 . Now apply the Riesz projection P to uBg to see that TuB (g) = P(uBg) = 0. For the other direction, simply reverse the argument.  Example 12.24 If u divides B = BΛ (a Blaschke product with simple zeros), then uB ∈ H ∞ and the kernel of the analytic Toeplitz operator TuB is indeed zero since it is just a multiplication operator on H 2 . So in this case, KΛ = Ku . Example 12.25 On the other extreme, if B divides u and u does not divide B, then u/B ∈ H ∞ with S ∗ (u/B)  0 and so   u u u u TuB S ∗ = TuB Tz = TuBz = P uBz = P(z) = 0. B B B B Thus KΛ  Ku .

12.8 Notes 12.8.1 References The material in this section is available in the book of Singer [181]. The Gram matrices and other operators introduced in this chapter are well known in the geometric studies of Hilbert spaces. More on this can be found in Akhiezer and Glazman [6]. The important classification Theorem 12.21 is due

12.8 Notes

279

to Bari [26] and partly to Boas [32]. A good general source for bases and frames is [40].

12.8.2 Kernels of Toeplitz operators Kernels of Toeplitz operators have been studied before and we refer the reader to the papers [66, 86, 164] for further details.

12.8.3 Completeness problems and Clark theory The original paper of Clark [46] examined kernel function completeness problems by using the Paley–Wiener theory, comparing a known orthonormal basis of eigenvectors for a Clark unitary operator with a given sequence of kernel functions. This idea was also explored by Sarason in [161].

12.8.4 Completeness problems for differential operators One can relate kernel completeness problems for model spaces with completeness theorems for solutions to differential operators such as the Schrödinger and Sturm–Liouville operators [130, 188].

12.8.5 Interpolating sequences Uniformly separated sequences are equivalently known as “interpolating sequences” for H 2 [63, 97].

12.8.6 Eigenvectors of complex symmetric operators Recall from Chapter 8 that a conjugation on a complex Hilbert space H is a conjugate-linear function C : H → H that is involutive and isometric (Definition 8.1). An operator T ∈ B(H) is called C-symmetric if T = CT ∗C. We say that T is a complex symmetric operator if there exists a conjugation C with respect to which T is C-symmetric. For example, Proposition 9.8 tells us that the compressed shift Su is a complex symmetric operator, since Su = CSu∗C, where C denotes the conjugation C f = f zu on Ku . Similarly, Su∗ = S ∗ |Ku is C-symmetric as well. Associated to each conjugation C on H is the bilinear form [x, y] = x, Cy.

(12.37)

280

Riesz bases

Indeed, since the standard sesquilinear form  · , ·  is conjugate-linear in the second position, it follows from the fact that C is conjugate-linear that [ · , · ] is linear in both positions. Although we have the Cauchy–Schwarz inequality |[x, y]|  x y, it turns out that [ · , · ] is not a true inner product since it is possible for [x, x] = 0 to hold even when x  0. Two vectors x and y are C-orthogonal if [x, y] = 0 (denoted by x ⊥C y). We say that two subspaces E1 and E2 are C-orthogonal (denoted E1 ⊥C E2 ) if [x1 , x2 ] = 0 for every x1 in E1 and x2 in E2 . With respect to the bilinear form [ · , · ], it turns out that C-symmetric operators superficially resemble self-adjoint operators. For instance, an operator T is C-symmetric if and only if [T x, y] = [x, T y] for all x, y in H. As another example, the eigenvectors of a C-symmetric operator corresponding to distinct eigenvalues are orthogonal with respect to [ · , · ], even though they are not necessarily orthogonal with respect to the original sesquilinear form  · , · . Suppose that {xn }n1 is a complete system of C-orthonormal vectors in H: [xi , xj ] = δi j .

(12.38)

In other words, suppose that {xn }n1 and {Cxn }n1 are complete biorthogonal

N cn xn sequences in H. In light of (12.38), a finite linear combination x = n=1 has coefficients given by cn = [x, xn ] and hence each such x can be recovered via the skew Fourier expansion  [x, xn ]xn , x= 1nN

where N depends on x. As an example, suppose that u is a Blaschke product whose zeros {λn }n1 are all simple. The preceding lemma tells us that the eigenvectors kλn of the C-symmetric operator Su∗ satisfy [kλi , kλj ] = 0 if i  j. This can be verified directly: [kλi , kλj ] = kλi , Ckλj  = kλi , Ckλj  = Ckλj , kλi  = Ckλj (λi ) ⎧ ⎪ ⎪ if i  j, ⎨0 =⎪ ⎪ ⎩u (λi ) if i = j. In particular, for any determination of (u (λn ))1/2 it follows that xn =

kλn (u (λn ))1/2

is a complete C-orthonormal system in Ku .

12.8 Notes

281

A key object in the study of the C-orthonormal systems is the densely defined operator     cn xn = cnCxn . A0 1nN

1nN

The following general theorem tells us precisely when a C-orthonormal system forms a Riesz basis for H [84, 88]. Theorem 12.26 If {xn }n1 is a complete C-orthonormal system in H, then the following are equivalent: (i) (ii) (iii) (iv)

{xn }n1 is a Bessel sequence with Bessel bound M; {xn }n1 is a Riesz basis with lower and upper bounds M −1 and M; A0 extends to a bounded linear operator on H satisfying A0   M; There exists an M > 0 satisfying        cn xn   M  cn xn , 1nN

1nN

for every finite sequence c1 , c2 , . . . , cN ; (v) The Gram matrix (xj , xk )j,k1 dominates its transpose: (M 2 xj , xk  − xk , xj )j,k1  0 for some M > 0; (vi) The Gram matrix G = (xj , xk )j,k1 is bounded on 2 (N) and orthogonal (GT G = I as matrices). Furthermore, G  M; (vii) The skew Fourier expansion  [x, xn ]xn n1

converges in norm for each x ∈ H and  1 x2  |[x, xn ]|2  Mx2 . M n1 In all cases, the infimum over all such M equals the norm of A0 . A non-trivial application of the above result to free interpolation in the Hardy space of the disk is described in [84]. Further information about complex symmetric operators in general can be found in [55, 80–83, 88].

13 Truncated Toeplitz operators

We have already seen truncated Toeplitz operators in Theorem 10.9 when we identified the commutant {S u } of the compressed shift S u . This led us to the operators Auϕ f = Pu (ϕ f ), where ϕ ∈ H ∞ . We also encountered them in Theorem 9.28 when we identified the C ∗ -algebra generated by S u , which led us to the operators Auϕ , where ϕ ∈ C(T). In this chapter, we consider the class of truncated Toeplitz operators Auϕ with symbols ϕ ∈ L2 . For these operators, several interesting technical difficulties arise. First, the symbol ϕ for Auϕ is never unique. Indeed, there are many symbols that represent Auϕ . Second, there are bounded operators Auϕ for ϕ ∈ L2 for which there is no bounded symbol ψ for which Auϕ = Auψ . Though initially disheartening, there will be a wonderful structure to this class of operators and, since the field is relatively new, there is a lot of work to be done. To give these operators some context, the reader might want to review our discussion of Toeplitz operators from Chapter 4. Finally, we point out that many of the proofs in the early part of this chapter are based on Sarason’s original paper [166].

13.1 The basics Definition 13.1 A truncated Toeplitz operator Auϕ with symbol ϕ ∈ L2 on the model space Ku is the operator initially defined on the dense linear manifold Ku ∩ H ∞ (Proposition 5.21) by the formula Auϕ f := Pu (ϕ f ), where Pu is the orthogonal projection of L2 onto Ku (Proposition 5.13). 282

(13.1)

13.1 The basics

283

In certain situations, one can enlarge the initial domain of definition Ku ∩ H ∞ . For example, when ϕ ∈ L∞ we can define Auϕ on all of Ku since ϕ f ∈ L2 for all f ∈ Ku and so Pu (ϕ f ) ∈ Ku . Note that Auz = S u is the compressed shift operator from Chapter 9 while Auz = S ∗ |Ku is the restriction of the backward shift to Ku (equivalently S u∗ ). Definition 13.2 Let Tu denote the set of all truncated Toeplitz operators, initially defined on Ku ∩ H ∞ , that extend to be bounded operators on Ku . The operator Auϕ can alternatively be understood as follows. By Proposition 5.13, the orthogonal projection Pu g of g ∈ L2 onto Ku is given by the formula  gkλu dm, λ ∈ D. (Pu g)(λ) = T



For each λ ∈ D, we have ∈ H . For fixed ζ ∈ T, the function λ → kλu (ζ) is a conjugate-analytic function on D. Thus the preceding integral formula still makes sense, and defines an analytic function on D, even when g belongs to the larger space L1 . Hence for any ϕ ∈ L2 , we have ϕ f ∈ L1 for all f ∈ Ku . Thus we can define the linear transformation kλu

Auϕ : Ku → O(D) (here O(D) denotes the analytic functions on D) by the integral formula  ϕ f kλu dm, f ∈ Ku , λ ∈ D. (13.2) (Auϕ f )(λ) := T

The Cauchy–Schwarz Inequality yields |(Auϕ f )(λ)|   f  ϕ kλu ∞ ,

λ ∈ D.

Proposition 13.3 Let u be inner and ϕ ∈ L2 . If Auϕ f ∈ Ku for every f ∈ Ku , then Auϕ ∈ Tu . Furthermore, when Auϕ ∈ Tu , the definitions in (13.1) and (13.2) for Auϕ coincide. Proof To show that the operator defined by (13.2) is bounded, we will use the Closed Graph Theorem (Theorem 1.28). Suppose { fn }n1 ⊂ Ku with fn → f in Ku and Auϕ fn → g in Ku . Then fn → f weakly in L2 and so for each λ ∈ D, we use the fact that kλu ∈ L∞ to get (Auϕ fn )(λ) = ϕ fn , kλu  =  fn , ϕkλu  →  f, ϕkλu  = ϕ f, kλ  = (Auϕ f )(λ). Since Auϕ fn → g in Ku we conclude that Auϕ fn → g pointwise on D (Proposition 3.4). Thus Auϕ f = g. By the Closed Graph Theorem, Auϕ ∈ Tu .

284

Truncated Toeplitz operators

For the proof of the second part of the theorem, observe that the (bounded) operators defined by the formulas in (13.1) and (13.2) agree on the dense set  Ku ∩ H ∞ and therefore must be equal. At this point, the reader might be confused as to why we bother defining Auϕ for ϕ ∈ L2 (and possibly unbounded) when one can define Auϕ everywhere on Ku if ϕ ∈ L∞ . We did not go through all this trouble when defining Toeplitz operators on H 2 . As we will see in a moment, there are some substantial differences between Toeplitz operators on H 2 and truncated Toeplitz operators on Ku . For Toeplitz operators, the symbol is unique in that Tϕ1 = Tϕ2 if and only if ϕ1 = ϕ2 almost everywhere. Furthermore, one can show that for a symbol ϕ ∈ L2 , the densely defined operator Tϕ f = P(ϕ f ) on H ∞ has a bounded extension to H 2 if and only if ϕ ∈ L∞ . For truncated Toeplitz operators, the symbol is never unique (Theorem 13.6). Moreover, there are Auϕ with ϕ ∈ L2 that extend to be bounded operators on Ku but for which there is no bounded symbol that represents Auϕ . The skeptical reader might still question the need to consider truncated Toeplitz operators with unbounded symbols. From Corollary 4.18, the set of bounded Toeplitz operators on H 2 forms a weakly closed linear space in B(H 2 ). To have the same result for truncated Toeplitz operators (Theorem 13.11), we need to include such unusual operators. Finally, we are deliberately using the term truncated Toeplitz operator and not compressed Toeplitz operator since, though the formula Auϕ f = Pu (ϕ f ) suggests a compression of the Toeplitz operator Tϕ to Ku , we are not assuming the extra condition (Auϕ )n f = Pu (ϕn f ) for all n ∈ N needed to make Auϕ a true compression of Tϕ to Ku (see Remark 9.2). For certain ϕ, for example if ϕ ∈ H ∞ , then Auϕ is indeed a compression of Tϕ to Ku . However, for general ϕ, this is not always the case since the product of two truncated Toeplitz operators is not always a truncated Toeplitz operator. Example 13.4 When u = zn , the set {1, z, . . . , zn−1 } is an orthonormal basis for Ku . Moreover, any operator in Tu , when represented with respect to this basis, yields a Toeplitz matrix. Conversely, any n × n Toeplitz matrix gives rise to a truncated Toeplitz operator on Ku . Indeed, for 0  j, k  n − 1, Auϕ z j , z k  = Pu (ϕz j ), z k  = ϕz j , Pu z k  = ϕz j , z k  = ϕ, zk− j  = ϕ(k − j)

13.1 The basics

and so the matrix representation {1, z, . . . , zn−1 } is the Toeplitz matrix ⎡ ⎢⎢⎢   ϕ(0) ϕ(−1)  ϕ(−2) ⎢⎢⎢⎢ ⎢⎢⎢   ϕ(0)  ϕ(−1) ⎢⎢⎢ ϕ(1) ⎢⎢⎢ .. ⎢⎢⎢  .  ϕ(2) ϕ(1) ⎢⎢⎢ .. .. .. ⎢⎢⎢ . . . ⎢⎢⎢ ⎢⎢⎢ . . ⎢⎢⎢ .. .. ⎢⎢⎣  ϕ(n − 1) ··· ···

285

[Auϕ ] of Auϕ with respect to the basis ⎤  ϕ(−n + 1)⎥⎥⎥ ⎥⎥⎥ .. ⎥⎥⎥ . ⎥⎥⎥ ⎥⎥⎥ .. .. ⎥⎥⎥ . . ⎥⎥⎥ . ⎥⎥   ϕ(−1) ϕ(−2) ⎥⎥⎥⎥ ⎥⎥⎥ ⎥   ϕ(1)  ϕ(0) ϕ(−1) ⎥⎥⎥⎥⎥ ⎦   ϕ(2)  ϕ(1) ϕ(0) ··· .. . .. . .. .

···

(13.3)

It is also worth pointing out in this particular case that since only a finite number of Fourier coefficients are involved, we have Auϕ ∈ Tu for all ϕ ∈ L2 . There are some similarities between truncated Toeplitz operators on Ku and Toeplitz operators on H 2 . Here is one example (see Proposition 4.14 for comparison). Proposition 13.5 Proof

If Auϕ ∈ Tu , then Auϕ ∈ Tu and (Auϕ )∗ = Auϕ .

For f, g ∈ Ku ∩ H ∞ , we have Auϕ f, g = Pu (ϕ f ), g = ϕ f, g =  f, ϕg = Pu f, ϕg =  f, Pu (ϕg) =  f, Auϕ g.

This proves that Auϕ ∈ Tu and (Auϕ )∗ = Auϕ .



Another similarity is that Tu forms a linear space since u , αAuϕ + βAuψ = Aαϕ+βψ

α, β ∈ C, ϕ, ψ ∈ L2 .

There are, however, many differences between Toeplitz operators and truncated Toeplitz operators. For instance, the symbol ϕ of a Toeplitz operator Tϕ is unique (Corollary 4.13). For truncated Toeplitz operators, the story is much different. Theorem 13.6 (Sarason)

A truncated Toeplitz operator Auϕ is identically zero

if and only if ϕ ∈ uH 2 + uH 2 . Consequently, Auϕ1 = Auϕ2 if and only if ϕ1 − ϕ2 ∈ uH 2 + uH 2 .

286

Truncated Toeplitz operators

Before getting to the proof, let us decompose L2 as L2 = H 2 ⊕ ζH 2 = uH 2 ⊕ Ku ⊕ ζH 2 and observe that uKu ⊂ ζH 2 .

(13.4)

Indeed, for any f ∈ Ku , we have 0 =  f, uh = u f, h for all h ∈ H 2 . Thus u f ∈ (H 2 )⊥ = ζH 2 . Proof of Theorem 13.6 Suppose that ϕ = uh1 + uh2 , where h1 , h2 ∈ H 2 . Then, for f ∈ Ku ∩ H ∞ , ϕ f = uh1 f + uh2 f. The first term belongs to uH 2 and so Pu (uh1 f ) = 0. By (13.4), the second term belongs to zH 2 and thus Pu (uh2 f ) = 0. Hence Auϕ f = Pu (ϕ f ) = Pu (uh1 f ) + Pu (uh2 f ) = 0. For the other direction, we need to recall two facts. The first is the pair of operator identities I − S u S u∗ = k0 ⊗ k0 ,

I − S u∗ S u = Ck0 ⊗ Ck0

from Lemma 9.9. The second is the fact that if ψ ∈ H 2 , one can prove a version of (9.12) to show that Auψ commutes with S u on H ∞ ∩ Ku . This will be true whether or not Auψ extends to be bounded on Ku . A similar argument will show that Au commutes with S u∗ on Ku ∩ H ∞ . ψ

Write the symbol ϕ ∈ L2 for Auϕ ∈ Tu as ϕ = ψ + χ, where ϕ, χ ∈ H 2 . Note that this decomposition is not unique since there is an additive constant involved. Since we are assuming that Auϕ is the zero operator, this means that Auψ = −Auχ . Using the fact, from the discussion in the previous paragraph, that Auψ S u = S u Auψ ,

(13.5)

we also see, from the identities Auψ = −Auχ and Auχ S u∗ = S u∗ Auχ , that Auψ and Auχ commute with both S u and S u∗ . Thus Auψ (k0 ⊗ k0 ) = Auψ (I − S u S u∗ ) = (I − S u S u∗ )Auψ = (k0 ⊗ k0 )Auψ . Evaluating the operator identity Auψ (k0 ⊗ k0 ) = (k0 ⊗ k0 )Auψ at k0 yields k0 2 Auψ k0 = Auψ k0 , k0 k0

(13.6)

13.2 A characterization

287

and so Auψ k0 = ck0 for some constant c ∈ C. From here we have 0 = (Auψ − cI)k0 = Pu ((ψ − c)(1 − u(0)u)) = Pu (ψ − c) − u(0)Pu ((ψ − c)u) = Pu (ψ − c) − 0. Thus ψ − c ∈ uH 2 and so Auψ = cI, which also implies that Auχ = −cI. Now repeat a version of the same argument to see that χ+c ∈ uH 2 and so χ+c ∈ uH 2 . Therefore ϕ = (ψ − c) + (χ + c) ∈ uH 2 + uH 2 , 

which completes the proof. Corollary 13.7

If A ∈ Tu , there are ψ, χ ∈ Ku such that A = Auψ+χ .

Proof Write the symbol ϕ ∈ L2 for A = Auϕ ∈ Tu as ϕ = ϕ1 + ϕ2 , where ϕ1 , ϕ2 ∈ H 2 . Now define ψ = Pu ϕ1 and χ = Pu ϕ2 . Then ϕ1 − ψ and ϕ2 − χ both belong to uH 2 . Furthermore, ϕ = ψ + χ + (ϕ1 − ψ) + (ϕ2 − χ). Theorem 13.6 ensures that Auϕ1 −ψ = Auϕ

2 −χ

= 0 and so Auϕ = Auψ+χ .



Theorem 13.6 implies that the symbol ϕ for Auϕ actually belongs to a set of symbols: ϕ + uH 2 + uH 2 . Thus, even when ϕ ∈ L∞ , there are unbounded symbols ψ for which Auϕ = Auψ . This leads to a natural question: if Auϕ is bounded, does there exist a ψ ∈ L∞ for which Auϕ = Auψ ? We will address this issue in the end notes of this chapter.

13.2 A characterization The Brown–Halmos Theorem (Theorem 4.16) says that T ∈ B(H 2 ) is a Toeplitz operator if and only if T = S T S ∗ , where S is the unilateral shift on H 2 . The following is the truncated Toeplitz operator analogue of this result, where the compressed shift S u plays the role of S .

288

Truncated Toeplitz operators

Theorem 13.8 (Sarason) A bounded operator A on Ku belongs to Tu if and only if there are ψ, χ ∈ Ku such that A = S u AS u∗ + ψ ⊗ k0 + k0 ⊗ χ, in which case, A = Auψ+χ . Proof

First let us prove the identity Auψ+χ − S u Auψ+χ S u∗ = ψ ⊗ k0 + k0 ⊗ χ,

ψ, χ ∈ Ku .

(13.7)

Since ψ ∈ Ku , we have Auψ k0 = ψ.

(13.8)

Indeed, Auψ k0 = Pu ((1 − u(0)u)ψ) = Pu (ψ) − u(0)Pu (uψ) = ψ. We now compute Auψ+χ − S u Auψ+χ S u∗ as follows: Auψ+χ − S u Auψ+χ S u∗ = (Auψ + Auχ ) − S u (Auψ + Auχ )S u∗ = (Aψ − S u Aψ S u∗ ) + (Aχ − S u Aχ S u∗ ) = (Auψ − Auψ S u S u∗ ) + (Auχ − S u S u∗ Auχ ) =

Auψ (I

=

Auψ (k0



S u S u∗ )

+ (I −

(by (13.5), (13.6))

S u S u∗ )Auχ

⊗ k0 ) + (k0 ⊗ k0 )Auχ

(Lemma 9.9)

= (Auψ k0 ) ⊗ k0 + k0 ⊗ (Auχ k0 )

(Proposition 1.32)

= ψ ⊗ k0 + k0 ⊗ χ.

(by (13.8))

This verifies (13.7). Next we want to prove, for fixed ψ, χ ∈ Ku , that    f, S un k0 S un ψ, g +  f, S un χS un k0 , g Auψ+χ f, g =

(13.9)

n0

for every f, g ∈ Ku ∩ H ∞ . To see this, use (13.7) along with Proposition 1.32 to obtain the identities S un Auψ+χ S u∗n − S un+1 Auψ+χ S u∗(n+1) = (S un ψ ⊗ S un k0 ) + (S un k0 ⊗ S un χ),

n  0.

Now sum both sides of the equation above from n = 0 to n = N (and use a telescoping series argument) to get    (S un ψ ⊗ S un k0 ) + (S un k0 ⊗ S un χ) + S uN+1 Auψ+χ S u∗(N+1) . (13.10) Auψ+χ = 0nN

13.2 A characterization

289

For any f, g ∈ Ku ∩ H ∞ we obtain, using (13.5) and (13.6), S uN Auψ+χ S u∗N f, g = S u∗N f, S u∗N Auψ g,  + S u∗N Auχ f, S u∗N g. Since S ∗N → 0 in the strong operator topology (Lemma 9.12), both of the terms on the right-hand side of the preceding identity approach zero as N → ∞. The identity in (13.9) now follows by first taking inner products in (13.10) and then taking limits as N → ∞. We are now ready for the main body of the proof. Indeed, one direction follows directly from the identity in (13.7). For the other direction, assume that A = S u AS u∗ + ψ ⊗ k0 + k0 ⊗ χ for some ψ, χ ∈ Ku . We then apply the proof of (13.9) to obtain  A= ((S un ψ ⊗ S un k0 ) + (S un k0 ⊗ S un χ)) + S uN+1 AS u∗(N+1) . 0nN

Again use the fact that S u∗N → 0 in the strong operator topology to see that   A= (13.11) (S un ψ ⊗ S un k0 ) + (S un k0 ⊗ S un χ) , n0

where the series in (13.11) converges in the strong operator topology. To finish,  use (13.9) to see that the right-hand side of (13.11) is equal to Auψ+χ . Remark 13.9

Note that

Auψ−ck0 = Auψ − cAu1−u(0)u = Auψ − cI − u(0)Auu = Auψ − cI and Auχ+ck = Auχ + cAu1−u(0)u = Auχ + cI. 0

Thus Auψ+χ = Au

(ψ−ck0 )+(χ+ck0 )

,

which means, if desired, we can adjust the value of either ψ or χ at the origin.  Example 13.10 Let u = z3 and note that Ku = {1, z, z2 } (Proposition 5.16). From (13.3), the truncated Toeplitz operators can be viewed as Toeplitz matrices. Suppose that A is a generic 3 × 3 Toeplitz matrix ⎡ ⎤ ⎢⎢⎢a d e ⎥⎥⎥ ⎢⎢⎢⎢b a d⎥⎥⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ c b a

290

Truncated Toeplitz operators

The compressed shift S u has the matrix representation ⎡ ⎤ ⎢⎢⎢0 0 0⎥⎥⎥ ⎢⎢⎢ ⎥ S u = ⎢⎢⎢1 0 0⎥⎥⎥⎥⎥ ⎣ ⎦ 0 1 0 and a computation will show that

⎡ ⎢⎢⎢−a −d ⎢ S u AS u∗ − A = ⎢⎢⎢⎢⎢−b 0 ⎣ −c 0

⎤ −e⎥⎥⎥ ⎥ 0 ⎥⎥⎥⎥⎥ . ⎦ 0

With respect to the basis {1, z, z2 }, the function k0 = 1 can be written vector (1, 0, 0). Furthermore, one can check the identities ⎤ ⎡ ⎡ ⎢⎢⎢z1 0 0⎥⎥⎥ ⎢⎢⎢z1 z2 ⎥ ⎢ ⎢ (z1 , z2 , z3 ) ⊗ (1, 0, 0) = ⎢⎢⎢⎢⎢z2 0 0⎥⎥⎥⎥⎥ , (1, 0, 0) ⊗ (z1 , z2 , z3 ) = ⎢⎢⎢⎢⎢ 0 0 ⎦ ⎣ ⎣ z3 0 0 0 0

as the ⎤ z3 ⎥⎥⎥ ⎥ 0 ⎥⎥⎥⎥⎥ . ⎦ 0

Hence S u AS u∗ − A = (− a2 , −b, −c) ⊗ (1, 0, 0) + (1, 0, 0) ⊗ (− 2a , −d, −e). Describing this in terms of elements of Ku , we have the formula S u AS u∗ = A + (− 2a − bz − cz2 ) ⊗ 1 + 1 ⊗ (− 2a − dz − ez2 ). Notice the rank-two perturbation of A in the formula above. Recall from Theorem 4.18 that the set of Toeplitz operators forms a linear space that is closed in the weak operator topology. The same is true for the truncated Toeplitz operators. Theorem 13.11 (Sarason)

Tu is closed in the weak operator topology.

Proof Suppose that a net {Auα }α∈I ⊂ Tu converges in the weak operator topology to A ∈ B(Ku ). By Theorem 13.8, there are ψα , χα ∈ Ku for which Auα − S u Auα S u∗ = (ψα ⊗ k0 ) + (k0 ⊗ χα ).

(13.12)

By Remark 13.9, we can assume that χα (0) = 0 for every α ∈ I. Hence inserting k0 into (13.12), we see that Auα k0 − S u Auα S u∗ k0 = k0 2 ψα . This implies that the net ψα converges weakly to ψ=

1 (Ak0 − S u AS u∗ k0 ) ∈ Ku . k0 

13.3 C-symmetric operators

291

Furthermore, the net of rank-one operators ψα ⊗ k0 converges in the weak operator topology to ψ ⊗ k0 . Using this fact and (13.12), we see that χα converges weakly to some χ ∈ Ku and hence the net k0 ⊗χα converges in the weak operator topology to k0 ⊗ χ. Putting this all together, we obtain A − S u AS u∗ = (ψ ⊗ k0 ) + (k0 ⊗ χ), whence, by Theorem 13.8, A ∈ Tu .



13.3 C-symmetric operators Recall from Chapter 8 that a conjugation on a complex Hilbert space H is a map C : H → H that is conjugate-linear (C(ax + y) = aCx + Cy), involutive (C 2 = I), and isometric (Cx, Cy = y, x). We say T ∈ B(H) is C-symmetric if T = CT ∗C and complex symmetric if there exists a conjugation C with respect to which T is C-symmetric. For example, the conjugation (C f )(x) = f (1 − x) on L2 [0, 1] from Example 8.4 satisfies CVC = V ∗, where  x 2 2 f (t)dt, V : L [0, 1] → L [0, 1], (V f )(x) = 0

is the classical Volterra operator. Thus V is a complex symmetric operator. With the Toeplitz conjugation C on Cn defined by C(z1 , . . . , zn ) = (zn , . . . , z1 ) from Example 8.3, one can show that T = CT ∗C for any Toeplitz matrix T . Thus any Toeplitz matrix defines a complex symmetric operator on Cn . Let C denote the conjugation (8.6) on Ku . The next result from [80] says that truncated Toeplitz operators are complex symmetric operators. We have already seen a version of this for the special truncated Toeplitz operator Auz = S u , the compressed shift (Proposition 9.8). Theorem 13.12 Proof

For any A ∈ Tu , we have A = CA∗C.

Recall that C f = ζ f u on T and so for f, g ∈ Ku ∩ H ∞ we obtain CAuϕC f, g = Cg, AuϕC f  = Cg, Pu (ϕC f ) = Cg, ϕC f  = ζgu, ϕζ f u = g, ϕ f  = ϕ f, g

292

Truncated Toeplitz operators = ϕ f, Pu g = Pu (ϕ f ), g = Auϕ f, g = (Auϕ )∗ f, g.



It is suspected that the truncated Toeplitz operators might serve as some sort of model operator for various classes of complex symmetric operators. See the end notes of this chapter for further references. For the moment, let us note that the matrix representation of a truncated Toeplitz operator Auϕ with respect to a modified Clark basis (see Example 8.15) is complex symmetric (self-transpose). This was first observed in [80] and developed further in [78].

13.4 The spectrum of Auϕ The Livšic–Möller Theorem (Theorem 9.22) characterized the spectrum of the compressed shift S u = Auz as the spectrum (see Proposition 7.19) σ(u) = λ ∈ D− : lim |u(z)| = 0 z→λ

of the inner function u. The following result of P. Fuhrmann [76, 77] generalizes this for the analytic truncated Toeplitz operators {Auϕ : ϕ ∈ H ∞ }. Theorem 13.13

Let u be inner and ϕ ∈ H ∞ . Then   σ(Auϕ ) = λ ∈ C : inf |u(z)| + |ϕ(z) − λ| = 0 . z∈D

Proof Let ψ ∈ H ∞ and observe, by the H ∞ -functional calculus for the compressed shift (Theorem 9.20), that (Auψ )∗ = Tψ |Ku . For each w ∈ D, we can write the reproducing kernel kw for Ku as kw (z) =

1 − u(w)u(z) = cw (z) + u(w)u(z)cw (z). 1 − wz

Combine these two facts to get (Auψ )∗ kw = Aψu kw = P(ψkw ) = P(ψcw ) − u(w)P(ψucw ) = Tψ cw − u(w)P(ψucw ) = ψ(w)cw − u(w)P(ψucw ).

(by Proposition 4.19)

13.4 The spectrum of Auϕ

293

Thus (Auψ )∗ kw   |ψ(w)| cw  + |u(w)| ψ∞ cw   (|ψ(w)| + |u(w)|)(1 + ψ∞ )cw . Hence, with κw = kw /kw , we get 1 + ψ∞ (|ψ(w)| + |u(w)|). (Auψ )∗ κw    1 − |u(w)|2

(13.13)

Let λ ∈ C satisfy inf (|u(z)| + |ϕ(z) − λ|) = 0. z∈D

Then there is a sequence {zn }n1 ⊂ D such that u(zn ) → 0

and

ϕ(zn ) → λ.

Applying (13.13) to ψ = ϕ − λ, we have (Auϕ − λI)κw  = Auψ κw  = (Auψ )∗ κw  

 (1 + |λ|) + ϕ∞  |ϕ(w) − λ| + |u(w)| .  2 1 − |u(w)|

Thus if we let w = zn and n → ∞, we obtain inf (Auϕ − λI) f  : f ∈ Ku ,  f  = 1 = 0.

(13.14)

The previous identity says that the operator Auϕ − λI is not bounded below and hence is not invertible. Thus λ ∈ σ(Auϕ ) = σ((Auϕ )∗ ) = σ(Auϕ ), and so

λ ∈ C : inf (|u(z)| + |ϕ(z) − λ|) = 0 ⊂ σ(Auϕ ). z∈D

To prove the reverse inclusion, we will now show that inf (|u(z)| + |ϕ(z) − λ|) > 0 =⇒ λ  σ(Auϕ ). z∈D

To do this, we appeal to the Corona Theorem [97]. This deep result of Carleson ensures that when inf (|u(z)| + |ϕ(z) − λ|) > 0, z∈D

there are f, g ∈ H ∞ such that f (z)u(z) + g(z)(ϕ(z) − λ) = 1,

z ∈ D.

294

Truncated Toeplitz operators

By Theorem 9.19 (the H ∞ -functional calculus for S u ), the preceding function identity implies the operator identity Afu Auu + Agu (Auϕ − λI) = I. Since Auu = 0 (Theorem 9.20), the operator identity above reduces to Agu (Auϕ − λI) = I. But Agu and Auϕ − λI = Auϕ−λ are analytic truncated Toeplitz operators and hence  they commute (see (9.12)). Thus (Auϕ − λI)Agu = I and so λ  σ(Auϕ ). We now give an alternate description of σ(Auϕ ) in terms of cluster sets for ϕ. In the course of the proof of Theorem 13.13, we used the fact that inf (|u(z)| + |ϕ(z) − λ|) = 0 z∈D

if and only if there is a sequence {zn }n1 ⊂ D such that u(zn ) → 0 and

ϕ(zn ) → λ.

By the Bolzano–Weierstrass Theorem, we may pass to a subsequence, if necessary, and assume that {zn }n1 converges to some point z0 ∈ D− . By Proposition 7.19, we know that z0 ∈ σ(u). Certainly if z0 ∈ D, the continuity of ϕ near z0 will imply that λ = ϕ(z0 ). However, the situation becomes more complicated if z0 ∈ T since ϕ ∈ H ∞ and is not necessarily continuous near the boundary point z0 . This naturally leads us to the notion of cluster sets. The text [50] is a good source for further information about this. Let B(z, r) = {w : |w − z| < r}. For f ∈ H ∞ and ζ ∈ T, the cluster set C( f, ζ) and the range R( f, ζ) of f at ζ are defined as L f (D ∩ B(ζ, r))− C( f, ζ) = r>0

and R( f, ζ) =

L

f (D ∩ B(ζ, r)) .

r>0

One can see that C( f, ζ) is a non-empty, compact, connected subset of C. Moreover, f is continuous at ζ if and only if C( f, ζ) is a singleton. On the other hand, by the Open Mapping Theorem, R( f, ζ) is a connected Gδ subset of C (possibly empty). Observe that w ∈ C( f, ζ) if and only if there is a sequence {zn }n1 ⊂ D such that lim zn = ζ

n→∞

and

lim f (zn ) = w,

n→∞

13.4 The spectrum of Auϕ

295

while w ∈ R( f, ζ) if and only if there is a sequence {zn }n1 ⊂ D such that lim zn = ζ

and

n→∞

f (zn ) ≡ w.

This means that R( f, ζ) is the set of values assumed by f infinitely many times when we approach ζ from within D, while C( f, ζ) is the set of values that can be approximated by f as we get closer to ζ. Using this interpretation, one can show that if f is not a constant function and f has an analytic extension to a neighborhood of ζ, then C( f, ζ) = { f (ζ)} and

R( f, ζ) = ∅.

On the other extreme we have the following. Proposition 13.14 then

If u is a non-constant inner function and ζ ∈ σ(u) ∩ T,

C(u, ζ) = D− where

and

D \ E ⊂ R( f, ζ) ⊂ D

w−u is not a Blaschke product . E= w∈D: 1 − wu

We remark that E is the exceptional set for u. It is a set of logarithmic capacity zero (Remark 2.23). Proof First note that the containment R(u, ζ) ⊂ D follows from the fact that u(D) ⊂ D since u is non-constant and inner. By Theorem 2.22, there is an exceptional set E ⊂ D such that for any w ∈ D \ E, the Frostman shift w−u uw = τ1,w ◦ u = 1 − wu is a Blaschke product. We also remind the reader that the singular points of a Blaschke product (that is, those points ξ on T where the Blaschke product does not analytically continue to a neighborhood of ξ) are precisely the accumulation points of its zero set on T (Theorem 7.18 and Proposition 7.20). Since we are assuming that ζ ∈ σ(u) ∩ T, the inner function u cannot be analytically continued to a neighborhood of ζ (Proposition 7.20). Hence, for any w ∈ D, the function uw also fails to have an analytic continuation to a neighborhood of ζ, since otherwise, u = τ1,w ◦ uw would have an analytic continuation in a neighborhood of ζ. This means that ζ is also a singular point of uw and thus, for w ∈ D \ E, must be an accumulation point of the zeros of uw . Hence there is a sequence {zn }n1 ⊂ D such that zn → ζ and uw (zn ) =

w − u(zn ) = 0. 1 − w u(zn )

Therefore, w ∈ R(u, ζ), which establishes D \ E ⊂ R(u, ζ).

296

Truncated Toeplitz operators

Since E has no interior (Theorem 2.22) and C(u, ζ) is a compact subset of D− , the first assertion follows from the containment D \ E ⊂ R(u, ζ) ⊂ C(u, ζ) ⊂ D− .



The preceding result shows that a non-constant inner function assumes all values in D, except possibly those in a small exceptional subset, infinitely many times in each neighborhood of any of its singular points. Consider a Blaschke product whose zeros accumulate at all points of T, or perhaps a singular inner function with corresponding singular measure whose support is all of T. Such a function exhibits bizarre behavior at all points of T. On one hand, according to Fatou’s Theorem, this function has unimodular non-tangential limits almost everywhere. On the other hand, this function assumes nearly every value in D infinitely many times in the vicinity of any boundary point. The above discussion shows that Theorem 13.13 can be rewritten as σ(Auϕ ) = {ϕ(z) : z ∈ σ(u) ∩ D} ∪ {C(ϕ, ζ) : ζ ∈ σ(u) ∩ T}.

(13.15)

For certain ϕ, we have the following spectral mapping result. Recall from Definition 5.23 that the disk algebra A is the set of all analytic functions on D which extend to be continuous on D− . Corollary 13.15

Let u be inner and let ϕ ∈ A. Then       σ Auϕ = ϕ σ(Auz ) = ϕ σ(u) .

Proof Since ϕ is continuous on D− , we have C(ϕ, ζ) = {ϕ(ζ)}. The result now follows from (13.15).  Recall Theorem 9.28, where a similar result holds for the essential spectrum of Auϕ . The following result complements Theorem 13.13 by giving a complete description of the point spectrum of Auϕ . Theorem 13.16

Let u be inner and ϕ ∈ H ∞ . Fix λ ∈ C and set v = gcd ((ϕ − λ)i , u) ,

where (ϕ − λ)i is the inner part of ϕ − λ. Then u Kv v

(13.16)

ker(Auϕ − λI) = Kv .

(13.17)

ker(Auϕ − λI) = and

13.4 The spectrum of Auϕ

297

In particular, the following are equivalent: (i) λ ∈ σp (Auϕ ); (ii) λ ∈ σp (Auϕ );   (iii) v = gcd (ϕ − λ)i , u is not constant. Proof

Let 3 u = u/v. For f ∈ Ku we have f ∈ ker(Auϕ − λI) ⇐⇒ f ∈ ker Auϕ−λ   ⇐⇒ Pu (ϕ − λ) f = 0 ⇐⇒ u|(ϕ − λ) f ⇐⇒ 3 u| f.

Therefore, uH 2 ker(Auϕ − λI) = Ku ∩ 3 uH 2 = (K3u ⊕ 3 uKv ) ∩ 3

(Lemma 5.10)

=3 uKv , which proves (13.16). To prove (13.17), we use Theorem 13.12 to see that if C f = z f u is the conjugation operator on Ku we have CAuϕC = Auϕ . From here it follows that ker(Auϕ − λI) = C ker(Auϕ − λI) u  = C Kv v u  = C f : f ∈ Kv v u  = C zgv : g ∈ Kv v u zgvuz : g ∈ Kv = v = Kv .

(by Proposition 5.4)

The equivalence of statements (i), (ii), and (iii) now follow.



Corollary 13.17 Let u be inner and let z0 ∈ D be such that u(z0 ) = 0. For any ϕ ∈ H ∞ we have ϕ(z0 ) ∈ σp (Auϕ )

and

ϕ(z0 ) ∈ σp (Auϕ ).

(13.18)

298

Truncated Toeplitz operators

If ϕ is univalent, then   ker Auϕ − ϕ(z0 )I = CQz0 u and

  ker Auϕ − ϕ(z0 )I = Ccz0 .

(13.19)

Proof The proof of (13.18) follows directly from Theorem 13.16. To prove the first identity in (13.19), observe that since ϕ is univalent, we get (ϕ − ϕ(z0 ))i =

z − z0 . 1 − z0 z

Furthermore, the single Blaschke factor on the right-hand side of the above is an inner factor of u and so z − z0 v = gcd((ϕ − ϕ(z0 ))i , u) = . 1 − z0 z Finally, from Proposition 5.16 we have Kv = C

1 1 − z0 z

and so, by (13.16), we see that ker(Auϕ − ϕ(z0 )I) = u

1 − z0 z 1 = CQz0 u. C z − z0 1 − z0 z

One can prove the other identity in (13.19) by noting that Qz0 u = Ckz0 and kz0 = cz0 (since u(z0 ) = 0), together with the identity ker(Auϕ − ϕ(z0 )I) = C ker(Auϕ − ϕ(z0 )I).



For Blaschke products, we can be even more specific. We write BΛ for the Blaschke product corresponding to the Blaschke sequence Λ. Corollary 13.18 Let ϕ ∈ H ∞ and let B = BΛ be a Blaschke product with simple zeros. Then λ ∈ σp (AϕB )

⇐⇒

Λ = ϕ−1 ({λ}) ∩ Λ  ∅.

Moreover, for each λ ∈ σp (AϕB ), ker(AϕB − λI) = BΛ KBΛ ,

(13.20)

ker(AϕB − λI) = KBΛ ,

(13.21)

and where Λ = Λ \ Λ . In particular, for each z0 ∈ Λ,  B  B AϕB = ϕ(z0 ) z − z0 z − z0

(13.22)

13.5 An operator disintegration formula

299

and AϕB cz0 = ϕ(z0 )cz0 .

(13.23)

Proof The first part follows from Theorem 13.16 and the fact that the only divisors of a Blaschke product are its partial products. The equations (13.20) and (13.21) follow respectively from equations (13.16) and (13.17). Finally, for each z0 ∈ Λ and λ := ϕ(z0 ), we have z0 ∈ Λ . Thus, both functions kz0 = cz0 B and z−z belong to KBΛ . This observation implies (13.22) and (13.23).  0

13.5 An operator disintegration formula Here is a fascinating operator integral formula that relates truncated Toeplitz operators, the Clark unitary operators Uα (see (11.6)), and the Aleksandrov disintegration formula (Theorem 11.16). The Spectral Theorem ensures that if ϕ is a bounded Borel function on T then ϕ(Uα ) is a well-defined normal operator. Theorem 13.19 If ϕ is a bounded Borel function on T and u is inner, then Aϕu can be written as  Aϕu =

T

ϕ(Uα ) dm(α).

By this we mean that for any f, g ∈ Ku  Aϕu f, g = ϕ(Uα ) f, g dm(α). T

Proof

Let f, g ∈ Ku . Then Aϕu f, g

= Pu (ϕ f ), g = ϕ f, Pu g = ϕ f, g  = ϕ f gdm T    ϕ f gdσα dm(α) = T T    = ϕVα f Vα gdσα dm(α) T T = Mϕ Vα f, Vα gL2 (σα ) dm(α) T = Vα∗ Mϕ Vα f, Vα∗ Vα gH 2 dm(α) T = ϕ(Uα ) f, gH 2 dm(α). T

(Remark 11.17) (Theorem 11.14) (Theorem 11.4) (Theorem 11.6) 

300

Truncated Toeplitz operators

There are problems that arise when ϕ is unbounded. For example, how does one define ϕ(Uα ) in this case? The paper [166] carefully documents some of the technical difficulties that arise when ϕ is unbounded.

13.6 Norm of a truncated Toeplitz operator For Toeplitz operators, recall that Tϕ  = ϕ∞ for each ϕ ∈ L∞ . In contrast to this, we can say little more than 0  Aϕu   ϕ∞

(13.24)

for general truncated Toeplitz operators with bounded symbols. In fact, computing, or at least estimating, the norm of a truncated Toeplitz operator is a difficult problem. In a way, the problem seems unfair since, as we have seen in Theorem 13.6, there are many symbols that represent the same truncated Toeplitz operator. This makes it difficult to pose theorems about norms in terms of the symbol as one did so neatly for Toeplitz operators (and composition operators). However, it is possible to obtain lower estimates of Auϕ  for general ϕ in L2 . This can be helpful, for instance, in determining whether or not a given truncated Toeplitz operator is unbounded. Although a variety of lower bounds on Auϕ  are provided in [92], we focus here on perhaps the most useful of these. Fatou’s Theorem (Theorem 1.10) says that limr→1− P(ϕ)(rξ) = ϕ(ξ) whenever ϕ is continuous at ξ, or, more generally, whenever ξ is a Lebesgue point of ϕ. Here ξ is a Lebesgue point of ϕ if  1 |ϕ(ζ) − ϕ(ξ)| dm(ζ) = 0, lim δ→0+ m(Iδ,ξ ) I δ,ξ where Iδ,ξ is the arc of T subtended by the points e−iδ ξ and eiδ ξ. The Lebesgue Differentiation Theorem says that almost every point of T is a Lebesgue point for ϕ [158, p. 165]. Theorem 13.20

If ϕ ∈ L2 and u is inner, then Aϕu   sup{|P(ϕ)(λ)| : λ ∈ D : u(λ) = 0},

where the supremum above is regarded as 0 if u never vanishes on D. Proof The proof is similar to the one used in Proposition 4.12 to compute the norm of a Toeplitz operator. When u(λ) = 0, the corresponding

13.7 Notes

301

kernel function kλ turns out to be cλ . Now follow the rest of the proof of Proposition 4.12.  Corollary 13.21 If ϕ ∈ C(T) and u is an inner function whose zeros accumulate almost everywhere on T, then Auϕ  = ϕ∞ . At the expense of wordiness, the hypothesis of the previous corollary can be considerably weakened. We only need ζ ∈ T to be a limit point of the zeros of u, the symbol ϕ ∈ L∞ to be continuous on an open arc containing ζ, and |ϕ(ζ)| = ϕ∞ .

13.7 Notes Although we covered the basics of truncated Toeplitz operators, the subject is quickly growing [93]. Below are some topics of interest in the literature.

13.7.1 The bounded symbol problem From the estimate Auϕ   ϕ∞ we see that Auϕ ∈ Tu whenever ϕ ∈ L∞ . Howu ever, one can take any unbounded ψ ∈ uH 2 + uH 2 and observe that Auϕ = Aϕ+ψ . Thus bounded truncated Toeplitz operators always have unbounded symbols. If a truncated Toeplitz operator is a bounded operator, does it have a bounded symbol? The answer, in general, is no and a specific example was provided in [24]. There are, however, technical conditions one can impose on the inner function u to guarantee that every operator from Tu has a bounded symbol [25].

13.7.2 Finite-rank operators By Theorem 4.15, there are no non-zero compact Toeplitz operators on H 2 . In contrast to this, there are many examples of finite-rank, hence compact, truncated Toeplitz operators. In fact, the rank-one truncated Toeplitz operators were first identified by Sarason [166, Thm. 5.1] who proved that kλ ⊗ Ckλ = Auu , z−λ

Ckλ ⊗ kλ = Auu , z−λ

kζ ⊗ kζ = Aku +k −1 , ζ

ζ

302

Truncated Toeplitz operators

where λ ∈ D and u has a finite angular derivative, in the sense of Carathéodory, at ζ ∈ T. Furthermore, any truncated Toeplitz operator of rank-one is a scalar multiple of one of the above operators. We should also mention the somewhat more involved results of Sarason [166, Thms. 6.1 & 6.2] which identify a variety of natural finite-rank truncated Toeplitz operators. The full classification of the finite-rank truncated Toeplitz operators was given by Bessonov in [29].

13.7.3 Topologies on Tu Baranov, Bessonov, and Kapustin identified the predual of Tu and discussed the weak-∗ topology on Tu [25]. Consider the space   Xu := F = fn gn : fn , gn ∈ Ku ,  fn  gn  < ∞ n1

n1

with norm FXu := inf



 fn  gn  : F =

n1



fn gn .

n1

It turns out that Xu is a Banach space of functions on D and, in terms of nontangential boundary values, Xu ⊂ uzH 1 ∩ uzH 1 . Furthermore, each element of Xu can be written as a linear combination of four elements of the form f g, where f, g ∈ Ku . It turns out that Xu∗ , the dual space of Xu , is isometrically isomorphic to Tu via the dual pairing   A fn , gn , F = fn gn ∈ Xu , A ∈ Tu . (F, A) := n1

n1

Moreover, if Tu c denotes the compact truncated Toeplitz operators, then (Tu c )∗ , the dual of Tu c , is isometrically isomorphic to Xu . These results go on to say that the weak topology and the weak-∗ topology on Tu are the same. Moreover, the norm closed linear span of the rank-one truncated Toeplitz operators is Tu c and Tu c is weakly dense in Tu . Finally, the truncated Toeplitz operators with bounded symbols are weakly dense in Tu .

13.7.4 The spatial isomorphism problem For two inner functions u1 and u2 , when is Tu1 spatially isomorphic to Tu2 ? In other words, when does there exist a unitary operator U : Ku1 → Ku2

13.7 Notes

303

such that UTu1 U ∗ = Tu2 ? This is evidently a stronger condition than isometric isomorphism since one insists that the isometric isomorphism is implemented in a particularly restrictive manner. A concrete solution to the spatial isomorphism problem posed above was given in [45]. To describe its solution, let us consider several special cases. (i) Theorem 6.8 says that if ψ is a disk automorphism, then the weighted composition operator Uψ : Ku → Ku◦ψ ,

Uψ f =



ψ ( f ◦ ψ),

is unitary. Moreover, the map Λψ (A) = Uψ AUψ∗ ,

Λψ : Tu → Tu◦ψ ,

u◦ψ satisfies Λψ (Aϕu ) = Aϕ◦ψ and thus implements a spatial isomorphism between Tu and Tu◦ψ . (ii) From Theorem 6.7, recall that for each a ∈ D and disk automorphism τ1,a , one can verify that the Crofoot transform

 Ja : Ku → Kτ1,a ◦u ,

Ja f =

1 − |a|2 f, 1 − au

is unitary. Moreover, the map Λa (A) = Ja AJa∗ ,

Λa : Tu → Tτ1,a ◦u ,

implements a spatial isomorphism between Tu and Tτ1,a ◦u . (iii) From Theorem 6.9, recall the unitary operator U# : Ku → Ku# defined by [U# f ](ζ) = ζ f (ζ)u# (ζ),

u# (z) := u(z).

The map Λ# : Tu → Tu# ,

Λ# (A) = U# AU#∗ ,

#

satisfies Λ# (Aϕu ) = Au # whence Tu is spatially isomorphic to Tu# . ϕ

It turns out that any spatial isomorphism between truncated Toeplitz operator spaces can be written in terms of the three basic types described above. Indeed, for any two inner functions u1 and u2 , the spaces Tu1 and Tu2 are spatially isomorphic if and only if either u1 = ψ ◦ u2 ◦ ϕ or u1 = ψ ◦ u#2 ◦ ϕ for some disk automorphisms ϕ, ψ. Moreover, any spatial isomorphism Λ : Tu1 → Tu2 can be written as Λ = Λa Λψ or Λa Λ# Λψ , where we allow a = 0 or ψ(z) = z.

304

Truncated Toeplitz operators

13.7.5 Algebras of truncated Toeplitz operators Recall that Tu is a weakly closed subspace of B(Ku ) (Theorem 13.11). Although Tu is not an algebra, there are many interesting (weakly closed) algebras contained within Tu . In fact, the thesis [169] and subsequent paper [170] of Sedlock describes them all. We discuss the properties of these so-called Sedlock algebras below, along with several further results from [96]. For the sake of comparison, recall Theorem 4.22 which describes when the product of two Toeplitz operators is another Toeplitz operator. What is the corresponding result for Tu ? To begin, we require the following generalization (see [166, Sect. 10]) of the Clark unitary operators defined earlier in (11.6): S ua = S u +

a 1 − u(0)a

k0 ⊗ Ck0 ,

a ∈ D− .

(13.25)

These operators turn out to be fundamental to the study of Sedlock algebras. Before proceeding, let us recall a few facts from Chapter 1. For A ∈ Tu , the commutant {A} of A is defined to be the set of all B ∈ B(Ku ) such that AB = BA. The (WOT) closed linear span of {An : n  0} will be denoted by W(A). Note that W(A) ⊂ {A} and that {A} is a (WOT) closed subset of B(Ku ). The relevance of these concepts lies in the following two results from [166, p. 515] and [96], respectively. Namely that for each a ∈ D− , {S ua } ⊂ Tu

and

{S ua } = W(S ua ).

The preceding results tell us that W(S ua ) and W((S ub )∗ ), where a, b belong to D− , are weakly closed commutative algebras contained in Tu . We adopt the following notation introduced by Sedlock [170]: ⎧ a ⎪ ⎪ if a ∈ D− , ⎨W(S u ) a Bu := ⎪ ⎪ ⎩W((S u1/a )∗ ) if a ∈  C \ D− . Note that Bu0 is the algebra of analytic truncated Toeplitz operators (Bu0 = W(Azu ) = {Auz } ) while Bu∞ is the algebra of conjugate analytic truncated Toeplitz operators (Bu∞ = W(Azu ) = {Azu } ). Sedlock’s result says that the C are the only maximal abelian algebras in Tu . algebras Bua for a ∈ 

13.7.6 Norms of truncated Toeplitz operators The norm of Auϕ is related to the norm of a Hankel operator as well as to certain classical extremal problems. We refer the reader to the papers [38, 91] for more on this.

13.8 For further exploration

305

13.8 For further exploration 13.8.1 Models for complex symmetric operators The class of complex symmetric operators has undergone much study recently [39, 55, 78–83, 88–90, 98, 115, 116, 123, 191, 192, 194–196] and is deserving of further study. We have seen that every truncated Toeplitz operator is complex symmetric (Theorem 13.12). An increasingly long list of complex symmetric operators have been proven to be unitarily equivalent to truncated Toeplitz operators [93]. Is every complex symmetric operator unitarily equivalent to a truncated Toeplitz operator? If not, which ones are? One can also explore when a bounded operator is similar to a truncated Toeplitz operator. In finite dimensions, every operator is similar to a truncated Toeplitz operator [45]. What happens in infinite dimensions?

13.8.2 Operator theory for truncated Toeplitz operators Some natural operator theoretic questions about truncated Toeplitz operators (when are they normal, self-adjoint, unitary?) are beginning to be answered [37]. Others, such as a description of the spectrum and the norm, remain open.

13.8.3 Unitary equivalence When are Auϕ and Avψ unitarily equivalent? This problem was solved for compressed shifts Su = Auz and S v = Avz (see the end notes for Chapter 9). What happens in general? Be aware that problems like this could be tricky to answer due to the fact that there is an entire family of symbols that represent the operator (Theorem 13.6). In either case, the answer to these types of operator theoretic questions about truncated Toeplitz operators should depend on the relationship between the inner function (which defines the space) and the family of symbols (which define the operator).

13.8.4 Products of truncated Toeplitz operators As discussed earlier in the end notes for this chapter, Tu is not an algebra. This leads us to the question as to what types of operators in B(Ku ) can be written as a finite product of elements from Tu . One could also take infinite products but, as expected, there are convergence issues to overcome. When u = zn , Tu can be identified with the n × n Toeplitz matrices (see (13.3)) and it is known that every matrix can be written as a product of Toeplitz matrices [124]. There

306

Truncated Toeplitz operators

are even bounds on how many Toeplitz matrices are needed to do this. It seems reasonable that the same holds (every operator on Ku can be written as a finite product of elements from Tu ) when u is a finite Blaschke product. For general inner functions u, how far can one go with this?

13.8.5 Weak factorization The authors in [25] considered the Banach space   fn gn : fn , gn ∈ Ku ,  fn  gn  < ∞ Ku % Ku := n1

n1

of analytic functions on D with the norm    f Ku %Ku := inf  fn  gn  : f = fn gn . n1

n1

For certain inner functions u, there is a description of Ku % Ku that relates to the problem of whether or not every operator in Tu has a bounded symbol. For general u, the space Ku % Ku is not completely understood. Does anything interesting come out of exploring Ku % Kv for two inner functions u and v?

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Index

Abakumov, E., 148, 168 absolutely continuous measure, 2, 3 adjoint, 25 Agler, J., 81, 112 Ahern, P., 42, 50, 56, 110, 158, 207 Akhiezer, N., 279 Aleksandrov Clark measure, 248, 251 disintegration theorem, 245, 299 operator, 248 Aleksandrov, A., 119, 148, 168, 245, 250, 252, 256 Aleman, A., 103, 125, 168 analytic continuation, 153, 155, 156 self-map, 126 Toeplitz operator, 90 angular derivative, 46, 50, 158, 232 anti-unitary operator, 170 Arithmetic–Geometric Mean Inequality, 76 Aronszajn, N., 112 Arov, D., 184 Arveson, W., 206 associated inner function, 178 Atkinson’s Theorem, 30 automorphisms of the disk, 32

Riesz, 271, 277, 281 TMW, 120 Bercovici, H., 195, 213, 230 Bergman space, 81, 82, 125 Berman, R., 168 Bessonov, R., 301, 302 Beurling’s Theorem, 85, 102 Beurling, A., 85 bilateral shift, 93 biorthogonal sequence, 260, 262 Blaschke condition, 37, 80 product, 37, 80 Boas, R., 279 Borel σ-algebra, 1 function, 1 measure, 1 set, 1 Bottcher, A., 90 bounded analytic functions, 33 operator, 24 type, 74, 144 variation, 4 Brown, A., 91–94, 96, 201, 301

backward shift, 99, 104 Bagemihl, F., 55, 145 Ball, J., 252 Banach–Alaoglu Theorem, 4, 7, 23 Baranov, A., 301, 302 Bari, N., 279 basis C-real orthonormal, 172

Calkin algebra, 30 Carathéodory Extension Theorem, 5 Carathéodory, C., 50 Cargo, G., 168 Carleson measure, 250, 267 Carleson, L., 8, 250, 293 carrier of a measure, 1 Cauchy

318

Index

Integral Formula, 60, 66 kernel, 60 Schwarz inequality, 23 transform, 242, 250 Chalendar, I., 301, 305 characteristic function, 214 Chevrot, N., 183, 305 Christensen, O., 115, 279 Cima, J., 57, 119, 122, 131, 135, 156, 157, 213, 245, 248, 249, 256, 257, 303 Clark measure, 232, 250, 258, 299 unitary operator, 212, 236, 299, 304 Clark, D., 42, 50, 56, 110, 158, 207, 236, 239, 252, 256, 257, 279 Closed Graph Theorem, 24 closed span, 24 cluster set, 294 Coburn, L., 206 Cohn, W., 168, 250 Collingwood, E., 32, 34, 42, 45, 55, 131, 294 commutant, 26, 96, 229, 304 commutant lifting theorem, 215, 227, 229 commutator ideal, 206 compact operator, 29 complete sequence, 260 completely non-unitary, 195 completeness problems, 277 complex symmetric operator, 190, 279, 291 composition operator, 126, 130, 247–249 compressed shift, 187, 189, 197, 215, 229, 235, 279, 283, 286 compression of an operator, 188 conjugate analytic Toeplitz operator, 90 conjugation, 170, 173, 180, 276, 291 constant of uniform minimality, 263 contraction operator, 25, 27, 28, 195 Conway, J., 31, 206, 211 Corona Theorem, 293 Cowen, C., 127, 249 Crofoot transform, 135, 139, 142, 255, 303 Crofoot, B., 142 cyclic, 26, 86, 107, 151, 152, 167, 194 Darlington synthesis, 184 Davidson, K., 206, 211 de Branges, L., 124 de Branges–Rovnyak space, 123, 252 defect operator, 28, 217 derivative of a measure, 2 difference quotient operator, 101, 132, 204

319

dilation of an operator, 188, 216 dimension, 22 Dirichlet problem, 10 Dirichlet space, 82, 125, 142 discrete measure, 3 disk algebra, 119, 198, 296 disk automorphisms, 32 divisors of an inner function, 41 Douglas, R., 129, 149, 156, 157, 168, 178, 179, 184, 206 Duren, P., 11, 31, 32, 37, 58, 78, 82, 102, 127, 141, 279 Dyakonov, K., 119, 125, 142, 279 Ebbinghaus, H., 184 El-Fallah, O., 82, 142 Erdélyi, I., 27 essential norm, 30, 249 spectrum, 31, 204, 207 supremum, 7 exceptional set, 52, 295 exponential sequence, 265 exposed points, 185 extremal problems, 304 F-property, 106 F. and M. Riesz Theorem, 19 Fabry, E., 148 factorization of inner functions, 40 Fatou function, 19 Lemma, 15 Theorem, 11, 55 Fatou, P., 11, 19, 55 Feldman, N., 103 Fermat’s Last Theorem, 125 final space, 27 finite-dimensional model space, 115 finite-rank operator, 29, 301 Foia¸s, C., 195, 198, 213, 215, 230 Fourier coefficient of a function, 8 of a measure, 10 Fourier-Plancherel transform, 79 Fredholm operator, 31 Fricain, E., 123, 183, 301, 304, 305 Frostman shift, 52, 178, 295 Theorem, 50, 52, 55, 295 Frostman, O., 42, 50

320

Index

Fuhrmann, P., 292 functional calculus, 197, 198 Gao, Z., 305 gap theorems, 148 Garcia, S., 135, 142, 172, 178, 179, 183–185, 190, 212, 279, 281, 291, 300, 303–305 Garnett, J., 32, 58, 67, 72, 78, 80, 267, 279, 293 generalized analytic continuation, 168 Gilbreth, T., 305 Glazman, I., 279 Gohberg, I., 214 Gram matrix, 270, 281 greatest common divisor, 88, 109 Hadamard, J., 148 Hahn–Jordan Decomposition Theorem, 2 Halmos, P., 91–94, 96, 301 Hankel operator, 304 Hardy space, 58, 78, 79, 144 Hartmann, A., 165, 168, 169 Havin, V., 124 Hayashi, E., 142 Hedenmalm, H., 82, 102 Helly selection theorem, 7 Helson, H., 181, 186 Helton, W., 184 Herglotz function, 258 integral, 234 Representation Theorem, 16 Herglotz, G., 17, 18 Hermes, H„ 184 Hirzebruch, F., 184 Hitt, D., 103, 142 Hoffman, K., 11, 31, 37, 40, 58 Hrušˇcev, S., 57 indestructible Blaschke products, 56 initial space, 27 inner function, 36, 80, 85, 182, 213 Inoue, J., 186 integral means, 62, 79 interpolating sequences, 279 invertible, 25 irreducible, 194 isolated winding point, 146 isometric, 25 dilation, 216 embedding, 251 multiplier, 142

Jensen’s formula, 67 Ji, Y., 305 Julia, G., 50 Julia–Carathéodory Theorem, 234 K. Zhu, 82 Kérchy, L., 195, 198, 213, 230 Kapustin, V., 301, 302 Kellay, K., 82, 142 Kelley, J., 26 kernel of a Toeplitz operator, 108, 139, 142, 278, 279 Khavinson, D., 119 Köbe function, 185 Koecher, M., 184 Koosis, P., 55, 58, 81 Korenblum, B., 102 Krein, M., 258 Kriete, T., 110, 168 Krupnik, N., 214 Lacey, M., 251 Lanucha, B., 213 least common multiple, 88, 109 Lebesgue Decomposition Theorem, 3, 5 measure, 1 point, 300 Stieltjes integral, 5, 11 Li, C., 305 Lim, L., 305 liminf zero set, 154 Lindelöf’s Theorem, 34, 131 Lindelöf, E., 34 linear functional, 23 Littlewood Subordination Principle, 127, 141 Littlewood, J., 141 Livšic, M., 201, 214 Livšic–Möller Theorem, 201, 292 logarithmic capacity, 55 Lohwater, A., 32, 34, 42, 45, 55, 131, 294 Lotto, B., 186 Lubin, A., 252 Lusin, N., 55, 245 MacCluer, B., 127, 249 Mainzer, K., 184 Makarov, N., 279 Malmquist–Walsh Lemma, 120 Martin, R., 257–259 Martínez-Avendaño, R., 90, 94, 97

Index

Mashreghi, J., 19, 21, 56, 81, 82, 123, 124, 130, 142, 301 Mason, R., 125 Matheson, A., 57, 119, 131, 245, 248, 249, 256, 257 matrix-valued inner function, 213 Maximum modulus theorem, 15 McCarthy, J., 81, 112, 186 Mean Value Property, 15 measure preserving, 131, 257 minimal dilation, 216 sequence, 260, 263 Möbius transformation, 32 model space, 104, 122 model theory for contractions, 195 Moeller, J., 201 Morera’s Theorem, 157 multiplier, 137 nets, 26 Neukirch, J., 184 Nevanlinna counting function, 249 Nevanlinna, R., 40 Nikolskii, N., 55, 120, 156, 195, 207, 212 non-tangential limit, 14 norm composition operator, 127, 129 operator norm, 24 shift operator, 83 Toeplitz operator, 91 topology, 23 truncated Toeplitz operator, 300 operator norm, 24 orthogonal complement, 24 orthogonal projection, 25 orthogonalizer, 271 outer function, 69, 78, 86 Parseval’s theorem, 8 partial isometry, 27, 213, 223 Paulsen, V., 81, 112 PCBT, 145 Peller, V., 103 Piranian, G., 245 point spectrum, 25, 96, 203, 296 Poisson Integral Formula, 10 integral of a measure, 9 kernel, 9 Poltoratski, A., 186, 245, 256, 279

321

Prestel, A., 184 Principle of Uniform Boundedness, 23 Privalov’s Uniqueness Theorem, 55, 145 Privalov, I., 55 probability measure, 231 projection onto model space, 112 pseudocontinuation, 145, 168, 174 Putinar, M., 172, 183, 190, 281, 291, 305 quaternions, 183 Radjavi, H., 103 Radon–Nikodym derivative, 2 Theorem, 2 rank of an operator, 27 rank-one operator, 27 Ransford, T., 82, 142 real Smirnov, 181 regular measure, 2 reproducing kernel, 60, 111, 123 Richter, S., 102, 103, 125, 168 Riesz basis, 271, 277, 281 projection, 66 Representation Theorem, 3, 6, 24 sequence, 271 Riesz, F., 19, 24, 40, 95, 272 Riesz, M., 19 rigid function, 185 Rosenblum, M., 195 Rosenfeld, P., 90, 94, 97 Rosenthal, P., 103 Ross, W., 56, 57, 103, 119, 122, 125, 131, 135, 146, 156, 157, 165, 168, 169, 212, 213, 245, 248, 256, 257, 300, 303–305 Rovnyak, J., 124, 195 Royden, H., 31 Rudin, W., 13, 20, 31, 32, 34, 37, 40, 67, 76, 154, 263, 271, 300 Saksman, E., 249, 256 Sarason, D., 123, 124, 135, 142, 158, 185, 186, 202, 204, 212, 215, 230, 252, 256, 279, 282, 285, 288, 290, 300–302, 304 Schuster, A., 102 Schwarz Lemma, 32 Sedlock algebra, 304 Sedlock, N., 304 Seidel, W, 55 self-map, 126 separation constant, 265

322

Index

Shabankhah, M., 130 Shapiro, H., 125, 129, 144, 146, 148, 149, 156, 157, 168, 178, 179 Shapiro, J., 127 Shields, A., 129, 149, 156, 157, 178, 179, 201 shift operator, 83, 216 Shimorin, S., 102 Shirokov, N., 106 Silbermann, B., 90, 97 Simon, B., 171, 248 Singer, I., 279 singular inner function, 40, 80 measure, 2, 3 Smirnov class, 74, 144, 181 Theorem, 76 Smirnov, V., 76 span, 24 spatial isomorphism, 302 Spectral Theorem, 26 spectrum Clark unitary operator, 236 compressed shift, 201 definition, 25 essential spectrum, 31, 204, 207 inner function, 152 point spectrum, 25, 83, 203, 236, 296 shift operator, 83 truncated Toeplitz operator, 292 Stegenga, D., 142 Stolz domain, 14 Stothers, W., 125 strong operator topology, 26, 192, 200 subspace, 22 Sundberg, C., 168 support of a measure, 1 symmetric derivative of a measure, 2 Sz.-Nagy, B., 95, 195, 198, 213, 215, 230, 272 Sz.-Nagy–Foia¸s model, 189, 195, 197 Szeg˝o kernel, 60 Takenaka, S., 120

Teschl, G., 279 tight frame, 115 Timotin, D., 183, 301, 304, 305 Toeplitz algebra, 206 matrix, 91, 285, 289, 306 operator, 90, 106–108, 124, 138, 139, 278, 279, 282, 285 total variation, 5 Treil, S., 250 truncated Toeplitz operator, 282 Uniform Boundedness Principle, 23 uniform Frostman Blaschke products, 57 uniformly minimal sequence, 263, 277 uniformly separated sequence, 265, 277 unilateral shift, 83, 187, 216 unitarily equivalent, 25 unitary, 25 vector-valued Hardy space, 213 vector-valued model space, 213 Vinogradov, S., 57 Volberg, A., 250 Volterra operator, 171 Wang, X., 305 weak factorization, 306 operator topology, 26, 200, 290, 302 star topology, 3 topology, 23, 302 Wick, B., 251 Wiener algebra, 199 Wogen, W., 135, 212, 213, 303–305 Wolff, T., 248 Yabuta, K., 186 Ye, K., 305 Zagorodhyuk, S., 305 Zhu, K., 102 Zhu, S., 305