Spectral theory of bounded linear operators 9783030331481, 9783030331498

Table of contents :
Preface......Page 7
Contents......Page 10
1.1 Background: Notation and Terminology......Page 12
1.2 Inverse Theorems on Banach Spaces......Page 14
1.3 Orthogonal Structure in Hilbert Spaces......Page 16
1.4 Orthogonal Projections on Inner Product Spaces......Page 18
1.5 The Adjoint Operator on Hilbert Space......Page 20
1.6 Normal and Hyponormal Operators......Page 23
1.7 Orthogonal Reducing Eigenspaces......Page 27
1.8 Compact Operators and the Fredholm Alternative......Page 28
1.9 Supplementary Propositions......Page 32
Suggested Readings......Page 37
2.1 Basic Spectral Properties......Page 38
2.2 A Classical Partition of the Spectrum......Page 41
2.3 Spectral Mapping Theorems......Page 46
2.4 Spectral Radius and Normaloid Operators......Page 49
2.5 Numerical Radius and Spectraloid Operators......Page 52
2.6 Spectrum of a Compact Operator......Page 55
2.7 Supplementary Propositions......Page 58
Suggested Readings......Page 64
3.1 Spectral Theorem for Compact Operators......Page 65
3.2 Diagonalizable Operators......Page 69
3.3 Spectral Measure......Page 73
3.4 Spectral Theorem: The General Case......Page 78
3.5 Fuglede Theorems and Reducing Subspaces......Page 89
3.6 Supplementary Propositions......Page 94
Suggested Readings......Page 100
4.1 Rudiments of Banach and C*-Algebras......Page 101
4.2 Functional Calculus for Normal Operators......Page 105
4.3 Analytic Functional Calculus: Riesz Functional Calculus......Page 112
4.4 Riesz Idempotents and the Riesz Decomposition Theorem......Page 125
4.5 Supplementary Propositions......Page 136
Suggested Readings......Page 140
5.1 Fredholm and Weyl Operators — Fredholm Index......Page 141
5.2 Essential (Fredholm) Spectrum and Spectral Picture......Page 151
5.3 Riesz Points and Weyl Spectrum......Page 163
5.4 Browder Operators and Browder Spectrum......Page 172
5.5 Remarks on Browder and Weyl Theorems......Page 188
5.6 Supplementary Propositions......Page 194
Suggested Readings......Page 196
Appendices......Page 197
A.1 Quotient Space......Page 199
A.2 Complementation......Page 206
A.3 Range-Kernel Complementation......Page 210
A.4 Upper-Lower and Left-Right Approaches......Page 214
A.5 Essential Spectrum and Spectral Picture Again......Page 219
Suggested Readings......Page 227
B.1 Meanings of Multiplicity......Page 228
B.2 Complete Set of Unitary Invariants......Page 229
B.3 Multiplicity Function for Compact Operators......Page 230
B.4 Multiplicity-Free Normal Operators......Page 233
B.5 Multiplicity Function for Normal Operators......Page 236
Suggested Readings......Page 240
References......Page 241
List of Symbols......Page 247
Index......Page 251

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Carlos S. Kubrusly

Spectral Theory of Bounded Linear Operators

Carlos S. Kubrusly

Spectral Theory of Bounded Linear Operators

Carlos S. Kubrusly Institute of Mathematics Federal University of Rio de Janeiro Rio de Janeiro, Brazil

ISBN 978-3-030-33148-1 ISBN 978-3-030-33149-8 (eBook) https://doi.org/10.1007/978-3-030-33149-8 Mathematics Subject Classification (2010): 47-00, 47-01, 47-02, 47A10, 47A12, 47A16, 47A25, 47A53, 47A60, 47B15, 47B20, 47B48, 47C05 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Alan and Jessica

The positivists have a simple solution: the world must be divided into that which we can say clearly and the rest, which we had better pass over in silence. But can anyone conceive of a more pointless philosophy, seeing that what we can say clearly amounts to next to nothing? If we omitted all that is unclear, we would probably be left with completely uninteresting and trivial tautologies. Werner Heisenberg [62, Chapter 17, p. 213]

Mathematics is not a language, it’s an adventure. [...] Most mathematics is done with a friend over a cup of coffee, with a diagram scribbled on a napkin. Mathematics is and always has been about ideas, and a valuable idea transcends the symbols with which you choose to represent it. As Carl Friedrich Gauss once remarked, “What we need are notions, not notations.” Paul Lockhart [92, Part I, p. 53]

Preface

The book introduces spectral theory for bounded linear operators, giving a modern text for a graduate course focusing on two basic aspects. On the one hand, the spectral theory for normal operators acting on Hilbert spaces is comprehensively investigated, emphasizing recent aspects of the theory with detailed proofs, including Fredholm and multiplicity theories. On the other hand, Riesz–Dunford functional calculus for Banach-space operators is also meticulously presented, as well as Fredholm theory in Banach spaces (which is compared and clearly distinguished from its Hilbert-space counterpart). Seven portions somewhat uniformly distributed constitute the book contents: five chapters and two chapter-size appendices. The first five chapters consist of a revised, corrected, enlarged, updated, and thoroughly rewritten text based on the author’s 2012 book [80], which was heavily focused on Hilbert-space operators. The subsequent appendix draws a parallel between the Hilbert-space and the Banach-space ways of handling Fredholm theory. The final appendix gives a unified approach on multiplicity theory; precisely, on spectral multiplicity. Prerequisites for a smooth reading and for efficient learning are to some extent modest. A first course in elementary functional analysis, in particular in operator theory, will naturally be helpful. This encompasses a formal acquaintance with an introduction to analysis, including some measure theory and functions of a complex variable. An effort, however, has been made to introduce some essential results in order to make the book as self-contained as possible. Along this line, Chapter 1 supplies some basic concepts from operator theory which will be necessary in subsequent chapters. It does not intend to replace an introductory course on operator theory, but it summarizes notions and fundamental results needed for further chapters, and also establishes notations and terminologies used throughout the book. Chapter 2 deals with standard spectral results on the Banach algebra of bounded linear operators acting on Banach spaces, including the classical vii

viii

Preface

partition of the spectrum and spectral properties for classes of operators. Chapter 3 is central. It is entirely dedicated to the Spectral Theorem for normal operators on Hilbert spaces. After an initial consideration of the compact case, the general case is investigated in detail. Proofs of both versions of the Spectral Theorem for the general case are fully developed. The chapter closes with the Fuglede Theorems. Chapter 4 splits the functional calculus into two distinct but related parts. The functional calculus for normal operators is presented in the first part, which depends of course on the Spectral Theorem. The second part considers the analytic functional calculus (Riesz–Dunford functional calculus) for Banach-space operators. Chapter 5 focuses on Fredholm theory exclusively for Hilbert-space operators. This concentrates on compact perturbations of the spectrum, where a finer analysis is worked out leading to the notions of essential (Fredholm), Weyl, and Browder spectra and, consequently, getting to Weyl’s and Browder’s Theorems, including recent results. Each chapter has a final section on Supplementary Propositions consisting of (i) auxiliary results required in the sequel and (ii) additional results extending the theory beyond the bounds of the main text. These are followed by a set of Notes where each proposition is briefly discussed and references are provided indicating proofs for all of them. Such Supplementary Propositions can also be thought of as a section of proposed problems, and their respective Notes can be viewed as hints for solving them. This in turn is followed by a list of Suggested Readings indicating a collection of books on the subject of the chapter where different approaches, proofs, and further results can be found. Appendix A is the counterpart of Chapter 5 for operators acting on Banach spaces. This certainly is not a mere generalization: unlike Hilbert spaces, Banach spaces are not complemented, and this forces a rather different attitude for approaching Fredholm theory on Banach spaces. The appendix carries out a comparison between these two approaches, referred to as left-right and upper-lower ways of handling semi-Fredholm operators, in the course of which the origins and consequences of the lack of complementation in Banach spaces are carefully analyzed. Appendix B extends Chapter 3 on the Spectral Theorem by presenting an introductory view of multiplicity theory as a key factor for establishing a complete set of unitary invariants for normal operators. This closes the Spectral Theorem saga by stating that multiplicity function and spectral measure form a complete set of unitary invariants for normal operators on separable Hilbert spaces. The same level of detail devoted to proofs of theorems in the main chapters is maintained throughout the appendices. The idea behind such highly detailed proofs is to discuss and explain delicate points, stressing some usually hidden features. The subject matter of the book has been prepared to be covered in a onesemester graduate course (but the pace of lectures may depend on the boundary conditions). The whole book is the result of an effort to meet the needs of a contemporary course on spectral theory for mathematicians that will also be accessible to a wider audience of students in mathematics, statistics,

Preface

ix

economics, engineering, and physics, while requiring only a modest prerequisite. It may also be a useful resource for working mathematicians dealing with operator theory in general, and for scientists wishing to apply the theory to their field. I have tried to answer a continual request from students who ask for a text containing complete and detailed proofs, including discussions on a number of questions they have raised throughout the years. The logical dependence of the various sections and chapters is roughly linear, reflecting approximately a minimum amount of material needed to proceed further. There is an updated bibliography at the end of the book, including only references cited within the text, which contains most of the classics (of course) as well as some recent texts reveling modern approaches and present-day techniques. Since I have been lecturing on this subject for a long time, I have benefited from the help of many friends among students and colleagues, and I am grateful to all of them. In particular, I wish to thank Gustavo C. Amaral and El Hassan Benabdi, who helped with the tracing down of typos and imprecisions. Special thanks are due to Joaquim D. Garcia for his conscientious reading of the whole manuscript, correcting many typos and inaccuracies, and offering hints for sensible improvements. Special thanks are also due the copyeditor Brian Treadway who did an impressive job, including relevant mathematical corrections. I am also grateful to a pair of anonymous reviewers who raised reasonable suggestions and pointed out some necessary modifications. Rio de Janeiro, June 2019 Carlos S. Kubrusly

Contents

VII

Preface 1 Introductory Results 1.1 Background: Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . 1.2 Inverse Theorems on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Orthogonal Structure in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Orthogonal Projections on Inner Product Spaces . . . . . . . . . . . . . . 1.5 The Adjoint Operator on Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . 1.6 Normal and Hyponormal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Orthogonal Reducing Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Compact Operators and the Fredholm Alternative . . . . . . . . . . . . 1.9 Supplementary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 7 9 12 16 17 21

2 Spectrum of an Operator 2.1 Basic Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A Classical Partition of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Spectral Mapping Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Spectral Radius and Normaloid Operators . . . . . . . . . . . . . . . . . . . . 2.5 Numerical Radius and Spectraloid Operators . . . . . . . . . . . . . . . 2.6 Spectrum of a Compact Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Supplementary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 30 35 38 41 44 47

3 The Spectral Theorem

55

3.1 3.2 3.3 3.4 3.5 3.6

Spectral Theorem for Compact Operators . . . . . . . . . . . . . . . . . . . . . Diagonalizable Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Theorem: The General Case . . . . . . . . . . . . . . . . . . . . . . . . . Fuglede Theorems and Reducing Subspaces . . . . . . . . . . . . . . . . . . . Supplementary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 59 63 68 79 84 xi

Contents

xii

91

4 Functional Calculi 4.1 4.2 4.3 4.4 4.5

Rudiments of Banach and C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . 91 Functional Calculus for Normal Operators . . . . . . . . . . . . . . . . . . . . 95 Analytic Functional Calculus: Riesz Functional Calculus . . . . . . 102 Riesz Idempotents and the Riesz Decomposition Theorem . . . . 115 Supplementary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5 Fredholm Theory in Hilbert Space 5.1 5.2 5.3 5.4 5.5 5.6

Fredholm and Weyl Operators — Fredholm Index . . . . . . . . . . . . Essential (Fredholm) Spectrum and Spectral Picture . . . . . . . . . . Riesz Points and Weyl Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Browder Operators and Browder Spectrum . . . . . . . . . . . . . . . . . . . Remarks on Browder and Weyl Theorems . . . . . . . . . . . . . . . . . . . . Supplementary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 141 153 162 178 184

Appendices A Aspects of Fredholm Theory in Banach Space A.1 A.2 A.3 A.4 A.5

Quotient Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Range-Kernel Complementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upper-Lower and Left-Right Approaches . . . . . . . . . . . . . . . . . . . . . Essential Spectrum and Spectral Picture Again . . . . . . . . . . . . . .

B A Glimpse at Multiplicity Theory B.1 B.2 B.3 B.4 B.5

Meanings of Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complete Set of Unitary Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplicity Function for Compact Operators . . . . . . . . . . . . . . . . . Multiplicity-Free Normal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplicity Function for Normal Operators . . . . . . . . . . . . . . . . . .

189 189 196 200 204 209 219 219 220 221 224 227

References

233

List of Symbols

239

Index

243

1 Introductory Results

This chapter summarizes the essential background required for the rest of the book. Its purpose is notation, terminology, and basics. As for basics, we mean well-known theorems from operator theory needed in the sequel.

1.1 Background: Notation and Terminology Let ∅ be the empty set. Denote the sets of nonnegative, positive, and all integers by N 0 , N , Z ; the fields of rational, real, and complex numbers by Q , R , C ; and an arbitrary field by F . Use the same notation for a sequence and its range (e.g., {xn } means x(·) : N → X and also {x ∈ X: x = x(n) for some n ∈ N }). We assume the reader is familiar with the notions of linear space (or vector space), normed space, Banach space, inner product space, Hilbert space, as well as algebra and normed algebra. All spaces considered in this book are complex (i.e., over the complex field C ). Given a inner product space X , the sesquilinear form (linear in the first argument) · ; · : X × X → C on the Cartesian product X ×X stands for the inner product in X . We do not distinguish notation for norms. Thus  ·  denotes the norm on a normed space X (in particular, the norm generated by the inner product in an inner product space X — i.e., x2 = x ; x for all x in X ) and also the (induced uniform) operator norm in x B[X , Y ] (i.e., T  = supx=0 T x for all T in B[X , Y ] ), where B[X , Y ] stands for the normed space of all bounded linear transformations of a normed space X into a normed space Y. The induced uniform norm has the operator norm property; that is, if X , Y, and Z are normed spaces over the same scalar field, and if T ∈ B[X , Y ] and S ∈ B[Y, Z], then S T ∈ B[X , Z] and S T  ≤ ST . Between normed spaces, continuous linear transformation and bounded linear transformation are synonyms. A transformation T : X → Y between normed spaces X and Y is bounded if there exists a constant β ≥ 0 for which T x ≤ β x for every x in X . It is said to be bounded below if there is a constant α > 0 for which αx ≤ T x for every x in X . An operator on a normed space X is precisely a bounded linear (i.e., a continuous linear) transformation of X into itself. Set B[X ] = B[X , X ] for short: the normed algebra © Springer Nature Switzerland AG 2020 C. S. Kubrusly, Spectral Theory of Bounded Linear Operators, https://doi.org/10.1007/978-3-030-33149-8_1

1

2

1. Introductory Results

of all operators on X . If X = {0}, then B[X ] contains the identity operator I and I = 1 and so B[X ] is a unital normed algebra. If X = {0} is a Banach space, then B[X ] is a unital (complex ) Banach algebra. Since the induced uniform norm has the operator norm property, T n  ≤ T n for every operator T in B[X ] on a normed space X and every integer n ≥ 0. A transformation T ∈ B[X , Y ] is a contraction if T  ≤ 1 or, equivalently, if T x ≤ x for every x ∈ X . It is a strict contraction if T  < 1. If X = {0}, then a transformation T is a contraction (or a strict contraction) if and only if supx=0 (T x/x) ≤ 1 (or supx=0 (T x/x) < 1). If X = {0}, then T  < 1

=⇒

T x < x for every 0 = x ∈ X

=⇒

T  ≤ 1.

If T satisfies the middle inequality, then it is called a proper contraction. Let X be a Banach space. A sequence {Tn } of operators in B[X ] converges uniformly to an operator T in B[X ] if Tn − T  → 0, and {Tn } converges strongly to T if (Tn − T )x → 0 for every x in X . If X is a Hilbert space, then {Tn } converges weakly to T if (Tn − T )x ; y → 0 for every x, y in X (or, equivalently, (Tn − T )x ; x → 0 for every x in X if the Hilbert space is coms w u T , and Tn −→ T , respectively. T , Tn −→ plex). These will be denoted by Tn −→ The sequence {Tn } is bounded if supn Tn  < ∞. As is readily verified, u Tn −→ T

=⇒

s Tn −→ T

=⇒

w Tn −→ T

=⇒

supn Tn  < ∞

(the last implication is a consequence of the Banach–Steinhaus Theorem). A linear manifold M of a linear space X over a field F is a subset of X for which u + v ∈ M and α u ∈ M for every u, v ∈ M and α ∈ F . A subspace of a normed space X is a closed linear manifold of it. The closure of a linear manifold is a subspace. A subspace of a Banach space is a Banach space. A linear manifold M of a linear space X is nontrivial if {0} = M = X . A subset A of X is F -invariant for a function F : X → X (or A is an invariant set for F ) if F (A) ⊆ A. An invariant linear manifold (invariant subspace) for T ∈ B[X ] is a linear manifold (subspace) of X which, as a subset of X , is T -invariant. The zero space {0} and the whole space X (i.e., the trivial subspaces) are trivially invariant for every T in B[X ]. If M is an invariant linear manifold for T , then its closure M− is an invariant subspace for T . Let X and Y be linear spaces. The kernel (or null space) of a linear transformation T : X → Y is the inverse image of {0} under T ,   N (T ) = T −1 ({0}) = x ∈ X : T x = 0 , which is a linear manifold of X . The range of T is the image of X under T,   R(T ) = T (X ) = y ∈ Y : y = T x for some x ∈ X , which is a linear manifold of Y. If X and Y are normed spaces and T lies in B[X , Y ] (i.e., if the linear transformation T is bounded), then N (T ) is a subspace of X (i.e., if T is bounded, then N (T ) is closed).

1.2 Inverse Theorems on Banach Spaces

3

If M and N are linear manifolds of a linear space X , then the ordinary sum (or simply, the sum) M + N is the linear manifold of X consisting of all sums u + v with u in M and v in N . The direct sum M ⊕ N is the linear space of all ordered pairs (u, v) in M ×N (i.e., u in M and v in N ), where vector addition and scalar multiplication are defined coordinatewise. Although ordinary and direct sums are different linear spaces, they are isomorphic if M ∩ N = {0}. A pair of linear manifolds M and N are algebraic complements of each other if M + N = X and M ∩ N = {0}. In this case, each vector x in X can be uniquely written as x = u+v with u in M and v in N . If two subspaces (linear manifolds) of a normed space are algebraic complements of each other, then they are called complementary subspaces (complementary linear manifolds).

1.2 Inverse Theorems on Banach Spaces A fundamental theorem of (linear) functional analysis is the Open Mapping Theorem: a continuous linear transformation of a Banach space onto a Banach space is an open mapping (see, e.g., [78, Section 4.5] — a mapping is open if it maps open sets onto open sets). A crucial corollary of it is the next result. Theorem 1.1. (Inverse Mapping Theorem). If X and Y are Banach spaces and if T ∈ B[X , Y ] is injective and surjective, then T −1 ∈ B[Y, X ]. Proof. An invertible transformation T is precisely an injective and surjective one. Since the inverse T −1 of an invertible linear transformation is linear, and since an invertible transformation is open if and only if it has a continuous inverse, the stated result follows from the Open Mapping Theorem.  Thus an injective and surjective bounded linear transformation between Banach spaces has a bounded (linear) inverse. Let G[X , Y ] denote the collection of all invertible (i.e., injective and surjective) elements from B[X , Y ] with a bounded inverse. (Recall: inverse of linear transformation is linear, thus G[X , Y ] is the set of all topological isomorphism of X onto Y.) By the above theorem, if X and Y are Banach spaces, then every invertible transformation in B[X , Y ] lies in G[X , Y ]. Set G[X ] = G[X , X ], the group (under multiplication) of all invertible operators from B[X ] with a bounded inverse (or of all invertible operators from B[X ] if X is a Banach space). If X and Y are normed spaces and if there exists T −1 ∈ B[R(T ), X ] such that T −1 T = I ∈ B[X ], then T T −1 = I ∈ B[R(T )] (i.e., if y = T x for some x ∈ X , then T T −1 y = T T −1 T x = T x = y). In this case T is said to have a bounded inverse on its range. Theorem 1.2. (Bounded Inverse Theorem). Let X and Y be Banach spaces and take any T ∈ B[X , Y ]. The following assertions are equivalent. (a) T has a bounded inverse on its range. (b) T is bounded below . (c) T is injective and has a closed range.

4

1. Introductory Results

Proof. Part (i). The equivalence between (a) and (b) still holds if X and Y are just normed spaces. Indeed, if there exists T −1 ∈ B[R(T ), X ], then there exists a constant β > 0 for which T −1 y ≤ βy for every y ∈ R(T ). Take an arbitrary x ∈ X so that T x ∈ R(T ). Thus x = T −1 T x ≤ βT x, and so β1 x ≤ T x. Hence (a) implies (b). Conversely, if (b) holds true, then 0 < T x for every nonzero x in X , and so N (T ) = {0}. Then T has a (linear) inverse on its range — a linear transformation is injective if and only if it has a null kernel . Take an arbitrary y ∈ R(T ) so that y = T x for some x ∈ X . Thus T −1 y = T −1 T x = x ≤ α1 T x = α1 y for some constant α > 0. Hence T −1 is bounded. Thus (b) implies (a). Part (ii). Take an arbitrary R(T )-valued convergent sequence {yn }. Since each yn lies in R(T ), then there exists an X -valued sequence {xn } for which yn = T xn for each n. Since {T xn } converges in Y, then it is a Cauchy sequence in Y. Thus if T is bounded below, then there exists α > 0 such that 0 ≤ αxm − xn  ≤ T (xm − xn ) = T xm − T xn  for every m, n. Hence {xn } is a Cauchy sequence in X , and so it converges in X to, say, x ∈ X if X is a Banach space. Since T is continuous, it preserves convergence and hence yn = T xn → T x. Then the (unique) limit of {yn } lies in R(T ). Conclusion: R(T ) is closed in Y by the classical Closed Set Theorem. That is, R(T )− = R(T ) whenever X is a Banach space, where R(T )− stands for the closure of R(T ). Moreover, since (b) trivially implies N (T ) = {0}, it follows that (b) implies (c). On the other hand, if N (T ) = {0}, then T is injective. If in addition the linear manifold R(T ) is closed in the Banach space Y, then it is itself a Banach space and so T : X → R(T ) is an injective and surjective bounded linear transformation of the Banach space X onto the Banach space R(T ). Hence its inverse T −1 lies in B[R(T ), X ] by the Inverse Mapping Theorem (Theorem 1.1). Thus (c) implies (a).  If one of the Banach spaces X or Y is zero, then the results in the previous theorems are either void or hold trivially. Indeed, if X = {0} and Y = {0}, then the unique operator in B[X , Y ] is not injective (thus not bounded below and not invertible); if X = {0} and Y = {0}, then the unique operator in B[X , Y ] is not surjective but is bounded below (thus injective) and has a bounded inverse on its singleton range; if X = {0} and Y = {0} the unique operator in B[X , Y ] has an inverse in B[X , Y ]. The next theorem is a rather useful result establishing a power series expansion for (λI − T )−1. This is the Neumann expansion (or Neumann series) due to C.G. Neumann. The condition T  < |λ| will be weakened in Corollary 2.12. Theorem 1.3. (Neumann Expansion). If T ∈ B[X ] is an operator on a Banach space X , and if λ is any scalar such that T  < |λ|, then λI − T has an inverse in B[X ] given by the following uniformly convergent series: ∞   k T (λI − T )−1 = λ1 . λ k=0

1.3 Orthogonal Structure in Hilbert Spaces

5

operator T ∈ B[X ] and Proof.Takean   scalar   nλ. IfT < |λ|, then na nonzero n supn k=0  Tλ k < ∞. Thus, since k=0  Tλ k  ≤ k=0  Tλ k for every     n  T k  is n ≥ 0, the increasing sequence of nonnegative numbers k=0 λ    n bounded, and hence it converges in R . Thus the B[X ]-valued sequence Tλ is absolutely summable, and so it is summable (since B[X ] is a Banach space ∞ — see, e.g., [78, Proposition 4.4] ). That is, the series k=0 Tλ k converges in  ∞  T k B[X ]. Equivalently, there is an operator in B[X ], say k=0 λ , for which n   k u  ∞   k T T −→ . λ λ k=0

k=0

Now, as is also readily verified by induction, for every n ≥ 0  n  k  n  k  n+1 T T (λI − T ) λ1 = λ1 (λI − T ) = I − Tλ . λ λ k=0

k=0

 n  n But λ −→ O (since  Tλ  ≤  Tλ  → 0 when T  < |λ|), and so  n  k u n   k u T T (λI − T ) λ1 −→ I and λ1 (λI − T ) −→ I, λ λ  T n

u

k=0

k=0

where O stands for null operator. Hence ∞   k ∞   k T T (λI − T ) λ1 = λ1 (λI − T ) = I, λ λ and so

1 λ

 ∞  T k k=0

λ

k=0

k=0

∈ B[X ] is the inverse of λI − T ∈ B[X ].



Remark. Under the hypothesis of Theorem 1.3, ∞ T  k  −1 1 (λI − T )−1  ≤ |λ| = |λ| − T  . |λ| k=0

1.3 Orthogonal Structure in Hilbert Spaces Two vectors x and y in an inner product space X are orthogonal if x ; y = 0. In this case we write x ⊥ y. Two subsets A and B of X are orthogonal (notation: A ⊥ B) if every vector in A is orthogonal to every vector in B. The orthogonal complement of a set A is the set A⊥ made up of all vectors in X orthogonal to every vector of A,   A⊥ = x ∈ X : x ; y = 0 for every y ∈ A , which is a subspace (i.e., a closed linear manifold) of X , and therefore A⊥ = (A⊥ )− = (A− )⊥ . If M and N are orthogonal (M ⊥ N ) linear manifolds of an inner product space X , then M ∩ N = {0}. In particular, M ∩ M⊥ = {0}. If M ⊥ N , then M ⊕ N is an orthogonal direct sum. If X is a Hilbert space, then M⊥⊥ = M−, and M⊥ = {0} if and only if M− = X . Moreover, if M and N are orthogonal subspaces of a Hilbert space X , then M + N is a subspace of X . In this case, M + N ∼ = M ⊕ N : ordinary and direct sums of orthogonal

6

1. Introductory Results

subspaces of a Hilbert space are unitarily equivalent — the inner product in M ⊕ N is given by (u1 , v1 ) ; (u2 , v2 ) = u1 ; u2  + v1 ; v2  for every (u1 , v1 ) and (u2 , v2 ) in M ⊕ N . Two linear manifolds M and N are orthogonal complementary manifolds if M + N = X and M ⊥ N . If M and N are orthogonal complementary subspaces of a Hilbert space X , then we identify the ordinary sum M + N with the orthogonal sum M ⊕ N , and write (an abuse of notation, actually) X = M ⊕ N (instead of X ∼ = M ⊕ N ). A central result of Hilbert-space geometry, the Projection Theorem, says: if H is a Hilbert space and M is a subspace of it, then (cf. [78, Theorem 5.20] ) H = M + M⊥ . (Thus every subspace of a Hilbert space is complemented — cf. Remark 5.3(a) and Appendix B.) Ordinary and direct sums of orthogonal subspaces are unitarily equivalent. So the Projection Theorem is equivalently stated in terms of orthogonal direct sum up to unitary equivalence, H ∼ = M ⊕ M⊥, often written H = M ⊕ M⊥ . This leads to the notation M⊥ = H  M, if M is a subspace of H. For a nonempty subset A of a normed space X , let span A denote the span (or linear span) of A: the linear manifold of X consisting of all (finite) linear combinations of vectors of A or, equivalently, the smallest linear manifold of X including A (i.e., the intersection of  all linear manifolds of X that include A). Its closure (span A)−, denoted by A (and sometimes also called the span of A), is a subspace of X . Let {Mγ }γ∈Γ be a family of subspaces of X indexed by (not  necessarily countable) nonempty index set Γ. The subspace an arbitrary  − M of X is the topological sum of {Mγ }γ∈Γ , usually denoted span γ   γ∈Γ −  or M by γ γ∈Γ γ∈Γ Mγ . Thus 

− 



−    Mγ . = Mγ = Mγ = span Mγ γ∈Γ

γ∈Γ

γ∈Γ

γ∈Γ

A pivotal result of Hilbert-space theory is the Orthogonal Structure orTheorem, which reads as follows. Let {Mγ }γ∈Γ be a family    of pairwise − there thogonal subspaces of a Hilbert space H. For every x ∈ γ∈Γ Mγ is a unique summable family {uγ }γ∈Γ of vectors uγ ∈ Mγ such that  uγ . x= γ∈Γ  Moreover, x2 = γ∈Γ uγ 2 . Conversely, if {uγ }γ∈Γ is a square-summable ∈ Mγ for each γ ∈ Γ , then {uγ }γ∈Γ is family of vectors  in H with uγ  − . (See, e.g., [78, Section 5.5].) u lies in summable and γ∈Γ Mγ γ∈Γ γ

For the definition of summable family and square-summable family see [78, Definition 5.26]. A summable family in a normed space hasonly a countable number of nonzero vectors, and so for each x the sum x = γ∈Γ uγ has only a countable number of nonzero summands uγ (see, e.g., [78, Corollary 5.28] ).

1.4 Orthogonal Projections on Inner Product Spaces

7

 Take the direct sum γ∈Γ Mγ of pairwise orthogonal subspaces Mγ of a Hilbert space H made up of square-summable  families of vectors. That is, Mα ⊥ Mβ for α = β, and x = {xγ }γ∈Γ ∈ γ∈Γ Mγ if and only if each xγ  lies in Mγ and γ∈Γ xγ 2 < ∞. This is a Hilbert space with inner product x ; y =

 γ∈Γ

xγ ; yγ 

 for every x = {xγ }γ∈Γ and y = {yγ }γ∈Γ in γ∈Γ Mγ . A consequence (a Theorem reads Structure Orthogonal restatement, actually) of the −   as follows.  The orthogonal direct sum γ∈Γ Mγ and the topological sum γ∈Γ Mγ are unitarily equivalent Hilbert spaces. In other words,

−   Mγ . Mγ ∼ = γ∈Γ

γ∈Γ

−  If, in addition, {Mγ }γ∈Γ spans H (i.e., = γ∈Γ Mγ = H), γ∈Γ Mγ then it is usual to express the above identification by writing  H= Mγ . 

γ∈Γ

 Similarly, the external orthogonal direct sum γ∈Γ Hγ of a collection of Hilbert spaces {Hγ }γ∈Γ (not necessarily subspaces of a larger Hilbert space) consists of square-summable families {xγ }γ∈Γ (i.e., γ∈Γ xγ 2 < ∞), with each xγ in Hγ . This is a Hilbert space with inner product given as above, and the Hilbert spaces {Hγ }γ∈Γ are referred to as being externally orthogonal . Let H and K be Hilbert spaces. The direct sum of T ∈ B[H] and S ∈ B[K] is the operator T ⊕ S ∈ B[H ⊕ K] such that (T ⊕ S)(x, y) = (T x, Sy) for every (x, y) ∈ H ⊕ K. Similarly, if {Hγ }γ∈Γ is a collection of Hilbert spaces and if {Tγ }γ∈Γ is a bounded family  Tγ ∈ B[Hγ ] (i.e., supγ∈Γ Tγ  < ∞),  of operators then their direct sum T : γ∈Γ Hγ → γ∈Γ Hγ is the mapping  T {xγ }γ∈Γ = {Tγ xγ }γ∈Γ for every {xγ }γ∈Γ ∈ γ∈Γ Hγ , which is an operator in B

 γ∈Γ

Hγ ] with T  = supγ∈Γ Tγ , denoted by

T =

 γ∈Γ

Tγ .

1.4 Orthogonal Projections on Inner Product Spaces A function F : X → X of a set X into itself is idempotent if F 2 = F , where F 2 stands for the composition of F with itself. A projection E is an idempotent (i.e., E = E 2 ) linear transformation E : X → X of a linear space X into itself. If E is a projection, then so is I − E, which is referred to as the complementary projection of E, and their null spaces and ranges are related as follows:

8

1. Introductory Results

R(E) = N (I − E)

and

N (E) = R(I − E).

The range of a projection is the set of all its fixed points,   R(E) = x ∈ X : Ex = x , which forms with the kernel a pair of algebraic complements, R(E) + N (E) = X

and

R(E) ∩ N (E) = {0}.

An orthogonal projection E : X → X on an inner product space X is a projection for which R(E) ⊥ N (E). If E is an orthogonal projection on X , then so is the complementary projection I − E : X → X . Every orthogonal projection is bounded (i.e., continuous — E ∈ B[H] ). In fact, every orthogonal projection is a contraction. Indeed, every x ∈ X can be (uniquely) written as x = u + v with u ∈ R(E) and v ∈ N (E), and so Ex2 = Eu + Ev2 = Eu2 = u2 ≤ x2 since x2 = u2 + v2 by the Pythagorean Theorem (as u ⊥ v). Hence R(E) is a subspace (i.e., it is closed, because R(E) = N (I − E) and I − E is bounded). Moreover, N (E) = R(E)⊥ ; equivalently, R(E) = N (E)⊥. Conversely, if any of these properties holds for a projection E, then it is an orthogonal projection. Two orthogonal projections E1 and E2 on X are said to be orthogonal to each other (or mutually orthogonal ) if R(E1 ) ⊥ R(E2 ) (equivalently, if E1 E2 = E2 E1 = O). Let Γ be an arbitrary index set (not necessarily countable). If {Eγ }γ∈Γ is a family of orthogonal projections on an inner product space X which are orthogonal to each other (R(Eα ) ⊥ R(Eβ ) for α = β), then {Eγ }γ∈Γ is an orthogonal family of orthogonal projections on X . An orthogonal sequence of orthogonal projections {Ek }∞ k=0 is similarly defined. If {Eγ }γ∈Γ is an orthogonal family of orthogonal projections and  Eγ x = x for every x ∈ X , γ∈Γ

then {Eγ }γ∈Γ  is called a resolution of the identity on X . (Recall: for each x in X the sum γ∈Γ Eγ x has only a countable number of nonzero vectors.) If {Ek }∞ k=0 is an infinite sequence, then the above identity in X means convergence in the strong operator topology: n s Ek −→ I as n → ∞. k=0

{Ek }nk=0

is a finite If nfamily, then the above identity in X obviously coincides with the identity k=0 Ek = I in B[X ] (e.g., {E1 , E2 } is a resolution of the identity on X whenever E2 = I − E1 is the complementary projection of the orthogonal projection E1 on X ). The Projection Theorem and the Orthogonal Structure Theorem of the previous section can be written in terms of orthogonal projections. Indeed, for

1.5 The Adjoint Operator on Hilbert Space

9

every subspace M of a Hilbert space H there is a unique orthogonal projection E ∈ B[H] such that R(E) = M, referred to as the orthogonal projection onto M. This is an equivalent version of the Projection Theorem (see, e.g., [78, Theorem 5.52] ). An equivalent version of the Orthogonal Structure Theorem reads as follows. If {Eγ }γ∈Γ is a resolution of the identity on H, then H=

 γ∈Γ

R(Eγ )

−

.

Conversely,if{Mγ }γ∈Γ is a family of pairwise orthogonal subspaces of H for − , then the family of orthogonal projections {Eγ }γ∈Γ which H = γ∈Γ Mγ with each Eγ ∈ B[H] onto each Mγ (i.e., with R(Eγ ) = Mγ ) is a resolution of the identity on H (see, e.g., [78, Theorem 5.59] ). Since {R(Eγ )}γ∈Γ is a family of pairwise orthogonal subspaces, the above orthogonal  decomposition of H is unitarily equivalent to the orthogonal direct sum γ∈Γ R(Eγ ), that −   ∼ is, = γ∈Γ R(Eγ ), which is commonly written as γ∈Γ R(Eγ ) H=

 γ∈Γ

R(Eγ ).

If {Eγ }γ∈Γ is a resolution of the identity made up of nonzero projections on a nonzero Hilbert space H, and if {λγ }γ∈Γ is a similarly indexed family of scalars, then consider the following subset of H:   D(T ) = x ∈ H : {λγ Eγ x}γ∈Γ is a summable family of vectors in H . The mapping T : D(T ) → H defined by  Tx = λγ Eγ x for every x ∈ D(T ) γ∈Γ

is called a weighted sum of projections. The domain D(T ) of a weighted sum of projections is a linear manifold of H, and T is a linear transformation. Moreover, T is bounded if and only if {λγ }γ∈Γ is a bounded family of scalars, which happens if and only if D(T ) = H. In this case T ∈ B[H] and T  = supγ∈Γ |λγ | (see, e.g., [78, Proposition 5.61] ).

1.5 The Adjoint Operator on Hilbert Space Throughout the text H and K are Hilbert spaces. Take any T ∈ B[H, K]. The adjoint T ∗ of T is the unique mapping of K into H for which T x ; y = x ; T ∗ y for every x ∈ H and y ∈ K. In fact, T ∗ ∈ B[K, H] (i.e., T ∗ is bounded and linear), T ∗∗ = T , and T ∗ 2 = T ∗ T  = T T ∗  = T 2 .

10

1. Introductory Results

Moreover, if Z is a Hilbert space and S ∈ B[K, Z], then S T ∈ B[H, Z] is such that (S T )∗ = T ∗ S ∗ (see, e.g., [78, Proposition 5.65 and Section 5.12] ). Lemma 1.4. For every T ∈ B[H, K], N (T ) = R(T ∗ )⊥ = N (T ∗ T )

and

R(T )− = N (T ∗ )⊥ = R(T T ∗ )− .

Proof. By definition x ∈ R(T ∗ )⊥ if and only if x ; T ∗ y = 0 or, equivalently, T x ; y = 0, for every y in K, which means T x = 0; that is, x ∈ N (T ). Hence R(T ∗ )⊥ = N (T ). Since T x2 = T x ; T x = T ∗ T x ; x for every x ∈ H, we get N (T ∗ T ) ⊆ N (T ). But N (T ) ⊆ N (T ∗ T ) trivially and so N (T ) = N (T ∗ T ). This completes the proof of the first identities. Since these hold for every T in B[H, K], they also hold for T ∗ in B[K, H] and for T T ∗ in B[K]. Thus R(T )− = R(T ∗∗ )⊥⊥ = N (T ∗ )⊥ = N (T ∗∗ T ∗ )⊥ = N (T T ∗ )⊥ = R((T T ∗ )∗ )⊥⊥ = R(T ∗∗ T ∗ )⊥⊥ = R(T T ∗ )− , because M⊥⊥ = M− for every linear manifold M and T ∗∗ = T .



Lemma 1.5. R(T ) is closed in K if and only if R(T ∗ ) is closed in H. Proof. Let T |M denote the restriction of T to any linear manifold M ⊆ H. Set T⊥ = T |N (T )⊥ in B[N (T )⊥, K]. Since H = N (T ) + N (T )⊥, every x ∈ H can be written as x = u + v with u ∈ N (T ) and v ∈ N (T )⊥ . If y ∈ R(T ), then y = T x = T u+T v = T v = T |N (T )⊥ v for some x ∈ H, and so y ∈ R(T |N (T )⊥). Hence R(T ) ⊆ R(T |N (T )⊥ ). Since R(T |N (T )⊥) ⊆ R(T ) we get R(T⊥ ) = R(T )

and

N (T⊥ ) = {0}.

If R(T ) = R(T )−, then take the inverse T⊥−1 in B[R(T ), N (T )⊥ ] (Theorem 1.2). For any w ∈ N (T )⊥ consider the functional fw : R(T ) → C defined by fw (y) = T⊥−1 y ; w for every y ∈ R(T ). Since T⊥−1 is linear and the inner product is linear in the first argument, then f is linear. Since |fw (y)| ≤ T⊥−1 wy for every y in R(T ), then f is also bounded. Moreover, R(T ) is a subspace of the Hilbert space K (it is closed in K), and so a Hilbert space itself. Then by the Riesz Representation Theorem in Hilbert space (see, e.g., [78, Theorem 5.62]) there exists a unique vector zw in the Hilbert space R(T ) for which fw (y) = y ; zw 

1.5 The Adjoint Operator on Hilbert Space

11

for every y ∈ R(T ). Thus, since an arbitrary x ∈ H is written as x = u + v with u ∈ N (T ) and v ∈ N (T )⊥ , we get x ; T ∗ zw  = T x ; zw  = T u ; zw  + T v ; zw  = T v ; zw  = fw (T v) = T⊥−1 T v ; w = T⊥−1 T⊥ v ; w = v ; w = u ; w + v ; w = x ; w. Hence x ; T ∗ zw − w = 0 for every x ∈ H; that is, T ∗ zw = w. So w ∈ R(T ∗ ). Therefore N (T )⊥ ⊆ R(T ∗ ). On the other hand, R(T ∗ ) ⊆ R(T ∗ )− = N (T )⊥ by Lemma 1.4 (since T ∗∗ = T ), and hence R(T ∗ ) = R(T ∗ )− . Thus R(T ) = R(T )− implies R(T ∗ ) = R(T ∗ )−, and consequently the converse also holds because T ∗∗ = T . So R(T ) = R(T )−

if and only if

R(T ∗ ) = R(T ∗ )− .



Take the adjoint operator T ∗ ∈ B[H] of an operator T ∈ B[H] on a Hilbert space H. Let M be a subspace of H. If M and its orthogonal complement M⊥ are both T -invariant (i.e., if T (M) ⊆ M and T (M⊥ ) ⊆ M⊥ ), then we say that M reduces T (or M is a reducing subspace for T ). An operator is reducible if it has a nontrivial reducing subspace (otherwise it is called irreducible), and reductive if all its invariant subspaces are reducing. Since M = {0} if and only if M⊥ = H, and M⊥ = {0} if and only if M = H, then an operator T on a Hilbert space H is reducible if there exists a subspace M of H such that both M and M⊥ are nonzero and T -invariant or, equivalently, if M is nontrivial and invariant for both T and T ∗ as we will see next. Lemma 1.6. Take an operator T on a Hilbert space H. A subspace M of H is T -invariant if and only if M⊥ is T ∗ -invariant. M reduces T if and only if M is invariant for both T and T ∗, which implies (T |M )∗ = T ∗ |M . Proof. Take any vector y ∈ M⊥. If T x ∈ M whenever x ∈ M, then x ; T ∗ y = T x ; y = 0 for every x ∈ M, and so T ∗ y ⊥ M; that is, T ∗ y ∈ M⊥. Thus T (M) ⊆ M implies T ∗ (M⊥ ) ⊆ M⊥. Since this holds for every operator in B[H], T ∗ (M⊥ ) ⊆ M⊥ implies T ∗∗ (M⊥⊥ ) ⊆ M⊥⊥, and hence T (M) ⊆ M (because T ∗∗ = T and M⊥⊥ = M− = M). Then T (M) ⊆ M if and only if T ∗ (M⊥ ) ⊆ M⊥, and so T (M⊥ ) ⊆ M⊥ if and only if T ∗ (M) ⊆ M. Therefore T (M) ⊆ M and T (M⊥ ) ⊆ M⊥ if and only if T (M) ⊆ M and T ∗ (M) ⊆ M,  and so (T |M )x ; y = T x ; y = x ; T ∗ y = x ; (T ∗ |M )y for x, y ∈ M. An operator T ∈ B[H] on a Hilbert space H is self-adjoint (or Hermitian) if it coincides with its adjoint (i.e., if T ∗ = T ). A characterization of self-adjoint operators reads as follows: on a complex Hilbert space H, an operator T is self-adjoint if and only if T x ; x ∈ R for every vector x ∈ H. Moreover, if

12

1. Introductory Results

T ∈ B[H] is self-adjoint, then T = O if and only if T x ; x = 0 for all x ∈ H. (See, e.g., [78, Proposition 5.79 and Corollary 5.80].) A self-adjoint operator T is nonnegative (notation: O ≤ T ) if 0 ≤ T x ; x for every x ∈ H, and it is positive (notation: O < T ) if 0 < T x ; x for every nonzero x ∈ H. An invertible positive operator T in G[H] is called strictly positive (notation: O ≺ T ). For every T in B[H], the operators T ∗ T and T T ∗ in B[H] are always nonnegative (reason: 0 ≤ T x2 = T x ; T x = T ∗ T x ; x for every x ∈ H). If S and T are operators in B[H], then we write S ≤ T if T − S is nonnegative (thus self-adjoint). Similarly, we also write S < T and S ≺ T if T − S is positive or strictly positive (thus self-adjoint). So if T − S is self-adjoint (in particular, if both T and S are), then S = T if and only if S ≤ T and T ≤ S.

1.6 Normal and Hyponormal Operators Let S and T be operators on the same space X . They commute if S T = T S. An operator T ∈ B[H] on a Hilbert space H is normal if it commutes with its adjoint (i.e., if T T ∗ = T ∗ T , or O = T ∗ T − T T ∗ ). Observe that T ∗ T − T T ∗ is always self-adjoint. An operator T ∈ B[H] is hyponormal if T T ∗ ≤ T ∗ T (i.e., if O ≤ T ∗ T − T T ∗ ), which is equivalent to saying that (λI − T )(λI − T )∗ ≤ (λI − T )∗ (λI − T ) for every scalar λ ∈ C . An operator T ∈ B[H] is cohyponormal if its adjoint is hyponormal (i.e., if T ∗ T ≤ T T ∗ , or O ≤ T T ∗ − T ∗ T ). An operator T is normal if and only if it is both hyponormal and cohyponormal. Thus every normal operator is hyponormal . If it is either hyponormal or cohyponormal, then it is called seminormal . Theorem 1.7. T ∈ B[H] is hyponormal if and only if T ∗ x ≤ T x for every x ∈ H. Moreover, the following assertions are pairwise equivalent. (a) T is normal . (b) T ∗ x = T x for every x ∈ H. (c) T n is normal for every positive integer n. (d) T ∗n x = T n x for every x ∈ H and every n ≥ 1. Proof. Take any T ∈ B[H]. The characterization for hyponormal operators goes as follows: T T ∗ ≤ T ∗ T if and only if T T ∗ x ; x ≤ T ∗ T x ; x or, equivalently, T ∗ x2 ≤ T x2 , for every x ∈ H. Since T is normal if and only if it is both hyponormal and cohyponormal, assertions (a) and (b) are equivalent. Thus, as T ∗n = T n∗ for each n ≥ 1, (c) and (d) are equivalent. If T ∗ commutes with T, then it commutes with T n and, dually, T n commutes with T ∗n = T n∗. Hence (a) implies (c), and (d) implies (b) trivially.  In spite of the equivalence between (a) and (c) above, the square of a hyponormal operator is not necessarily hyponormal. It is clear that every self-

1.6 Normal and Hyponormal Operators

13

adjoint operator is normal, and so is every nonnegative operator. Moreover, normality distinguishes the orthogonal projections among the projections. Theorem 1.8. If E ∈ B[H] is a nonzero projection, then the following assertions are pairwise equivalent. (a) E is an orthogonal projection. (b) E is nonnegative. (c) E is self-adjoint. (d) E is normal . (e) E = 1. (f) E ≤ 1. Proof. Let E be a projection (i.e., a linear idempotent). If E is an orthogonal projection (i.e., R(E) ⊥ N (E)), then I − E is again an orthogonal projection, thus bounded, and so R(E) = N (I − E) is closed. Hence R(E ∗ ) is closed by Lemma 1.5. Moreover, R(E) = N (E)⊥ . But N (E)⊥ = R(E ∗ )− by Lemma 1.4 (since E ∗∗ = E). Then R(E) = R(E ∗ ), and so for every x ∈ H there is a z ∈ H for which Ex = E ∗ z. Therefore, E ∗ Ex = (E ∗ )2 z = (E 2 )∗ z = E ∗ z = Ex. That is, E ∗ E = E. Hence E is nonnegative (it is self-adjoint, and E ∗ E is always nonnegative). Outcome: (a) ⇒ (b) ⇒ (c) ⇒ (d), R(E) ⊥ N (E)

=⇒

E≥O

E∗ = E

=⇒

=⇒

E ∗ E = EE ∗ .

Conversely, if E ∈ B[H] is normal, then E ∗ x = Ex for every x in H (Theorem 1.7) and so N (E ∗ ) = N (E). Hence R(E) = N (E ∗ )⊥ = N (E)⊥ by Lemma 1.4 (because R(E) = N (I − E) is closed). Therefore R(E) ⊥ N (E). Then E ∗ E = EE ∗

=⇒

R(E) ⊥ N (E).

Thus (d) ⇒ (a). To show that (c) ⇒ (e), which trivially implies (f), proceed as follows. If O = E = E ∗, then E2 = E ∗ E = E 2  = E = 0. So E = 1: E = E∗

=⇒

E = 1

=⇒

E ≤ 1.

Now take any v ∈ N (E)⊥. Then (I − E)v ∈ N (E) (since R(I − E) = N (E)). Hence (I − E)v ⊥ v and so 0 = (I − E)v ; v = v2 − Ev ; v. If E ≤ 1, then v2 = Ev ; v ≤ Evv ≤ Ev2 ≤ v2. Thus Ev = v = 1 Ev ; v 2 . So (I − E)v2 = Ev − v2 = Ev2 − 2 Re Ev ; v + v2 = 0, and hence v ∈ N (I − E) = R(E). Then N (E)⊥ ⊆ R(E), which implies that R(E ∗ ) ⊆ N (E ∗ )⊥ by Lemma 1.4, and therefore R(E ∗ ) ⊥ N (E ∗ ). But E ∗ is a projection whenever E is (E = E 2 implies E ∗ = E 2∗ = E ∗2 ). Thus E ∗ is an orthogonal projection. Then E ∗ is self-adjoint, and so is E. Hence (f) ⇒ (c): E ≤ 1

=⇒

E = E∗.



14

1. Introductory Results

Corollary 1.9. Every bounded weighted sum of projections is normal . Proof. Let T ∈ B[H] be a weighted sum of projections, which means T x =  γ∈Γ λγ Eγ x for every x in the Hilbert space H, where {λγ }γ∈Γ is a bounded family of scalars and {Eγ }γ∈Γ is a resolution of the identity on H. Since Eγ∗ = Eγ for every γ and R(Eα ) ⊥ R(Eβ ) whenever α = β, it is readily verified  that the adjoint T ∗ of T is given by T ∗ x = γ∈Γ λγ Eγ for every x in H (i.e., T x ; y = x ; T ∗ y for every x, y in H). Moreover, since Eγ2 = Eγ for every γ and Eα Eβ = Eβ Eα = O whenever α = β, we also get T T ∗ x = T ∗ T x for every x ∈ H. Hence T is normal.  An isometry between metric spaces is a map that preserves distance. Thus every isometry is injective. A linear transformation V between normed spaces is an isometry if and only if it preserves norm ( V x = x — hence V  = 1) and, between inner product spaces, if and only if it preserves inner product (V x ; V y = x ; y). An invertible linear isometry (i.e., a surjective linear isometry) is an isometric isomorphism. The inverse of an invertible linear isometry is again a linear isometry (V −1 x = V V −1 x = x). If W, X , Y, Z are normed spaces, S ∈ B[W, X ], T ∈ B[Y, Z], and V ∈ B[X, Y] is an isometry, then V S = S

and

T V  = T  if V is surjective.

A transformation V ∈ B[H, K] between Hilbert spaces is an isometry if and only if V ∗ V = I (identity on H), and a coisometry if its adjoint is an isometry (V V ∗ = I — identity on K). A unitary transformation is an isometry and a coisometry or, equivalently, an invertible transformation with V −1 = V ∗. Thus a unitary operator U ∈ B[H] is precisely a normal isometry: U U ∗ = U ∗ U = I; and so normality distinguishes the unitary operators among the isometries. Lemma 1.10. If X is a normed space and T ∈ B[X ], then the real-valued 1 sequence {T n  n} converges in R . Proof. Take any positive integer m. Every positive integer n can be written as n = m pn + qn for some nonnegative integers pn , qn with qn < m. Hence T n  = T mpn+qn = T mpn T qn ≤ T mpnT qn ≤ T m pn T qn. Suppose T = O. Set α = max 0≤k≤m−1 {T k } = 0. (Recall: qn ≤ m − 1.) Thus 1

T n  n ≤ T m  1

1

pn n

1

1

1

qn

α n = α n T m  m − mn .

qn

1

Since α n → 1 and T m  m − mn → T m  m as n → ∞, we get 1

1

lim sup T n  n ≤ T m  m . n

Therefore, since m is an arbitrary positive integer, 1

1

lim sup T n  n ≤ lim inf T n  n , 1

n

and so {T n  n} converges in R .

n



1.6 Normal and Hyponormal Operators

15

1

For each operator T ∈ B[X ] let r(T ) denote the limit of {T n  n}, 1

r(T ) = lim T n  n . n

1

As we saw before (proof of Lemma 1.10), r(T ) ≤ T n  n for every n ≥ 1. In 1 1 particular, r(T ) ≤ T . Also r(T k ) k = limn T kn  kn = r(T ) for each k ≥ 1 1 1 (as {T kn  kn} is a subsequence of the convergent sequence {T n  n}). Then r(T k ) = r(T )k for every positive integer k. Thus for every operator T ∈ B[X ] on a normed space X and for each nonnegative integer n, 0 ≤ r(T )n = r(T n ) ≤ T n  ≤ T n . An operator T ∈ B[X ] on a normed space X is normaloid if r(T ) = T . Here is an alternate definition of a normaloid operator. Theorem 1.11. r(T ) = T  if and only if T n  = T n for every n ≥ 1. Proof. Immediate by the above inequalities and the definition of r(T ).



Theorem 1.12. Every hyponormal operator is normaloid . Proof. Let T ∈ B[H] be a hyponormal operator on a Hilbert space H. Claim 1. T n 2 ≤ T n+1 T n−1  for every positive integer n. Proof. Indeed, for any operator T ∈ B[H], T n x2 = T n x ; T n x = T ∗ T n x ; T n−1 x ≤ T ∗ T n xT n−1 x for each n ≥ 1 and every x ∈ H. Now if T is hyponormal, then T ∗ T n xT n−1 x ≤ T n+1 xT n−1 x ≤ T n+1 T n−1 x2 by Theorem 1.7 and hence, for each n ≥ 1 and every x ∈ H, T n x2 ≤ T n+1 T n−1 x2 , which ensures the claimed result. This completes the proof of Claim 1. Claim 2. T k  = T k for every positive integer k ≤ n, for all n ≥ 1. Proof. The above result holds trivially if T = O and it also holds trivially for n = 1 (for all T ∈ B[H]). Let T = O and suppose the above result holds for some integer n ≥ 1. By Claim 1 we get T 2n = (T n )2 = T n 2 ≤ T n+1 T n−1  = T n+1 T n−1 . Therefore, as T n  ≤ T n for every n ≥ 1, and since T = O, T n+1 = T 2n (T n−1 )−1 ≤ T n+1  ≤ T n+1 .

16

1. Introductory Results

Hence T n+1  = T n+1 . Then the claimed result holds for n + 1 whenever it holds for n, which concludes the proof of Claim 2 by induction. Therefore, T n  = T n for every integer n ≥ 1 by Claim 2, and so T is normaloid by Theorem 1.11.  Since T ∗n  = T n  for each n ≥ 1, then r(T ∗ ) = r(T ). Thus an operator T is normaloid if and only if its adjoint T ∗ is normaloid, and so (by Theorem 1.12) every seminormal operator is normaloid. In particular, every normal operator is normaloid . Summing up: an operator T is normal if it commutes with its adjoint (i.e., T T ∗ = T ∗ T ), hyponormal if T T ∗ ≤ T ∗ T , and normaloid if r(T ) = T . These classes are related by proper inclusion: Normal

⊂ Hyponormal

⊂

Normaloid.

1.7 Orthogonal Reducing Eigenspaces Let I denote the identity operator on a Hilbert space H. A scalar operator on H is a scalar multiple of the identity, say λI for any scalar λ in C . (The terms “scalar operator” or “scalar type operator” are also employed with a different meaning — see the paragraph following Proposition 4.J in Section 4.5.) Take an arbitrary operator T on H. For every scalar λ in C consider the kernel N (λI − T ), which is a (closed) subspace of H (since T is bounded). If this kernel is not zero, then it is called an eigenspace of T . Lemma 1.13. Take any operator T ∈ B[H] and an arbitrary scalar λ ∈ C . (a) If T is hyponormal, then N (λI − T ) ⊆ N (λI − T ∗ ). (b) If T is normal, then N (λI − T ) = N (λI − T ∗ ). Proof. For any operator T ∈ B[H] and any scalar λ ∈ C , (λI − T )∗ (λI − T ) − (λI − T ) (λI − T )∗ = T ∗ T − T T ∗ . Thus λI − T is hyponormal (normal) if and only if T is hyponormal (normal). Hence if T is hyponormal, then so is λI − T , and therefore (λI − T ∗ )x ≤ (λI − T )x for every x ∈ H and every λ ∈ C by Theorem 1.7, which yields the inclusion in (a). If T is normal, then the preceding inequality becomes an identity yielding the identity in (b).  Lemma 1.14. Take T ∈ B[H] and λ ∈ C . If N (λI − T ) ⊆ N (λI − T ∗ ), then (a) N (λI − T ) ⊥ N (ζI − T ) whenever ζ = λ, (b) N (λI − T ) reduces T .

and

1.8 Compact Operators and the Fredholm Alternative

17

Proof. (a) Take any x ∈ N (λI − T ) and any y ∈ N (ζI − T ). Thus λx = T x and ζ y = T y. If N (λI − T ) ⊆ N (λI − T ∗ ), then x ∈ N (λI − T ∗ ) and so λx = T ∗ x. Hence ζ y ; x = T y ; x = y ; T ∗x = y ; λx = λy ; x and therefore (ζ − λ)y ; x = 0, which implies y ; x = 0 whenever ζ = λ. (b) If x ∈ N (λI − T ) ⊆ N (λI − T ∗ ), then T x = λx and T ∗ x = λx and hence λT ∗ x = λλx = λλx = λT x = T λx = T T ∗ x, which implies T ∗ x ∈ N (λI − T ). Thus N (λI − T ) is T ∗-invariant. But N (λI − T ) is clearly T -invariant (if T x = λx, then T (T x) = λ(T x)). So N (λI − T ) reduces T by Lemma 1.6.  Lemma 1.15. If T ∈ B[H] is hyponormal, then (a) N (λI − T ) ⊥ N (ζI − T ) for every λ, ζ ∈ C such that λ = ζ,

and

(b) N (λI − T ) reduces T for every λ ∈ C . Proof. Apply Lemma 1.13(a) to Lemma 1.14.



Theorem 1.16. If {λγ }γ∈Γ is a (nonempty) family of distinct complex numbers, and if T ∈ B[H] is hyponormal, then the topological sum 

− M= N (λγ I − T ) γ∈Γ

reduces T , and the restriction of T to it, T |M ∈ B[M], is normal . Proof. For each γ ∈ Γ = ∅ write Nγ = N (λγ I − T ), which is a T -invariant subspace of H. Thus T |Nγ lies in B[Nγ ] and coincides with the scalar operator λγ I on Nγ . By Lemma 1.15(a), {Nγ }γ∈Γ − of pairwise orthogonal  is a family N . According to the Orsubspaces of H. Take an arbitrary x∈ γ γ∈Γ thogonal Structure Theorem, x = γ∈Γ uγ with each uγ in Nγ . Moreover, T uγ and T ∗ uγ lie in Nγ for every γ ∈ Γ because Nγ reduces T (Lemmas and continuous we con1.6 and 1.15(b)).Thus since T and T ∗ are linear −   ∗ ∗ N and T x = T uγ lies clude that T x = γ∈Γ T uγ lies in γ γ∈Γ − − γ∈Γ   N . Then N reduces T (Lemma 1.6). Finally, since in γ γ γ∈Γ γ∈Γ we may iden{Nγ }γ∈Γ is a family of orthogonal subspaces (Lemma 1.15(a)),  tify the topological sum M with the orthogonal direct sum γ∈Γ Nγ by the Orthogonal Structure Theorem, where each Nγ reduces T |M (it reduces T by Lemma 1.15(b)), and so T |M is identified with the direct sum of operators  T  γ∈Γ |Nγ , which is normal because each T |Nγ is normal.

1.8 Compact Operators and the Fredholm Alternative A set A in a topological space is compact if every open covering of it has a finite subcovering. It is sequentially compact if every A-valued sequence has a subsequence that converges in A. It is relatively compact or conditionally compact if its closure is compact. A set in a metric space is totally bounded if for every ε > 0 there is a finite partition of it into sets of diameter less than ε. The

18

1. Introductory Results

Compactness Theorem says that a set in a metric space is compact if and only if it is sequentially compact, if and only if it is complete and totally bounded . In a metric space every compact set is closed and every totally bounded set is bounded. In a finite-dimensional normed space a set is compact if and only if it is closed and bounded. This is the Heine–Borel Theorem.

Let X and Y be normed spaces. A linear transformation T : X → Y is compact (or completely continuous) if it maps bounded subsets of X into relatively compact subsets of Y; that is, if T (A)− is compact in Y whenever A is bounded in X or, equivalently, if T (A) lies in a compact subset of Y whenever A is bounded in X . Every compact linear transformation is bounded, thus continuous. Every linear transformation on a finite-dimensional space is compact. Let B∞[X , Y ] denote the collection of all compact linear transformations of X into Y. Hence B∞[X , Y ] ⊆ B[X , Y ] (and B∞[X , Y ] = B[X , Y ] whenever X is finite-dimensional). B∞[X , Y ] is a linear manifold of B[X , Y ] which is closed in B[X , Y ] if Y is a Banach space. Set B∞[X ] = B∞[X , X ], the collection of all compact operators on X . B∞[X ] is an ideal (i.e., a two-sided ideal) of the normed algebra B[X ]. That is, B∞[X ] is a subalgebra of B[X ] for which the product of a compact operator with a bounded operator is again compact. Compact operators are supposed in this section to act on a complex nonzero Hilbert space H, although the theory for compact operators equally applies, and usually is developed, in Banach spaces. (But a Hilbert-space approach considerably simplifies proofs since Hilbert spaces are complemented — a normed space is complemented if every subspace has a complementary subspace.) Theorem 1.17. If T ∈ B∞[H] and λ ∈ C \{0}, then R(λI − T ) is closed . Proof. Let M be a subspace of a complex Banach space X . Take any compact transformation K in B∞[M, X ]. Let I be the identity on M, let λ be a nonzero complex number, and consider the operator λI − K in B[M, X ]. Claim. If N (λI − K) = {0}, then R(λI − K) is closed in X . Proof. Suppose N (λI − K) = {0}. If R(λI − K) is not closed in X , then λI − K is not bounded below (Theorem 1.2). Thus for every ε > 0 there is a nonzero vector xε ∈ M for which (λI − K)xε  < εxε . Then there is a sequence {xn } of unit vectors (xn  = 1) in M for which (λI − K)xn  → 0. Since K is compact and {xn } is bounded, Proposition 1.S ensures that {Kxn } has a convergent subsequence, say {Kxk }, and so Kxk → y ∈ X . However, λxk − y = λxk − Kxk + Kxk − y ≤ (λI − K)xk  + Kxk − y → 0. Thus the M-valued sequence {λxk } converges in X to y, and so y ∈ M (because M is closed in X ). Since y is in the domain of K, then y ∈ N (λI − K) (because Ky = K limk λxk = λ limk Kxk = λy since K is continuous). Moreover, y = 0 (because 0 = |λ| = λxk  → y). Hence N (λI − K) = {0}, which is a contradiction. Therefore R(λI − K) is closed in X , concluding the proof of the claimed result.

1.8 Compact Operators and the Fredholm Alternative

19

Take T ∈ B[H] and λ = 0. Consider the restriction (λI − T )|N (λI−T )⊥ of λI − T   to N (λI − T )⊥ in B[N (λI − T )⊥, H]. Clearly N (λI − T )|N (λI−T )⊥ = {0}. If T is compact, then so is the restriction T |N (λI−T )⊥ in B[N (λI − T )⊥, H] (Proposition 1.V in Section 1.9). Since (λI − T )|N (λI−T )⊥ = λI − T |N (λI−T )⊥ (where the symbol I in the right-hand side denotes the identity on N (λI − T )⊥) we get N (λI − T |N (λI−T )⊥) = {0}. Therefore R(λI − T |N (λI−T )⊥) is closed by    the above claim. But R (λI − T )|N (λI−T )⊥ = R(λI − T ). Theorem 1.18. If T ∈ B∞[H], λ ∈ C \{0}, and N (λI − T ) = {0}, then R(λI − T ) = H. Proof. Take any operator T ∈ B[H] and any scalar λ ∈ C . Consider the sequence {Mn }∞ n=0 of linear manifolds of H recursively defined by Mn+1 = (λI − T )(Mn ) for every n ≥ 0,

with

M0 = H.

As it can be verified by induction, Mn+1 ⊆ Mn for every n ≥ 0. In fact, M1 = R(λI − T ) ⊆ H = M0 and, if the above inclusion holds for some n ≥ 0, then Mn+2 = (λI − T )(Mn+1 ) ⊆ (λI − T )(Mn ) = Mn+1 , which concludes the induction. Suppose T ∈ B∞[H] and λ = 0 so that λI − T has a closed range (Theorem 1.17). If N (λI − T ) = {0}, then λI − T has a bounded inverse on its range (Theorem 1.2). Since (λI − T )−1 : R(λI − T ) → H is continuous, λI − T sends closed sets into closed sets. (A map between topological spaces is continuous if and only if the inverse image of closed sets is closed .) Hence, since M0 is closed, by another induction we get {Mn }∞ n=0 is a decreasing sequence of subspaces of H. Now suppose R(λI − T ) = H. If Mn+1 = Mn for some integer n, then (since M0 = H = R(λI − T ) = M1 ) there exists an integer k ≥ 1 for which Mk+1 = Mk = Mk−1 , and this leads to a contradiction, namely, if Mk+1 = Mk , then (λI − T )(Mk ) = Mk and so Mk = (λI − T )−1 (Mk ) = Mk−1 . Outcome: Mn+1 is properly included in Mn for each n; that is, Mn+1 ⊂ Mn for every n ≥ 0. Hence each Mn+1 is a nonzero proper subspace of the Hilbert space Mn . Then each Mn+1 has a nonzero complementary subspace in Mn , namely Mn  Mn+1 ⊂ Mn . Therefore, for each integer n ≥ 0 there is an xn ∈ Mn with xn  = 1 such that 21 < inf u∈Mn+1 xn − u. In light of this, take any pair of integers 0 ≤ m < n and set   x = xn + λ−1 (λI − T )xm − (λI − T )xn (recall: λ = 0). So T xn − T xm = λ(x − xm ). Since x lies in Mm+1 ,

20

1. Introductory Results 1 2 |λ|

< |λ|x − xm  = T xn − T xm ,

and the sequence {T xn } has no convergent subsequence (every subsequence of it is not Cauchy). Since {xn } is bounded, T is not compact (cf. Proposition 1.S again), which is a contradiction. Thus R(λI − T ) = H.  Theorem 1.19. If T ∈ B∞[H] and λ ∈ C \{0} (i.e., if T is compact and λ is nonzero), then dim N (λI − T ) = dim N (λI − T ∗ ) < ∞. Proof. Let dim X stand for the dimension of a linear space X as usual. Take an arbitrary compact operator T ∈ B∞[H] and an arbitrary nonzero scalar λ ∈ C . By Theorem 1.17 we get R(λI − T ) = R(λI − T )− and so R(λI − T ∗ ) = R(λI − T ∗ )− by Lemma 1.5. Thus (cf. Lemma 1.4), N (λI − T ) = {0}

if and only if

N (λI − T ∗ ) = {0}

if and only if

R(λI − T ∗ ) = H, R(λI − T ) = H.

Since T, T ∗ ∈ B∞[H] (Proposition 1.W in Section 1.9), then by Theorem 1.18 N (λI − T ) = {0} ∗

N (λI − T ) = {0}

implies

R(λI − T ) = H,

implies

R(λI − T ∗ ) = H.

Hence, N (λI − T ) = {0} if and only if N (λI − T ∗ ) = {0}. Equivalently, dim N (λI − T ) = 0 if and only if

dim N (λI − T ∗ ) = 0.

Now suppose dim N (λI − T ) = 0, and so dim N (λI − T ∗ ) = 0. The subspace N (λI − T ) = {0} is invariant for T (cf. Proposition 1.E(b) in Section 1.9) and T |N (λI−T ) = λI in B[N (λI − T )]. If T is compact, then so is T |N (λI−T ) (Proposition 1.V). Thus λI = O is compact on N (λI − T ) = {0}, and hence dim N (λI − T ) < ∞ (Proposition 1.Y). Dually, since T ∗ is compact, we get dim N (λI − T ∗ ) < ∞. Hence there are positive integers m and n for which dim N (λI − T ) = m

and

dim N (λI − T ∗ ) = n.

n We show next that m = n. Let {ei }m i=1 and {fi }i=1 be orthonormal bases for ∗ the Hilbert spaces N (λI − T ) and N (λI − T ). Set k = min{m, n} ≥ 1 and consider the mappings S : H → H and S ∗ : H → H defined by

Sx =

k i=1

x ; ei fi

and

S∗x =

k i=1

x ; fi ei

for every x ∈ H. Clearly, S and S ∗ lie in B[H] and S ∗ is the adjoint of S; that is, Sx ; y = x ; S ∗ y for every x, y ∈ H. Actually,   R(S) ⊆ {fi }ki=1 ⊆ N (λI − T ∗ ) and R(S ∗ ) ⊆ {ei }ki=1 ⊆ N (λI − T ), and so S and S ∗ in B[H] are finite-rank operators, thus compact (i.e., they lie in B∞[H] ), and hence T + S and T ∗ + S ∗ also lie in B∞[H] (since B∞[H]

1.9 Supplementary Propositions

21

is a subspace of B[H] ). First suppose m ≤ n (which implies k = m). If x lies in N (λI − (T + S)), then (λI − T )x = Sx. But R(S) ⊆ N (λI − T ∗ ) = Sx lies in R(λI − T )⊥ (Lemma 1.4), and hence (λI − T )x = Sx = 0 (since  m x = i=1 αi ei R(λI − T )). Then x ∈ N (λI − T ) = span {ei }m i=1 and hence  m (for some family of scalars {αi }m ). Therefore 0 = Sx = j=1 αj Sej = m m m i=1 α e ; e f = α f , and so α = 0 for every i = 1, . . . , m — j j i i i i i j=1 i=1 i=1 is an orthonormal set, thus linearly independent. That is, x = 0. Then {fi }m i=1 N (λI − (T + S)) = {0}. Thus, by Theorem 1.18, m ≤ n implies

R(λI − (T + S)) = H.

Dually, using exactly the same argument, n≤m

implies

R(λI − (T ∗ + S ∗ )) = H.

If m < n, then k = m < m + 1 ≤ n and fm+1 ∈ R(λI − (T + S)) = H so that there exists v ∈ H for which (λI − (T + S))v = fm+1 . Hence 1 = fm+1 ; fm+1  = (λI − (T + S))v ; fm+1  = (λI − T )v ; fm+1  − Sv ; fm+1  = 0, which is a contradiction. Indeed, (λI − T )v ; fm+1  = Sv ; fm+1  = 0 since fm+1 ∈ N (λI − T ∗ ) = R(λI − T )⊥ and Sv ∈ R(S) ⊆ span {fi }m i=1 . If n < m, then k = n < n + 1 ≤ m, and en+1 ∈ R(λI − (T ∗ + S ∗ )) = H so that there exists u ∈ H for which (λI − (T ∗ + S ∗ ))u = en+1 . Hence 1 = em+1 ; em+1  = (λI − (T ∗ + S ∗ ))u ; en+1  = (λI − T ∗ )u ; en+1  − S ∗ u ; en+1  = 0, which is again a contradiction (because en+1 ∈ N (λI − T ) = R(λI − T ∗ )⊥ and S ∗ u ∈ R(S ∗ ) ⊆ span {ei }ni=1 ). Therefore m = n.  Together, the results of Theorems 1.17 and 1.19 are referred to as the Fredholm Alternative, which is stated below. Corollary 1.20. (Fredholm Alternative). If T ∈ B∞[H] and λ ∈ C \{0}, then R(λI − T ) is closed and dim N (λI − T ) = dim N (λI − T ∗ ) < ∞.

1.9 Supplementary Propositions Let X be a complex linear space. A map f : X ×X → C is a sesquilinear form if it is additive in both arguments, homogeneous in the first argument, and conjugate homogeneous in the second argument. A Hermitian symmetric sesquilinear form that induces a positive quadratic form is an inner product. Proposition 1.A. (Polarization Identities). If f : X × X → C is a sesquilinear form on a complex linear space X , then for every x, y ∈ X

22

(a1 )

1. Introductory Results

f (x, y) =

1 4



f (x + y, x + y) − f (x − y, x − y)

 + i f (x + iy, x + iy) − i f (x − iy, x − iy) .

If X is a complex inner product space and S, T ∈ B[X ], then for x, y ∈ X  Sx ; T y = 14 S(x + y) ; T (x + y) − S(x − y) ; T (x − y) (a2 )  + iS(x + iy) ; T (x + iy) − iS(x − iy) ; T (x − iy) . (Parallelogram Law). If X is an inner product space and  ·  is the norm generated by the inner product, then for every x, y ∈ X   x + y2 + x − y2 = 2 x2 + y2 . (b) Proposition 1.B. A linear manifold of a Banach space is a Banach space if and only if it is a subspace. A linear transformation is bounded if and only if it maps bounded sets into bounded sets. If X and Y are nonzero normed spaces, then B[X , Y ] is a Banach space if and only if Y is a Banach space. A sum of two subspaces of a Hilbert space may not be a subspace (it may not be closed if the subspaces are not orthogonal). But if one of them is finitedimensional, then the sum is closed (see Proposition A.6 in Appendix A). Proposition 1.C. If M and N are subspaces of a normed space X , and if dim N < ∞, then the sum M + N is a subspace of X . Every subspace of a Hilbert space has a complementary subspace (e.g., its orthogonal complement). However, this is not true for every Banach space — see Remarks A.4 and A.5 in Appendix A. Proposition 1.D. If M and N are complementary subspaces of a Banach space X , then the unique projection E : X → X with R(E) = M and N (E) = N is continuous. Conversely, if E ∈ B[X ] is a (continuous) projection, then R(E) and N (E) are complementary subspaces of X . Proposition 1.E. Let T and S be arbitrary operators on a normed space X . (a) The subspaces N (T ) and R(T )− of X are invariant for T . If S and T commute, then the subspaces N (S), N (T ), R(S)− and R(T )− of X are invariant for both S and T . (b) If T is an operator on a normed space, then N (p(T )) and R(p(T ))− are n T -invariant subspaces for every polynomial p(T ) = i=0 αi T i in T — in − particular, N (λI − T ) and R(λI − T ) are T -invariant for every λ ∈ C . Proposition 1.F. Let T ∈ B[X ] be an operator on a Banach space X . If there exists an invertible operator S ∈ G[X ] for which S − T  < S −1 −1 , then T is itself invertible (i.e., T ∈ G[X ] and so T −1 ∈ B[X ] ). Proposition 1.G. Let M be a linear manifold of a normed space X , let Y be a Banach space, and take T ∈ B[M, Y].

1.9 Supplementary Propositions

23

(a) There exists a unique extension T ∈ B[M−, Y] of T ∈ B[M, Y] over the closure of M. Moreover , T = T . (b) If X is a Hilbert space, then TE ∈ B[X , Y] is a (bounded linear ) extension of T ∈ B[M, Y] over the whole space X , where E ∈ B[X ] is the orthogonal projection onto the subspace M− of X , so that TE ≤ T . Proposition 1.H. Let M and N be subspaces of a Hilbert space. (a) M⊥ ∩ N ⊥ = (M + N )⊥.     (b) dim N < ∞ =⇒ dim N  (M ∩ N ) = dim M⊥  (M + N )⊥ . Proposition 1.I. A subspace M of a Hilbert space H is invariant (or reducing) for a nonzero T ∈ B[H] if and only if the unique orthogonal projection E ∈ B[H] with R(E) = M is such that E T E = T E (or T E = E T ). Take T ∈ B[X ] and S ∈ B[Y] on normed spaces X and Y, and X ∈ B[X , Y ] of X into Y. If X T = SX, then X intertwines T to S (or T is intertwined to S through X). If there is an X with dense range intertwining T to S, then T is densely intertwined to S. If X has dense range and is injective, then it is quasiinvertible (or a quasiaffinity). If a quasiinvertible X intertwines T to S, then T is a quasiaffine transform of S. If T is a quasiaffine transform of S and S is a quasiaffine transform of T , then T and S are quasisimilar . If an invertible X (with a bounded inverse) intertwines T to S (and so X −1 intertwines S to T ), then T and S are similar (or equivalent). Unitary equivalence is a special case of similarity on inner product spaces through a (surjective) isometry: operators T and S on inner product spaces X and Y are unitarily equivalent (notation: T ∼ = S) if there is a unitary transformation X intertwining them. A linear manifold (or a subspace) of a normed space X is hyperinvariant for an operator T ∈ B[X ] if it is invariant for every operator in B[X ] that commutes with T . Obviously, hyperinvariant subspaces are invariant. Proposition 1.J. Similarity preserves invariant subspaces (if two operators are similar, and if one has a nontrivial invariant subspace, then so has the other ), and quasisimilarity preserves hyperinvariant subspaces (if two operators are quasisimilar, and if one has a nontrivial hyperinvariant subspace, then so has the other ). is an orthogonal sequence of orthogonal projections Proposition 1.K. If {Ek } n s on a Hilbert space H, then −E, where E ∈ B[H] is the orthogonal   k=1 Ek −→ R(E ) . projection with R(E) = k k∈N Proposition 1.L. A bounded linear transformation between Hilbert spaces is invertible if and only if its adjoint is (i.e., T ∈ G[H, K] ⇐⇒ T ∗ ∈ G[K, H] ). Moreover , (T ∗ )−1 = (T −1 )∗ .

24

1. Introductory Results

Proposition 1.M. Every nonnegative operator Q ∈ B[H] on a Hilbert space 1 1 H has a unique nonnegative square root Q 2 ∈ B[H] (i.e., (Q 2 )2 = Q), which commutes with every operator in B[H] that commutes with Q. Proposition 1.N. Every T ∈ B[H, K] between Hilbert spaces H and K has a unique polar decomposition T = W |T |, where W ∈ B[H, K] is a partial isom1 etry (i.e., W |N (W )⊥ : N (W )⊥ → K is an isometry) and |T | = (T ∗ T ) 2 . Proposition 1.O. Every operator T ∈ B[H] on a complex Hilbert space H has a unique Cartesian decomposition T = A + iB, with A = 12 (T + T ∗ ) and B = − 2i (T − T ∗ ) are self-adjoint operators on H. Moreover, the operator T is normal if and only if A and B commute and, in this case, T ∗ T = A2 + B 2 and max{A2 , B2 } ≤ T 2 ≤ A2 + B2 . Proposition 1.P. The restriction T |M of a hyponormal operator T to an invariant subspace M is hyponormal. If T |M is normal, then M reduces T . Proposition 1.Q. The restriction T |M of a normal operator T to an invariant subspace M is normal if and only if M reduces T . Recall from Section 1.5: a Hilbert-space operator is reductive if every invariant subspace is reducing (i.e., T is reductive if T (M) ⊆ M implies T (M⊥ ) ⊆ M⊥ or, equivalently, if T (M) ⊆ M implies T ∗ (M) ⊆ M). Therefore Proposition 1.Q can be restated as follows. A normal operator is reductive if and only if its restriction to every invariant subspace is normal . Proposition 1.R. A transformation U between Hilbert spaces is unitary if and only if it is an invertible contraction whose inverse also is a contraction (U  ≤ 1 and U −1  ≤ 1), which happens if and only if U  = U −1  = 1. Proposition 1.S. A linear transformation between normed spaces is compact if and only if it maps bounded sequences into sequences that have a convergent subsequence. Proposition 1.T. A linear transformation of a normed space into a Banach space is compact if and only if it maps bounded sets into totally bounded sets. Proposition 1.U. A linear transformation of a Hilbert space into a normed space is compact if and only if it maps weakly convergent sequences into strongly convergent sequences. Proposition 1.V. The restriction of a compact linear transformation to a linear manifold is again a compact linear transformation. Proposition 1.W. A linear transformation between Hilbert spaces is compact if and only if its adjoint is (i.e., T ∈ B∞[H, K] ⇐⇒ T ∗ ∈ B∞[K, H] ).

1.9 Supplementary Propositions

25

Proposition 1.X. A finite-rank (i.e., one with a finite-dimensional range) bounded linear transformation between normed spaces is compact. Proposition 1.Y. The identity I on a normed space X is compact if and only if dim X < ∞. Hence, if T ∈ B∞[X ] is invertible, then dim X < ∞. A sequence {xn } of vectors in a normed space X is a Schauder basis for X if  each vector x in X is uniquely written as countable linear combination x = k αk xk of {xn }. Banach spaces with a Schauder basis are separable (i.e., have a countable dense subset or, equivalently, are spanned by a countable set — see, e.g., [78, Problem 4.11, Theorem 4.9] ). Proposition 1.Z. (a) If a sequence of finite-rank bounded linear transformations between Banach spaces converges uniformly, then its limit is compact. (b) Every compact linear transformation between Banach spaces possessing a Schauder basis is the uniform limit of a sequence of finite-rank bounded linear transformations. Notes: These are standard results required in the sequel. Proofs can be found in many texts on operator theory (see Suggested Reading). Proposition 1.A can be found in [78, Propositions 5.4 and Problem 5.3], and Proposition 1.B in [78, Propositions 4.7, 4.12, 4.15, and Example 4.P]. For Proposition 1.C see [30, Proposition III.4.3] or [58, Problem 13] — see Proposition A.6 in Appendix A. For Proposition 1.D see [30, Theorem III.13.2] or [78, Problem 4.35] — see Remark A.5 in Appendix A. For Propositions 1.E(a,b) see [78, Problems 4.20 and 4.22]. Proposition 1.F is a consequence of the Neumann expansion (Theorem 1.3) — see, e.g., [58, Solution 100] or [30, Corollary VII.2.39b)]. Proposition 1.G(a) is called extension by continuity; see [78, Theorem 4.35]. It implies Proposition 1.G(b): an orthogonal projection is continuous and T = TE|R(E) in B[M−, H] is a restriction of TE in B[X , H] to R(E) = M−. Proposition 1.H will be needed in Theorem 5.4. For Proposition 1.H(a) see [78, Problems 5.8]. Here is a proof for Proposition 1.H(b). Proof of Proposition 1.H(b). Since (M ∩ N ) ⊥ (M⊥ ∩ N ), consider the orthogonal projection E ∈ B[H] onto M⊥ ∩ N (i.e., R(E) = M⊥ ∩ N ) along M ∩ N (i.e., N (E) = M ∩ N ). Let L : N  (M ∩ N ) → M⊥ ∩ N be the restriction of E to N  (M ∩ N ). It is clear that L = E|N (M∩N ) is linear, injective (N (L) = {0}), and surjective (R(L) = R(E) = M⊥ ∩ N ). Therefore N  (M ∩ N ) is isomorphic to M⊥ ∩ N . Since M is closed and N is finitedimensional, M + N is closed (even if M and N are not orthogonal — cf. Proposition 1.C). Then (Projection Theorem) H = (M + N )⊥ ⊕ (M + N ) and M⊥ = (M+N )⊥ ⊕[M⊥ (M+N )⊥ ], and hence M⊥  (M + N )⊥ coincides with M⊥ ∩ (M + N ) = M⊥ ∩ N . Therefore N  (M ∩ N ) is isomorphic to  M⊥  (M + N )⊥, and so they have the same dimension. For Proposition 1.I see, e.g., [75, Solutions 4.1 and 4.6], and for Proposition 1.J see, e.g., [74, Corollaries 4.2 and 4.8]. By the way, does quasisimilarity preserve

26

1. Introductory Results

nontrivial invariant subspaces? (See, e.g., [98, p. 194].) As for Propositions 1.K and 1.L see [78, Propositions 5.28 and p. 385]). Propositions 1.M, 1.N, and 1.O are the classical Square Root Theorem, the Polar Decomposition Theorem, and the Cartesian Decomposition Theorem (see, e.g., [78, Theorems 5.85 and 5.89, and Problem 5.46]). Propositions 1.P, 1.Q, 1.R can be found in [78, Problems 6.16 and 6.17, and Proposition 5.73]. Propositions 1.S, 1.T, 1.U, 1.V, 1.W, 1.X, 1.Y, and 1.Z are all related to compact operators (see, e.g., [78, Theorem 4.52(d,e), Problem 4.51, p. 258, Problem 5.41, Proposition 4.50, remark to Theorem 4.52, Corollary 4.55, Problem 4.58]).

Suggested Readings Akhiezer and Glazman [4, 5] Beals [13] Berberian [18, 19] Brown and Pearcy [23, 24] Conway [30, 31, 32, 33] Douglas [38] Dunford and Schwartz [45]

Fillmore [48] Halmos [54, 55, 58] Harte [60] Istrˇ a¸tescu [69] Kubrusly [74, 75, 78] Radjavi and Rosenthal [98] Weidmann [113]

2 Spectrum of an Operator

Let T : M → X be a linear transformation, where X is a nonzero normed space and M = D(T ) is the domain of T which is a linear manifold of X . Let F denote either the real field R or the complex field C , and let I be the identity operator on M. The general notion of spectrum applies to bounded or unbounded linear transformations T from M ⊆ X to X . The resolvent set ρ(T ) of T is the set of all scalars λ in F for which the linear transformation λI − T : M → X has a densely defined continuous inverse:   ρ(T ) = λ ∈ F : (λI − T )−1 ∈ B[R(λI − T ), M] and R(λI − T )− = X (see, e.g., [9, Definition 18.2] — there are different definitions of resolvent set for unbounded linear transformations, but they all coincide for the bounded case). We will, however, restrict the theory to bounded linear transformations of a complex Banach space into itself (i.e., to elements of the complex Banach algebra B[X ] ). The spectrum σ(T ) of T is the complement in C of ρ(T ).

2.1 Basic Spectral Properties Throughout this chapter T : X → X will be a bounded linear transformation of X into itself (i.e., an operator on X ), and so D(T ) = X where X = {0} is a complex Banach space. In other words, T ∈ B[X ] where X is a nonzero complex Banach space. In such a case (i.e., in the unital complex Banach algebra B[X ] ), the resolvent set ρ(T ) is precisely the set of all complex numbers λ for which λI − T ∈ B[X ] is invertible (i.e., has a bounded inverse on X ), according to Theorem 1.2. Thus (see also Theorem 1.1) the resolvent set is given by     ρ(T ) = λ ∈ C : λI − T ∈ G[X ] = λ ∈ C : λI − T has an inverse in B[X ]   = λ ∈ C : N (λI − T ) = {0} and R(λI − T ) = X , and the spectrum is   σ(T ) = C \ ρ(T ) = λ ∈ C : λI − T has no inverse in B[X ]   = λ ∈ C : N (λI − T ) = {0} or R(λI − T ) = X . © Springer Nature Switzerland AG 2020 C. S. Kubrusly, Spectral Theory of Bounded Linear Operators, https://doi.org/10.1007/978-3-030-33149-8_2

27

28

2. Spectrum of an Operator

Theorem 2.1. ρ(T ) is nonempty and open, and σ(T ) is compact. Proof. Take any T ∈ B[X ]. By the Neumann expansion (Theorem 1.3), if T  < |λ|, then λ ∈ ρ(T ). Equivalently, since σ(T ) = C \ρ(T ), |λ| ≤ T  for every λ ∈ σ(T ). Thus σ(T ) is bounded, and therefore ρ(T ) = ∅. Claim. If λ ∈ ρ(T ), then the open ball Bδ (λ) with center at λ and (positive) radius δ = (λI − T )−1 −1 is included in ρ(T ). Proof. If λ ∈ ρ(T ), then λI − T ∈ G[X ]. Hence (λI − T )−1 is nonzero and bounded. Thus 0 < (λI − T )−1 −1 < ∞. Set δ = (λI − T )−1 −1, let Bδ (0) be the nonempty open ball of radius δ about the origin of the complex plane C , and take any ζ in Bδ (0). Since |ζ| < (λI − T )−1 −1, then ζ(λI − T )−1  < 1. Therefore [I − ζ(λI − T )−1 ] ∈ G[X ] by Theorem 1.3, and so (λ − ζ)I − T = (λI − T )[I − ζ(λI − T )−1 ] ∈ G[X ]. Thus λ − ζ ∈ ρ(T ) so that   Bδ (λ) = Bδ (0) + λ = λ − Bδ (0) = λ − ζ ∈ C : for some ζ ∈ Bδ (0) ⊆ ρ(T ), which completes the proof of the claimed result. Thus ρ(T ) is open (it includes a nonempty open ball centered at each of its points) and so σ(T ) is closed. Compact in C means closed and bounded.  Remark. Since Bδ (λ) ⊆ ρ(T ), the distance of any λ in ρ(T ) to the spectrum σ(T ) is greater than or equal to δ; that is (compare with Proposition 2.E), λ ∈ ρ(T )

implies

(λI − T )−1 −1 ≤ d(λ, σ(T )).

The resolvent function RT : ρ(T ) → G[X ] of an operator T ∈ B[X ] is the mapping of the resolvent set ρ(T ) of T into the group G[X ] of all invertible operators from B[X ] defined by RT (λ) = (λI − T )−1 for every λ ∈ ρ(T ). Since RT (λ) − RT (ζ) = RT (λ)[RT (ζ)−1 − RT (λ)−1 ]RT (ζ), we get RT (λ) − RT (ζ) = (ζ − λ)RT (λ)RT (ζ) for every λ, ζ ∈ ρ(T ) (because RT (ζ)−1 − RT (λ)−1 = (ζ − λ)I). This is the resolvent identity. Swapping λ and ζ in the resolvent identity, it follows that RT (λ) and RT (ζ) commute for every λ, ζ ∈ ρ(T ). Also, T RT (λ) = RT (λ)T for every λ ∈ ρ(T ) (since RT (λ)−1 RT (λ) = RT (λ)RT (λ)−1 trivially). Let Λ be a nonempty open subset of the complex plane C . Take a function f : Λ → C and a point ζ ∈ Λ. Suppose there exists a complex number f  (ζ) with the following property. For every ε > 0 there is a δ > 0 for which

2.1 Basic Spectral Properties

29

   f (λ)−f (ζ) − f  (ζ) < ε for all λ in Λ such that 0 < |λ − ζ| < δ. If there exists λ−ζ such an f  (ζ) ∈ C , then it is called the derivative of f at ζ. If f  (ζ) exists for every ζ in Λ, then f : Λ → C is analytic or holomorphic (on its domain Λ). In this context the above terms are synonyms, implying infinite differentiability and power series expansion. A function f : C → C is entire if it is analytic on the whole complex plane C . To prove the next result we need the Liouville Theorem, which says that every bounded entire function is constant. Theorem 2.2. If X is nonzero, then the spectrum σ(T ) is nonempty. Proof. (Note: if X = {0}, then T = O = I, and so ρ(T ) = C .) Let T ∈ B[X ] be ∗ an operator on a nonzero complex Banach space X. Let B[X ] = B[B[X ], C ] be the Banach space of all bounded linear functionals on B[X ]; that is, the dual ∗ of B[X ]. Since X = {0}, then B[X ] = {O}, and so B[X ] = {0} (a consequence of the Hahn–Banach Theorem — see, e.g., [78, Corollary 4.64] ). Take any non∗ zero ξ in B[X ] (i.e., a nonzero bounded linear functional ξ : B[X ] → C ), and consider the composition of it with the resolvent function, ξ ◦ RT : ρ(T ) → C . Recall: ρ(T ) = C \σ(T ) is nonempty and open in C . Claim 1. If σ(T ) is empty, then ξ ◦ RT : ρ(T ) → C is bounded. Proof. The resolvent function RT : ρ(T ) → G[X ] is continuous (since scalar multiplication and addition are continuous mappings, and inversion is also a continuous mapping; see, e.g., [78, Problem 4.48] ). Thus RT ( · ) : ρ(T ) → R is continuous. Then sup|λ|≤T  RT (λ) < ∞ if σ(T ) = ∅. In fact if σ(T ) is empty, then ρ(T ) ∩ BT  [0] = BT  [0] = {λ ∈ C : |λ| ≤ T } is a compact set in C , so that the continuous function RT ( · ) attains its maximum on it by the Weierstrass Theorem: a continuous real-valued function attains its maximum and minimum on any compact set in a metric space. On the other hand, if T  < |λ|, then RT (λ) ≤ (|λ| − T )−1 (see the remark following Theorem 1.3), and so RT (λ) → 0 as |λ| → ∞. Thus, since RT ( · ) : ρ(T ) → R is continuous, supT ≤λ RT (λ) < ∞. Hence supλ∈ρ(T ) RT (λ) < ∞. Thus sup (ξ ◦ RT )(λ) ≤ ξ sup RT (λ) < ∞,

λ∈ρ(T )

λ∈ρ(T )

which completes the proof of Claim 1. Claim 2. ξ ◦ RT : ρ(T ) → C is analytic. Proof. If λ and ζ are distinct points in ρ(T ), then   RT (λ) − RT (ζ) + RT (ζ)2 = RT (ζ) − RT (λ) RT (ζ) λ−ζ by the resolvent identity. Set f = ξ ◦ RT : ρ(T ) → C . Let f  : ρ(T ) → C be defined by f  (λ) = −ξ(RT (λ)2 ) for each λ ∈ ρ(T ). Therefore,  f (λ) − f (ζ)        − f  (ζ) = ξ RT (ζ) − RT (λ) RT (ζ)   λ−ζ ≤ ξRT (ζ)RT (ζ) − RT (λ)

30

2. Spectrum of an Operator

so that f : ρ(T ) → C is analytic because RT : ρ(T ) → G[X ] is continuous, which completes the proof of Claim 2. Thus, by Claims 1 and 2, if σ(T ) = ∅ (i.e., if ρ(T ) = C ), then ξ ◦ RT : C → C is a bounded entire function, and so a constant function by the Liouville Theorem. But (see proof of Claim 1) RT (λ) → 0 as |λ| → ∞, and hence ξ(RT (λ)) → 0 as |λ| → ∞ (since ξ is continuous). Then ξ ◦ RT = 0 for all ξ ∗ in B[H] = {0} and so RT = O (by the Hahn–Banach Theorem). That is,  (λI − T )−1 = O for λ ∈ C , which is a contradiction. Thus σ(T ) = ∅. Remark. The spectrum σ(T ) is compact and nonempty, and so is its boundary ∂σ(T ). Hence, ∂σ(T ) = ∂ρ(T ) = ∅.

2.2 A Classical Partition of the Spectrum The spectrum σ(T ) of an operator T in B[X ] is the set of all scalars λ in C for which the operator λI − T fails to be an invertible element of the algebra B[X ] (i.e., fails to have a bounded inverse on R(λI − T ) = X ). According to the nature of such a failure, σ(T ) can be split into many disjoint parts. A classical partition comprises three parts. The set σP (T ) of those λ for which λI − T has no inverse (i.e., for which the operator λI − T is not injective) is the point spectrum of T ,   σP (T ) = λ ∈ C : N (λI − T ) = {0} . A scalar λ ∈ C is an eigenvalue of T if there exists a nonzero vector x in X such that T x = λx. Equivalently, λ is an eigenvalue of T if N (λI − T ) = {0}. If λ ∈ C is an eigenvalue of T , then the nonzero vectors in N (λI − T ) are the eigenvectors of T , and N (λI − T ) is the eigenspace (which is a subspace of X ), associated with the eigenvalue λ. The multiplicity of an eigenvalue is the dimension of the respective eigenspace. Thus the point spectrum of T is precisely the set of all eigenvalues of T . Now consider the set σC (T ) of those λ for which λI − T has a densely defined but unbounded inverse on its range,   σC (T ) = λ ∈ C : N (λI − T ) = {0}, R(λI − T )− = X and R(λI − T ) = X (see Theorem 1.3), which is referred to as the continuous spectrum of T . The residual spectrum of T is the set σR (T ) = σ(T )\(σP (T ) ∪ σC (T )) of all scalars λ such that λI − T has an inverse on its range which is not densely defined:   σR (T ) = λ ∈ C : N (λI − T ) = {0} and R(λI − T )− = X . The collection {σP (T ), σC (T ), σR (T )} is a partition of σ(T ): σP (T ) ∩ σC (T ) = σP (T ) ∩ σR (T ) = σC (T ) ∩ σR (T ) = ∅, σ(T ) = σP (T ) ∪ σC (T ) ∪ σR (T ).

2.2 A Classical Partition of the Spectrum

31

The diagram below, borrowed from [74], summarizes such a partition of the spectrum. The residual spectrum is split into two disjoint parts, σR(T ) = σR1(T ) ∪ σR2(T ), and the point spectrum into four disjoint parts, σP (T ) =

4 i=1 σPi(T ). We adopt the following abbreviated notation: Tλ = λI − T , Nλ = N (Tλ ), and Rλ = R(Tλ ). Recall that if N (Tλ ) = {0}, then its linear inverse Tλ−1 on Rλ is continuous if and only if Rλ is closed (Theorem 1.2). R− =X λ

R− = X λ

R− = Rλ R− = Rλ R− = Rλ R− = Rλ λ λ λ λ

Nλ = {0}

Tλ−1 ∈ B[Rλ ,X ]

ρ (T )

∅

∅

σR1(T )

Tλ−1 ∈ / B[Rλ ,X ]

∅

σC (T )

σR2(T )

∅

⎫ ⎬

σP1(T )

σP2(T )

σP3(T )

σP4(T )

⎭

Nλ = {0}





σCP (T )

σAP (T )



Fig. § 2.2. A classical partition of the spectrum According to Theorem 2.2, σ(T ) = ∅. But any of the above disjoint parts of the spectrum may be empty (Section 2.7). However, if σP (T ) = ∅, then a set of eigenvectors associated with distinct eigenvalues is linearly independent. Theorem 2.3. Let {λγ }γ∈Γ be a family of distinct eigenvalues of T . For each γ ∈ Γ let xγ be an eigenvector associated with λγ . The set {xγ }γ∈Γ is linearly independent. Proof. For each γ ∈ Γ take 0 = xγ ∈ N (λγ I − T ) = {0}, and consider the set {xγ }γ∈Γ (whose existence is ensured by the Axiom of Choice). Claim. Every finite subset of {xγ }γ∈Γ is linearly independent. Proof. Take any finite subset {xi }m i=1 of {xγ }γ∈Γ . The singleton {x1 } is trivfor some ially linearly independent. Suppose {xi }ni=1 is linearly independent n 1 ≤ n < m. If {xi }n+1 i=1 is linearly dependent, then xn+1 = i=1 αi xi , where the family {αi }ni=1 of complex numbers has at least one nonzero number. Thus n n λn+1 xn+1 = T xn+1 = αi T x i = αi λi x i . i=1

i=1

If λn+1 = 0, λi = 0 for every i = n + 1 (because the eigenvalues are disthen n not linearly independent, which tinct) and i=1 αi λi xi = 0 so that {xi }ni=1 is n is a contradiction. If λn+1 = 0, then xn+1 = i=1 αi λ−1 n+1 λi xi , and therefore n −1 n α (1 − λ λ )x = 0 so that {x } is not linearly independent (since i i=1 n+1 i i i=1 i λi = λn+1 for every i = n + 1 and αi = 0 for some i), which is again a contradiction. This completes the proof by induction:{xi}n+1 i=1 is linearly independent. However, if every finite subset of {xγ }γ∈Γ is linearly independent, then so is  the set {xγ }γ∈Γ itself (see, e.g., [78, Proposition 2.3] ).

32

2. Spectrum of an Operator

There are some overlapping parts of the spectrum which are commonly used: for instance, the compression spectrum σCP (T ) and the approximate point spectrum (or approximation spectrum) σAP (T ), which are defined by   σCP (T ) = λ ∈ C : R(λI − T ) is not dense in X = σP3(T ) ∪ σP4(T ) ∪ σR (T ),   σAP (T ) = λ ∈ C : λI − T is not bounded below = σP (T ) ∪ σC (T ) ∪ σR2(T ) = σ(T )\σR1(T ). The points of σAP (T ) are referred to as the approximate eigenvalues of T . Theorem 2.4. The following assertions are pairwise equivalent. (a) For every ε > 0 there is a unit vector xε in X such that (λI −T )xε  < ε. (b) There is a sequence {xn } of unit vectors in X such that (λI −T )xn  → 0. (c) λ ∈ σAP (T ). Proof. Clearly (a) implies (b). If (b) holds, then there is no constant α > 0 for which α = αxn  ≤ (λI − T )xn  for all n. Thus λI − T is not bounded below, and so (b) implies (c). Conversely, if λI − T is not bounded below, then there is no constant α > 0 such that αx ≤ (λI − T )x for all x ∈ X or, equivalently, for every ε > 0 there exists a nonzero yε in X for which  (λI − T )yε  < εyε . Set xε = yε −1 yε , and hence (c) implies (a). Theorem 2.5. The approximate point spectrum σAP (T ) is a nonempty closed subset of C that includes the boundary ∂σ(T ) of the spectrum σ(T ). Proof. By Theorems 2.1 and 2.2, ρ(T ) is nonempty and σ(T ) is nonempty and compact. Hence ∅ = ∂σ(T ) = ∂ρ(T ) = ρ(T )− ∩ σ(T ). Take an arbitrary λ in ∂σ(T ). Then λ is a point of adherence of ρ(T ) and so there is a ρ(T )-valued sequence {λn } (thus λn I − T ∈ G[X ]) such that λn → λ. Moreover, for each n (λn I − T ) − (λI − T ) = (λn − λ)I. Claim. supn (λn I − T )−1  = ∞. Proof. Since λn ∈ ρ(T ), then (λn I − T )−1 ∈ B[X ]. So 0 < (λn I − T )−1  < ∞ for each n. Suppose supn (λn I − T )−1  < ∞. Thus there is a β > 0 for which 0 < (λn I − T )−1  < β or, equivalently, β −1 < (λn I − T )−1 −1 < ∞, for all n. Since λn ∈ ρ(T ) → λ ∈ ρ(T ), take n large enough so that 0 < |λn − λ| < β −1. Hence by the above displayed identity (λn I − T ) − (λI − T ) < β −1 < (λn I − T )−1 −1 . Thus according to Proposition 1.F in Section 1.9 λI − T is invertible (i.e., λI − T ∈ G[X ] ) and so λ lies in ρ(T ), which is a contradiction because λ was taken in ∂σ(T ) ⊆ σ(T ). This completes the proof of the claimed result.

2.2 A Classical Partition of the Spectrum

33

Since (λn I − T )−1  = supy=1 (λn I − T )−1 y, then for each positive integer n we may take a unit vector yn in X such that (λn I − T )−1  −

1 n

≤ (λn I − T )−1 yn  ≤ (λn I − T )−1 .

But supn (λn I − T )−1 yn  = ∞ according to the preceding claim, and therefore inf n (λn I − T )−1 yn −1 = 0. So there exist subsequences {λk } and {yk } of {λn } and {yn } for which (λk I − T )−1 yk −1 → 0. Set xk = (λk I − T )−1 yk −1 (λk I − T )−1 yk and get a sequence {xk } of unit vectors in X such that (λk I − T )xk  = (λk I − T )−1 yk −1 . Hence (λI − T )xk  = (λk I − T )xk − (λk − λ)xk  ≤ (λk I − T )−1yk −1 + |λk − λ|. Since λk → λ and (λk I − T )−1 yk −1 → 0, then (λI − T )xk  → 0 and so λ ∈ σAP (T ) by Theorem 2.4. Thus ∂σ(T ) ⊆ σAP (T ). Then σAP (T ) = ∅. Finally, take an arbitrary λ ∈ C \σAP (T ). By definition λI − T is bounded below. Hence there exists an α > 0 for which αx ≤ (λI − T )x = (λI − ζI + ζI − T )x ≤ (ζI − T )x + (λ − ζ)x and so

(α − |λ − ζ|)x ≤ (ζI − T )x

for every x ∈ X and every ζ ∈ C . Then ζI − T is bounded below for every ζ such that 0 < α − |λ − ζ|. Equivalently, ζ ∈ C \σAP (T ) for every ζ sufficiently close to λ (i.e., if |λ − ζ| < α). Thus the nonempty open ball Bα (λ) centered at λ is included in C \σAP (T ). So C \σAP (T ) is open, and σAP (T ) is closed.  Remark. Therefore, since σAP (T ) = σ(T )\σR1(T ) is closed in C and includes the common boundary ∂σ(T ) = ∂ρ(T ), C \σR1(T ) = ρ(T ) ∪ σAP (T ) = ρ(T ) ∪ ∂ρ(T ) ∪ σAP (T ) = ρ(T )− ∪ σAP (T )

is closed in C . Outcome: σR1(T ) is an open subset of C . Next we assume T is an operator acting on a nonzero complex Hilbert space H. The Hilbert-space structure brings forth important simplifications. A particularly useful example of such simplifications is the formula for the residual spectrum in terms of point spectra in Theorem 2.6 below. This considerably simplifies the notion of residual spectrum for Hilbert-space operators and gives

34

2. Spectrum of an Operator

a complete characterization for the spectrum of T ∗ and its parts in terms of the spectrum of T and its parts. First recall the following piece of notation. If Λ is any subset of C , then set   Λ∗ = λ ∈ C : λ ∈ Λ . Clearly, Λ∗∗ = Λ, (C \Λ)∗ = C \Λ∗, and (Λ1 ∪ Λ2 )∗ = Λ∗1 ∪ Λ∗2 . Theorem 2.6. If T ∗ ∈ B[H] is the adjoint of T ∈ B[H], then ρ(T ) = ρ(T ∗ )∗ ,

σ(T ) = σ(T ∗ )∗ ,

σC (T ) = σC (T ∗ )∗ ,

and the residual spectrum of T is given by the formula σR (T ) = σP (T ∗ )∗ \σP (T ). As for the subparts of the point and residual spectra, σP1(T ) = σR1(T ∗ )∗ ,

σP2(T ) = σR2(T ∗ )∗ ,

σP3(T ) = σP3(T ∗ )∗ ,

σP4(T ) = σP4(T ∗ )∗ .

For the compression and approximate point spectra we get σCP (T ) = σP (T ∗ )∗ ,

σAP (T ∗ )∗ = σ(T )\σP1(T ),

∂σ(T ) ⊆ σAP (T ) ∩ σAP (T ∗ )∗ = σ(T )\(σP1(T ) ∪ σR1(T )). Proof. Since S ∈ G[H] if and only if S ∗ ∈ G[H], we get ρ(T ) = ρ(T ∗ )∗ . Hence σ(T )∗ = (C \ρ(T ))∗ = C \ρ(T ∗ ) = σ(T ∗ ). Recall: R(S)− = R(S) if and only if R(S ∗ )− = R(S ∗ ), and N (S) = {0} if and only if R(S ∗ )⊥ = {0} (Lemmas 1.4 and 1.5) which means R(S ∗ )− = H. So σP1(T ) = σR1(T ∗ )∗, σP2(T ) = σR2(T ∗ )∗, σP3(T ) = σP3(T ∗ )∗, and σP4(T ) = σP4(T ∗ )∗. Using the same argument, σC (T ) = σC (T ∗ )∗ and σCP (T ) = σP (T ∗ )∗ . Hence σR (T ) = σCP (T )\σP (T )

implies

σR (T ) = σP (T ∗ )∗ \σP (T ).

Moreover, since σ(T ∗ )∗ = σ(T ) and σR1(T ∗ )∗ = σP1(T ), and also recalling that σAP (T ) = σ(T )\σR1(T ), we get σAP (T ∗ )∗ = σ(T ∗ )∗ \σR1(T ∗ )∗ = σ(T )\σP1(T ). Thus σAP (T ∗ )∗ ∩ σAP (T ) = σ(T )\(σP1(T ) ∪ σR1(T )). But σ(T ) is closed and σR1(T ) is open (and so is σP1(T ) = σR1(T ∗ )∗ ) in C . Then σP1(T ) ∪ σR1(T ) ⊆ σ(T )◦ , where σ(T )◦ denotes the interior of σ(T ). Therefore ∂σ(T ) ⊆ σ(T )\(σP1(T ) ∪ σR1(T )).



Remark. Since σP1(T ) = σR1(T ∗ )∗ and since σR1(T ) is open in C for every operator T (as in the previous remark), then σP1(T ) is an open subset of C .

2.3 Spectral Mapping Theorems

35

2.3 Spectral Mapping Theorems Spectral Mapping Theorems are crucial results in spectral theory. Here we focus on the important particular case for polynomials. Further versions will be considered in Chapter 4. Let p : C → C be a polynomial with complex coefficients, take any subset Λ of C , and consider its image under p, viz.,   p(Λ) = p(λ) ∈ C : λ ∈ Λ . Theorem 2.7. (Spectral Mapping Theorem for Polynomials). Take an operator T ∈ B[X ] on a complex Banach space X . If p is an arbitrary polynomial with complex coefficients, then σ(p(T )) = p(σ(T )). Proof. To avoid trivialities, let p : C → C be an arbitrary nonconstant polynomial with complex coefficients, n αi λi with n ≥ 1 and αn = 0 p(λ) = i=0

for every λ ∈ C . Take an arbitrary ζ ∈ C and consider the factorization n (λi − λ), ζ − p(λ) = βn i=1

with βn = (−1)n+1 αn where {λi }ni=1 are the roots of ζ − p(λ). Hence ζI − p(T ) = ζI −

n i=0

αi T i = βn

n i=1

(λi I − T ).

n If λi ∈ ρ(T ) for every i = 1, . . . , n, then βn i=1 (λi I − T ) ∈ G[X ] so that ζ ∈ ρ(p(T )). Thus if ζ ∈ σ(p(T )), then there exists λj ∈ σ(T ) for some j = 1, . . . , n. However, λj is a root of ζ − p(λ); that is, ζ − p(λj ) = βn

n i=1

(λi − λj ) = 0,

and so p(λj ) = ζ. Thus if ζ ∈ σ(p(T )), then   ζ = p(λj ) ∈ p(λ) ∈ C : λ ∈ σ(T ) = p(σ(T )) because λj ∈ σ(T ). Therefore σ(p(T )) ⊆ p(σ(T )). Conversely, suppose ζ ∈ p(σ(T )) = {p(λ) ∈ C : λ ∈ σ(T )}. Then ζ = p(λ), and so ζ − p(λ) = 0, for some λ ∈ σ(T ). Hence there exists λ in σ(T ) for which λ = λj for some j = 1, . . . , n. Now consider the factorization

36

2. Spectrum of an Operator

ζI − p(T ) = βn

n i=1

= (λj I − T ) βn

(λi I − T )

n j=i=1

(λi I − T ) = βn

n j=i=1

(λi I − T ) (λj I − T ),

which holds true since (λj I − T ) commutes with (λi I − T ) for every i. If ζ ∈ ρ(p(T )), then (ζI − p(T )) ∈ G[X ] so that n  

(λi I − T ) ζI − p(T ) −1 (λj I − T ) βn j=i=1



= ζI − p(T ) =



ζI − p(T )

 n  ζI − p(T ) −1 βn

−1

j=i=1

    = I = ζI − p(T ) −1 ζI − p(T )

(λi I − T ) (λj I − T ).

Thus (λj I − T ) has a right and a left inverse (i.e., it is invertible), and so (λj I − T ) ∈ G[X ] by Theorem 1.1. Then λ = λj ∈ ρ(T ), which contradicts the fact that λ ∈ σ(T ). Conclusion: if ζ ∈ p(σ(T )), then ζ ∈ / ρ(p(T )). Equivalently, ζ ∈ σ(p(T )). Therefore p(σ(T )) ⊆ σ(p(T )).  In particular, σ(T n ) = σ(T )n for every n ≥ 0, which means: ζ ∈ σ(T )n = {λn ∈ C : λ ∈ σ(T )} if and only if ζ ∈ σ(T n ). Also σ(αT ) = α σ(T ) for every α ∈ C , which means: ζ ∈ α σ(T ) = {αλ ∈ C : λ ∈ σ(T )} if and only if ζ ∈ σ(αT ). Now set Λ−1 = {λ−1 ∈ C : λ ∈ Λ} if 0 ∈ Λ. In addition, if T ∈ G[X ], then we get σ(T −1 ) = σ(T )−1 (even though this is not a particular case of the Spectral Mapping Theorem for polynomials), which means: ζ ∈ σ(T )−1 = {λ−1 ∈ C : 0 = λ ∈ σ(T )} if and only if ζ ∈ σ(T −1 ). Indeed, if T ∈ G[X ] (i.e., 0 ∈ ρ(T )) and if ζ = 0, then −ζ T −1 (ζ −1 I − T ) = ζI − T −1 . Thus ζ −1 ∈ ρ(T ) if and only if ζ ∈ ρ(T −1 ). Moreover, if T ∈ B[H], then σ(T ∗ ) = σ(T )∗ by Theorem 2.6, where H is a complex Hilbert space and σ(T )∗ stands for the set of all complex conjugates of elements of σ(T ). The next result is an extension of the Spectral Mapping Theorem for polynomials which holds for normal operators acting on a Hilbert space. With Λ∗ = {λ ∈ C : λ ∈ Λ} for an arbitrary subset Λ of C , set

2.3 Spectral Mapping Theorems

37

  p(Λ, Λ∗ ) = p(λ, λ) ∈ C : λ ∈ Λ where n,m p(· , ·)i: Λj ×Λ → C is any polynomial in λ and λ of the form p(λ, λ) = i,j=0 αi,j λ λ , which can also be viewed as a function p(· , ·) : Λ → C on Λ. Theorem 2.8. (Spectral Mapping Theorem for Normal Operators). If T ∈ B[H] is normal on a complex Hilbert space H and p(· , ·) : Λ ×Λ → C is a polynomial in λ and λ, then   σ(p(T, T ∗ )) = p(σ(T ), σ(T ∗ )) = p(λ, λ) ∈ C : λ ∈ σ(T ) . n,m Proof. Take any normal operator T ∈ B[H]. If p(λ, λ) = i,j=0 αi,j λi λj , then n,m set p(T, T ∗ ) = i,j=0 αi,j T i T ∗j . Let P(T, T ∗ ) be the collection of all polyno∗ mials p(T, T ), which is a commutative subalgebra of the Banach algebra B[H] since T commutes with T ∗. Consider the collection T of all commutative subalgebras of B[H] containing T and T ∗, which is partially ordered (in the inclusion ordering) and nonempty (e.g., P(T, T ∗ ) ∈ T ). Moreover, every chain in T has an upper bound in T (the union of all subalgebras in a given chain of subalgebras in T is again a subalgebra in T ). Thus Zorn’s Lemma says that T has a maximal element, say A(T ). Outcome: If T is normal, then there is a maximal (thus closed) commutative subalgebra A(T ) of B[H] containing T and T ∗. Since P(T, T ∗ ) ⊆ A(T ) ∈ T, and since every p(T, T ∗ ) ∈ P(T, T ∗ ) is normal, then A(p(T, T ∗ )) = A(T ) for every nonconstant p(T, T ∗ ). Furthermore, Φ(p(T, T ∗ )) = p(Φ(T ), Φ(T ∗ )) for every homomorphism Φ : A(T ) → C . Thus, by Proposition 2.Q(b),    ) , σ(p(T, T ∗ )) = p(Φ(T ), Φ(T ∗ )) ∈ C : Φ ∈ A(T  ) is the collection of all algebra homomorphisms of A(T ) onto C . where A(T  )). ConTake a surjective homomorphism Φ : A(T ) → C (i.e., take any Φ ∈ A(T sider the Cartesian decomposition T = A + i B, where A, B ∈ B[H] are selfadjoint, and so T ∗ = A − i B (Proposition 1.O). Thus Φ(T ) = Φ(A) + i Φ(B) and Φ(T ∗ ) = Φ(A) − i Φ(B). Since A = 12 (T + T ∗ ) and B = − 2i (T − T ∗ ) lie in P(T, T ∗ ), we get A(A) = A(B) = A(T ). Also, since A and B are self-adjoint,  )} = σ(A) ⊂ R and {Φ(B) ∈ C : Φ ∈ A(T  )} = σ(B) ⊂ R {Φ(A) ∈ C : Φ ∈ A(T (Propositions 2.A and 2.Q(b)), and so Φ(A) ∈ R and Φ(B) ∈ R . Hence Φ(T ∗ ) = Φ(T ). (Φ(T ) is the complex conjugate of Φ(T ) ∈ C .) Thus since σ(T ∗ ) = σ(T )∗ for every T ∈ B[H] by Theorem 2.6, and recalling Proposition 2.Q(b), we get    ) σ(p(T, T ∗ )) = p(Φ(T ), Φ(T )) ∈ C : Φ ∈ A(T    )} = p(λ, λ) ∈ C : λ ∈ {Φ(T ) ∈ C : Φ ∈ A(T   = p(λ, λ) ∈ C : λ ∈ σ(T ) = p(σ(T ), σ(T )∗ ) = p(σ(T ), σ(T ∗ )).



38

2. Spectrum of an Operator

2.4 Spectral Radius and Normaloid Operators The spectral radius of an operator T ∈ B[X ] on a nonzero complex Banach space X is the nonnegative number rσ (T ) = sup |λ| = max |λ|. λ∈σ(T )

λ∈σ(T )

The first identity in the above expression defines the spectral radius rσ (T ), and the second one is a consequence of the Weierstrass Theorem (cf. proof of Theorem 2.2) since σ(T ) = ∅ is compact in C and the function | · | : C → R is continuous. A straightforward consequence of the Spectral Mapping Theorem for polynomials reads as follows. Corollary 2.9.

rσ (T n ) = rσ (T )n for every n ≥ 0.

Proof. Take any nonnegative integer n. By Theorem 2.7, σ(T n ) = σ(T )n. Thus ζ ∈ σ(T n ) if and only if ζ = λn for some λ ∈ σ(T ), and so supζ∈σ(T n ) |ζ| = n  supλ∈σ(T ) |λn | = supλ∈σ(T ) |λ|n = supλ∈σ(T ) |λ| .  If λ ∈ σ(T ), then |λ| ≤ T . This follows by the Neumann expansion of Theorem 1.3 (cf. proof of Theorem 2.1). Hence rσ (T ) ≤ T . Therefore, for every operator T ∈ B[X ] and for each nonnegative integer n, 0 ≤ rσ (T n ) = rσ (T )n ≤ T n  ≤ T n . Thus rσ (T ) ≤ 1 if T is power bounded (i.e., if supn T n  < ∞). Indeed, in this 1 case, rσ (T )n ≤ T n  ≤ supk T k  for n ≥ 1 and limn (supk T k ) n = 1. Then sup T n  < ∞

implies

n

rσ (T ) ≤ 1.

Remark. If T is a nilpotent operator (i.e., if T n = O for some n ≥ 1), then rσ (T ) = 0, and so σ(T ) = σP (T ) = {0} (cf. Proposition 2.J). An operator T ∈ B[X ] is quasinilpotent if rσ (T ) = 0 (i.e., if σ(T ) = {0}). Thus every nilpotent is quasinilpotent. Since σP (T ) may be empty for a quasinilpotent operator (see, e.g., Proposition 2.N), these classes are related by proper inclusion: Nilpotent

⊂

Quasinilpotent.

The next result is the well-known Gelfand–Beurling formula for the spectral radius. Its proof requires another piece of elementary complex analysis, namely, every analytic function has a power series representation. That is, if f : Λ → C is analytic, and if Bα,β (ζ) = {λ ∈ C : 0 ≤ α < |λ − ζ| < β} lies in < β, then f has a unique Laurent expanthe open set Λ ⊆ C for some 0 ≤ α  ∞ sion about the point ζ, viz., f (λ) = k=−∞ γk (λ − ζ)k for every λ ∈ Bα,β (ζ). Theorem 2.10. (Gelfand–Beurling Formula). 1

rσ (T ) = lim T n  n . n

2.4 Spectral Radius and Normaloid Operators

39

Proof. Since rσ (T )n ≤ T n  for every positive integer n, and since the limit 1 of the sequence {T n  n} exists by Lemma 1.10, we get 1

rσ (T ) ≤ lim T n  n . n

For the reverse inequality proceed as follows. Consider the Neumann expansion (Theorem 1.3) for the resolvent function RT : ρ(T ) → G[X ], ∞ RT (λ) = (λI − T )−1 = λ−1 T k λ−k k=0

for every λ ∈ ρ(T ) such that T  < |λ|, where the above series converges in the (uniform) topology of B[X ]. Take an arbitrary bounded linear functional ∗ ξ : B[X ] → C in B[X ] (cf. proof of Theorem 2.2). Since ξ is linear continuous, ∞ ξ(RT (λ)) = λ−1 ξ(T k )λ−k k=0

for every λ ∈ ρ(T ) such that T  < |λ|. Claim. The above displayed identity holds whenever rσ (T ) < |λ|. ∞ Proof. The series λ−1 k=0 ξ(T k )λ−k is a Laurent expansion of ξ(RT (λ)) about the origin for every λ ∈ ρ(T ) such that T  < |λ|. But ξ ◦ RT is analytic on ρ(T ) (cf. Claim 2 in Theorem 2.2) and so ξ(RT (λ)) has a unique Laurent expansion about the origin for every λ ∈ ρ(T ∞), and hence for every λ ∈ C such that rσ (T ) < |λ|. Then ξ(RT (λ)) = λ−1 k=0 ξ(T k )λ−k , which holds whenever rσ (T ) ≤ T  < |λ|, must be the Laurent expansion about the origin for every λ ∈ C such that rσ (T ) < |λ|. This proves the claimed result. ∞ Thus if rσ (T ) < |λ|, then the series of complex numbers k=0 ξ(T k )λ−k converges, and so ξ((λ−1 T )k ) = ξ(T k )λ−k → 0, for every ξ in the dual space ∗ B[X ] of B[X ]. This means that the B[X ]-valued sequence {(λ−1 T )k } converges weakly. Then it is bounded (in the uniform topology of B[X ] as a consequence of the Banach–Steinhaus Theorem). Thus the operator λ−1 T is power bounded. Hence |λ|−n T n  ≤ supk (λ−1 T )k  < ∞, so that  1 1 |λ|−1 T n  n ≤ sup (λ−1 T )k  n k 1

1

for every n. Therefore |λ|−1 limn T n  n ≤ 1, and so limn T n  n ≤ |λ|, for 1 every λ ∈ C such that rσ (T ) < |λ|. Then limn T n  n ≤ rσ (T ) + ε for every ε > 0. Outcome: 1  lim T n  n ≤ rσ (T ). n

What Theorem 2.10 says is rσ (T ) = r(T ), where rσ (T ) is the spectral radius 1 of T and r(T ) is the limit of the sequence {T n  n} (whose existence was proved in Lemma 1.10). In light of this, we will adopt one and the same notation (the simpler one, of course) for both of them. So from now on we write

40

2. Spectrum of an Operator 1

r(T ) = sup |λ| = max |λ| = lim T n  n λ∈σ(T )

λ∈σ(T )

n

for the spectral radius. A normaloid operator was defined in Section 1.6 as an operator T for which r(T ) = T . Thus a normaloid operator acting on a complex Banach space is precisely an operator whose norm coincides with the spectral radius, by Theorem 2.10. Moreover, T is normaloid if and only if T n  = T n for every n ≥ 0, by Theorem 1.11. Furthermore, since on a complex Hilbert space H every normal operator is normaloid, and so is every nonnegative operator, and since T ∗ T is always nonnegative, then r(T ∗ T ) = r(T T ∗ ) = T ∗ T  = T T ∗  = T 2 = T ∗ 2 for T ∈ B[H]. Further useful properties follow from Theorem 2.10. For instance, r(αT ) = |α| r(T ) for every α ∈ C . Also r(S T ) = r(T S) (see Proposition 2.C). In a Hilbert space, r(T ∗ ) = r(T ) 1 1 and r(Q 2 ) = r(Q) 2 if Q ≥ O (cf. Theorem 2.6 and Proposition 2.D). Note: although σ(T −1 ) = σ(T )−1, in general r(T −1 ) = r(T )−1 (e.g., T = diag(1, 2)). An important application of the Gelfand–Beurling formula is the characterization of uniform stability in terms of the spectral radius. A normed-space operator T ∈ B[X ] is uniformly stable if the power sequence {T n } converges u O, i.e., if T n  → 0). uniformly to the null operator (notation: T n −→ Corollary 2.11. If T ∈ B[X ] is an operator on a complex Banach space X , then the following assertions are pairwise equivalent. u (a) T n −→ O.

(b) r(T ) < 1. (c) T n  ≤ β αn for every n ≥ 0, for some β ≥ 1 and some α ∈ (0, 1). Proof. Since r(T )n = r(T n ) ≤ T n  for each n ≥ 1, if T n  → 0 then r(T ) < 1. Thus (a) ⇒ (b). Suppose r(T ) < 1 and take an arbitrary α in (r(T ), 1). Since 1 r(T ) = limn T n  n (Gelfand–Beurling formula), there is an integer nα ≥ 1 for which T n  ≤ αn whenever n ≥ nα . Thus T n  ≤ β αn for every n ≥ 0 with  β = max 0≤n≤nα T n α−nα , and so T n  → 0. Thus (b) ⇒ (c) ⇒ (a). A normed-space operator T ∈ B[X ] is strongly stable if the power sequence s {T n } converges strongly to the null operator (notation: T n −→ O, i.e., if T n x → 0 for every x ∈ X ), and T is weakly stable if {T n } converges weakly w O). In a Hilbert space H this means to the null operator (notation: T n −→ T n x ; y → 0 for every x, y ∈ H, or T n x ; x → 0 for every x ∈ H if H is complex (cf. Section 1.1). Thus from what we have seen so far, in a Hilbert space, u O =⇒ r(T ) < 1 ⇐⇒ T n −→ s w O =⇒ T n −→ O =⇒ sup T n  < ∞ =⇒ r(T ) ≤ 1. T n −→ n

2.5 Numerical Radius and Spectraloid Operators

41

The converses to the above one-way implications fail in general. The next result applies the characterization of uniform stability in Corollary 2.11 to extend the Neumann expansion of Theorem 1.3. Corollary 2.12. Let T ∈ B[X ] be an operator on a complex Banach space, and let λ ∈ C be any nonzero complex number . n  T k (a) r(T ) < |λ| if and only if converges uniformly. In this case we k=0 λ ∞   ∞   −1 get λ ∈ ρ(T ) and RT (λ) = (λI − T ) = λ1 k=0 Tλ k where k=0 Tλ k n  T k . denotes the uniform limit of k=0 λ  n  T  k  converges strongly, then λ ∈ ρ(T ) and (b) If r(T ) = |λ| and k=0 λ ∞   ∞   RT (λ) = (λI − T )−1 = λ1 k=0 Tλ k where k=0 Tλ k denotes the strong n  T k limit of . k=0 λ n  T k does not converge strongly. (c) If |λ| < r(T ), then k=0 λ  n u n  T k converges uniformly, then Tλ −→ O, and therefore Proof. If k=0 λ   |λ|−1 r(T ) = r Tλ < 1 by Corollary 2.11. On the other hand, if r(T ) < |λ|, T  −1 then λ ∈ ρ(T ) so that λI − T ∈ G[X ], and also r r(T ) < 1. Hence λ = |λ|  T n  ≤ β αn for every n ≥ 0, for some β ≥ 1 and α ∈ (0, 1) according  λ  ∞    to Corollary 2.11, and so k=0  Tλ k  < ∞, which means the B[X ]-valued  T n sequence λ is absolutely summable. Now follow the steps in the proof of n  T k Theorem 1.3 to conclude the results in (a). If strongly, k=0  λ  converges  T n n  then λ x → 0 inX for every x ∈ X and so supn  Tλ x < ∞ for every  n x ∈ X . Hence supn  Tλ  < ∞ (by the Banach–Steinhaus Theorem). Thus |λ|−1 r(T ) = r( Tλ ) ≤ 1, which proves (c). Moreover, (λI − T ) λ1

n

 T k

k=0 λ

=

1 λ

n

 T k (λI k=0 λ

− T) = I −

 T n+1 λ

s −→ I.

∞   ∞   Therefore (λI − T )−1 = λ1 k=0 Tλ k, where k=0 Tλ k ∈ B[X ] is the strong n  T k limit of , which concludes the proof of (b).  k=0 λ

2.5 Numerical Radius and Spectraloid Operators The numerical range W (T ) of an operator T ∈ B[H] acting on a nonzero complex Hilbert space H is the nonempty and convex set consisting of the inner products T x ; x for all unit vectors x ∈ H:   W (T ) = λ ∈ C : λ = T x ; x for some x with x = 1 . Indeed, the numerical range W (T ) is a convex set in C (see, e.g., [58, Problem 210] ) which is nonempty (W (T ) = ∅) since H = {0}, and clearly W (T ∗ ) = W (T )∗ .

42

2. Spectrum of an Operator

Theorem 2.13.

σP (T ) ∪ σR (T ) ⊆ W (T ) and σ(T ) ⊆ W (T )− .

Proof. Take any operator T ∈ B[H] on a nonzero complex Hilbert space H. If λ ∈ σP (T ), then there is a unit vector x ∈ H such that T x = λx. Hence T x ; x = λx2 = λ and so λ ∈ W (T ). If λ ∈ σR (T ), then λ ∈ σP (T ∗) by Theorem 2.6, and so λ ∈ W (T ∗ ). Thus λ ∈ W (T ). Therefore σP (T ) ∪ σR (T ) ⊆ W (T ). If λ ∈ σAP (T ), then there is a sequence {xn } of unit vectors in H for which (λI − T )xn  → 0 by Theorem 2.4. Hence 0 ≤ |λ − T xn ; xn | = |(λI − T )xn ; xn | ≤ (λI − T )xn  → 0 and hence T xn ; xn  → λ. Since each T xn ; xn  lies in W (T ), it follows by the classical Closed Set Theorem that λ ∈ W (T )−. Thus σAP (T ) ⊆ W (T )− and so (since σR (T ) ⊆ W (T )) σ(T ) = σR (T ) ∪ σAP (T ) ⊆ W (T )− .



The numerical radius of an operator T ∈ B[H] on a nonzero complex Hilbert space H is the nonnegative number w(T ) = sup |λ| = sup |T x ; x|. λ∈W (T )

x=1

As is readily verified, w(T ∗ ) = w(T )

and

w(T ∗ T ) = T 2 .

Unlike the spectral radius, the numerical radius is a norm on B[H]. That is, 0 ≤ w(T ) for every T ∈ B[H] and 0 < w(T ) if T = O, w(αT ) = |α|w(T ), and w(T + S) ≤ w(T ) + w(S) for every α ∈ C and every S, T ∈ B[H]. However, the numerical radius does not have the operator norm property. In other words, the inequality w(S T ) ≤ w(S) w(T ) is not true for all operators S, T ∈ B[H]. Nevertheless, the power inequality holds: w(T n ) ≤ w(T )n for all T ∈ B[H] and every positive integer n (see, e.g., [58, p. 118 and Problem 221] ). Moreover, the numerical radius is a norm equivalent to the (induced uniform) operator norm of B[H] and dominates the spectral radius, as in the next theorem. Theorem 2.14.

0 ≤ r(T ) ≤ w(T ) ≤ T  ≤ 2w(T ).

Proof. Take any operator T in B[H]. Since σ(T ) ⊆ W (T )− we get r(T ) ≤ w(T ). Moreover, w(T ) = supx=1 |T x ; x| ≤ supx=1 T x = T . Now, by the polarization identity (cf. Proposition 1.A),  T x ; y = 14 T (x + y) ; (x + y) − T (x − y) ; (x − y)  + iT (x + iy) ; (x + iy) − iT (x − iy) ; (x − iy)

2.5 Numerical Radius and Spectraloid Operators

43

for every x, y in H. Therefore, since |T z ; z| ≤ supu=1 |T u ; u|z2 = w(T )z2 for every z ∈ H,  |T x ; y| ≤ 41 |T (x + y) ; (x + y)| + |T (x − y) ; (x − y)|  + |T (x + iy) ; (x + iy)| + |T (x − iy) ; (x − iy)|   ≤ 14 w(T ) x + y2 + x − y2 + x + iy2 + x − iy2 . for every x, y ∈ H. So, by the parallelogram law (cf. Proposition 1.A),   |T x ; y| ≤ w(T ) x2 + y2 ≤ 2w(T ) whenever x = y = 1. Thus, since T  = supx=y=1 |T x ; y| (see, e.g., [78, Corollary 5.71]), we get T  ≤ 2w(T ).  An operator T ∈ B[H] is spectraloid if r(T ) = w(T ). The next result is a straightforward application of Theorem 2.14. Corollary 2.15. Every normaloid operator is spectraloid . In a normed space, an operator T is normaloid if r(T ) = T  or, equivalently, if T n  = T n for every n ≥ 1 by Theorem 1.11. In a Hilbert space, r(T ) = T  implies r(T ) = w(T ) as in Corollary 2.15. Moreover, r(T ) = T  also implies w(T ) = T  by Theorem 2.14 again. Thus w(T ) = T  is a property of every normaloid operator on a Hilbert space. In fact, this can be viewed as a third definition of a normaloid operator on a complex Hilbert space. Theorem 2.16. T ∈ B[H] is normaloid if and only if w(T ) = T . Proof. Half of the proof was given above. It remains to prove the other half: w(T ) = T 

implies

r(T ) = T .

Suppose w(T ) = T  (and T = O to avoid trivialities). The closure of the numerical range W (T )− is compact in C (since W (T ) is trivially bounded). Thus maxλ∈W (T )− |λ| = supλ∈W (T )− |λ| = supλ∈W (T ) |λ| = w(T ) = T , and so there exists λ ∈ W (T )− such that |λ| = T . Since W (T ) is nonempty, λ is a point of adherence of W (T ), and hence there exists a sequence {λn } with each λn in W (T ) for which λn → λ. This means there exists a sequence {xn } of unit vectors in H (i.e., xn  = 1) such that λn = T xn ; xn  → λ, where |λ| = T  = 0. Thus if S = λ−1 T ∈ B[H], then Sxn ; xn  → 1. Claim. Sxn  → 1 and Re Sxn ; xn  → 1. Proof. |Sxn ; xn | ≤ Sxn  ≤ S = 1 for each n. But Sxn ; xn  → 1 implies |Sxn ; xn | → 1 (and so Sxn  → 1) and also ReSxn ; xn  → 1 (since | · | and Re(·) are continuous functions), which concludes the proof of the claim.

44

2. Spectrum of an Operator

Then (I − S)xn 2 = Sxn − xn 2 = Sxn 2 − 2ReSxn ; xn  + xn 2 → 0, and so 1 ∈ σAP (S) ⊆ σ(S) (cf. Theorem 2.4). Hence r(S) ≥ 1 which implies r(T ) = r(λ S) = |λ| r(S) ≥ |λ| = T , and therefore r(T ) = T  (because  r(T ) ≤ T  for every operator T ). Remark. Take T ∈ B[H]. If w(T ) = 0, then T = O (since the numerical radius is a norm — also by Theorem 2.14). In particular, if w(T ) = r(T ) = 0, then T = O. Thus, if an operator is spectraloid and quasinilpotent, then it is the null operator . Therefore, the unique normal (or hyponormal, or normaloid, or spectraloid ) quasinilpotent operator is the null operator . Corollary 2.17. If there exists λ ∈ W (T ) such that |λ| = T , then T is normaloid and λ ∈ σP (T ). In other words, if there exists a unit vector x such that T  = |T x ; x|, then r(T ) = w(T ) = T  and T x ; x ∈ σP (T ). Proof. If λ ∈ W (T ) is such that |λ| = T , then w(T ) = T  (Theorem 2.14) and so T is normaloid (Theorem 2.16). Moreover, since λ = T x ; x for some unit vector x, we get T  = |λ| = |T x ; x| ≤ T xx ≤ T , and hence |T x ; x| = T xx (the Schwarz inequality becomes an identity), which implies T x = αx for some α ∈ C (see, e.g., [78, Problem 5.2] ). Thus α ∈ σP (T ).  But α = αx2 = αx ; x = T x ; x = λ.

2.6 Spectrum of a Compact Operator The spectral theory of compact operators plays a central role in the Spectral Theorem for compact normal operators of the next chapter. Normal operators were defined on Hilbert spaces; thus we keep on working with compact operators on Hilbert spaces, as we did in Section 1.8, although the spectral theory of compact operators can be equally developed on nonzero complex Banach spaces. So we assume all operators in this section acting on a nonzero complex Hilbert space H. The main result for characterizing the spectrum of a compact operator is the Fredholm Alternative of Corollary 1.20 which, in view of the classical partition of the spectrum in Section 2.2, can be restated as follows. Theorem 2.18. (Fredholm Alternative). Take T ∈ B∞[H]. If λ ∈ C \{0}, then λ ∈ ρ(T ) ∪ σP (T ). Equivalently, σ(T )\{0} = σP (T )\{0}. Moreover, if λ ∈ C \{0}, then dim N (λI − T ) = dim N (λI − T ∗ ) < ∞ and so λ ∈ ρ(T ) ∪ σP4(T ). Equivalently, σ(T )\{0} = σP4(T )\{0}. Proof. Take a compact operator T in B[H] and a nonzero scalar λ in C . By Corollary 1.20 and the diagram of Section 2.2,

2.6 Spectrum of a Compact Operator

45

λ ∈ ρ(T ) ∪ σP1(T ) ∪ σR1(T ) ∪ σP4(T ). Also by Corollary 1.20, N (λI − T ) = {0} if and only if N (λI − T ∗ ) = {0}, and so λ ∈ σP (T ) if and only if λ ∈ σP (T ∗ ). Thus λ ∈ σP1(T ) ∪ σR1(T ) by Theorem 2.6, and hence λ ∈ ρ(T ) ∪ σP4(T ) or, equivalently, λ ∈ ρ(T ) ∪ σP (T ) (since λ ∈ σP1(T ) ∪ σP2(T ) ∪ σP3(T )). Therefore σ(T )\{0} = σP (T )\{0} = σP4(T )\{0}.



The scalar 0 may be anywhere. In other words, if T ∈ B∞[H], then λ = 0 may lie in σP (T ), σR (T ), σC (T ), or ρ(T ). However, if T is a compact operator on a nonzero space H and 0 ∈ ρ(T ), then H must be finite-dimensional. Indeed, if 0 ∈ ρ(T ), then T −1 ∈ B[H] and so I = T −1 T is compact (since B∞[H] is an ideal of B[H]), which forces H to be finite-dimensional (cf. Proposition 1.Y in Section 1.9). The preceding theorem in fact is a rewriting of the Fredholm Alternative (and it is also referred to as the Fredholm Alternative). It will be applied often from now on. Here is a first application. Let B0 [H] denote the class of all finite-rank operators on H (i.e., the class of all operators from B[H] with a finite-dimensional range). Recall: B0 [H] ⊆ B∞[H] (finite-rank operators are compact — cf. Proposition 1.X in Section 1.9). Let #A stand for the cardinality of a set A. Thus #A < ∞ means “A is a finite set”. Corollary 2.19. If T ∈ B0 [H], then σ(T ) = σP (T ) = σP4(T )

and

#σ(T ) < ∞.

Proof. In a finite-dimensional normed space an operator is injective if and only if it is surjective (see, e.g., [78, Problem 2.18] ), and linear manifolds are closed (see, e.g., [78, Corollary 4.29] ). So σ(T ) = σP (T ) = σP4(T ) by the diagram of Section 2.2 (since σP1(T ) = σR1(T ∗ )∗ according to Theorem 2.6). On the other hand, suppose dim H = ∞. Since B0 [H] ⊆ B∞[H] we get, by Theorem 2.18, σ(T )\{0} = σP (T )\{0} = σP4(T )\{0}. Since dim R(T ) < ∞ and dim H = ∞, then R(T )−= R(T ) = H and N (T ) = {0} (by the rank and nullity identity: dim N (T ) + dim R(T ) = dim H — see, e.g., [78, Problem 2.17] ). Then 0 ∈ σP4(T ) (cf. diagram of Section 2.2) and so σ(T ) = σP (T ) = σP4(T ). If σP (T ) is an infinite set, then there is an infinite set of linearly independent eigenvectors of T (Theorem 2.3). Since every eigenvector of T lies in R(T ), this implies dim R(T ) = ∞ (because every linearly independent subset of a linear space is included in some Hamel basis — see, e.g., [78, Theorem 2.5] ),  which is a contradiction. Conclusion: σP (T ) must be a finite set. In particular, the above result clearly holds if H is finite-dimensional since, as we saw above, dim H < ∞ implies B[H] = B0 [H].

46

2. Spectrum of an Operator

Corollary 2.20. Take an arbitrary compact operator T ∈ B∞[H]. (a) An infinite sequence of distinct points of σ(T ) converges to zero. (b) 0 is the only possible accumulation point of σ(T ). (c) If λ ∈ σ(T )\{0}, then λ is an isolated point of σ(T ). (d) σ(T )\{0} is a discrete subset of C . (e) σ(T ) is countable. Proof. Let T be a compact operator on H. ∞ (a) Let {λn }n=1 be an infinite sequence of distinct points in the spectrum σ(T ). Without loss of generality, suppose every λn is nonzero. Since T is compact and 0 = λn ∈ σ(T ), we get λn ∈ σP (T ) by Theorem 2.18. Let {xn }∞ n=1 ∞ be a sequence of eigenvectors associated with the eigenvalues {λn }n=1 (i.e., T xn = λn xn with each xn = 0). This is a sequence of linearly independent vectors by Theorem 2.3. For each n ≥ 1, set n , Mn = span {xi }i=1

which is a subspace of H with dim Mn = n, and Mn ⊂ Mn+1 n+1 for every n ≥ 1 (because {xi }i=1 is linearly independent and so xn+1 lies in Mn+1 \Mn ). From now on the argument is similar to the argument in the proof of Theorem 1.18. Since each Mn is a proper subspace of the Hilbert space Mn+1 , there exists a vector yn+1 in Mn+1 with yn+1  = 1 for which n+1 1 i=1 αi xi in Mn+1 we get 2 < inf u∈Mn yn+1 − u. Writing yn+1 =

(λn+1 I − T )yn+1 =

n+1 n αi (λn+1 − λi )xi = αi (λn+1 − λi )xi ∈ Mn . i=1

i=1

Since λn = 0 for every n, take any pair of integers 1 ≤ m < n and set −1 y = ym − λ−1 m (λm I − T )ym + λn (λn I − T )yn , −1 so that T (λ−1 m ym ) − T (λn yn ) = y − yn . Since y lies in Mn−1 , 1 2

−1 < y − yn  = T (λ−1 m ym ) − T (λn yn ).

Thus the sequence {T (λ−1 n yn )} has no convergent subsequence. Then, since T y } has no bounded subsequence by Proposition 1.S. Hence is compact, {λ−1 n n y supk |λk |−1 = supk λ−1 k  = ∞, and so inf k |λk | = 0, for every subsequence k ∞ {λk }∞ of {λ } . Therefore λn → 0. n n=1 k=1 (b) Thus if λ = 0, then there is no sequence of distinct points in σ(T ) converging to λ. So λ = 0 is not an accumulation point of σ(T ).

2.7 Supplementary Propositions

47

(c) Therefore every λ in σ(T )\{0} is not an accumulation point of σ(T ). Equivalently, every λ in σ(T )\{0} is an isolated point of σ(T ). (d) Hence σ(T )\{0} consists entirely of isolated points. In other words, σ(T )\{0} is a discrete subset of C . (e) Since a discrete subset of a separable metric space is countable (see, e.g., [78, Example 3.Q] ), and since C is separable, σ(T )\{0} is countable.  Corollary 2.21. If an operator T ∈ B[H] is compact and normaloid, then σP (T ) = ∅ and there exists λ ∈ σP (T ) such that |λ| = T . Proof. Suppose T is normaloid (i.e., r(T ) = T ). Thus σ(T ) = {0} only if T = O. If T = O and H = {0}, then 0 ∈ σP (T ) and T  = 0. If T = O, then σ(T ) = {0} and T  = r(T ) = maxλ∈σ(T ) |λ|. Thus there exists λ = 0 in σ(T ) such that |λ| = T . Moreover, if T is compact and σ(T ) = {0}, then ∅ = σ(T )\{0} ⊆ σP (T ) by Theorem 2.18. Hence r(T ) = maxλ∈σ(T ) |λ| =  maxλ∈σP (T ) |λ| = T . So there exists λ ∈ σP (T ) with |λ| = T . Corollary 2.22. Every compact hyponormal operator is normal . Proof. Suppose T ∈ B[H] is a compact hyponormal operator on a nonzero complex Hilbert 2.21, σP (T ) = ∅. Consider the sub−   space H. By Corollary of Theorem 1.16 with {λγ }γ∈Γ = σP (T ). N (λI − T ) space M = λ∈σP (T ) Hence σP (T |M⊥ ) = ∅. In fact, if there is a λ ∈ σP (T |M⊥ ), then there exists 0 = x ∈ M⊥ for which λx = T |M⊥ x = T x, and so x ∈ N (λI − T ) ⊆ M, which is a contradiction. Moreover, T |M⊥ is compact and hyponormal (Propositions 1.O and 1.U). Thus if M⊥ = {0}, then σP (T |M⊥ ) = ∅ by Corollary 2.21, which is another contradiction. Therefore M⊥ = {0} and so M = H (see  Section 1.3). Then T = T |H = T |M is normal by Theorem 1.16.

2.7 Supplementary Propositions The residual spectrum of a normal operator on a Hilbert space is empty. This also happens for cohyponormal operators, as we will see in Proposition 2.A below. Such a result is a consequence of the inclusion N (λI − T ) ⊆ N (λI − T ∗ ) which holds for every hyponormal operator as in Lemma 1.13(a). So the identity N (λI − T ) = N (λI − T ∗ ) holds for every normal operator as in Lemma 1.13(b). This makes a difference as far as the Spectral Theorem for compact normal operators (next chapter) is concerned. Indeed, the inclusion in Lemma 1.13(a) is enough for proving Theorem 1.16, which in turn plays a crucial role in the Spectral Theorem for compact normal operators (Theorem 3.3, Chapter 3). However, every compact hyponormal operator is normal, as we saw above in Corollary 2.22 (and so the inclusion becomes an identity when applied to the Spectral Theorem for compact normal operators). An eigenvalue for any Hilbert-space operator satisfying the inclusion N (λI − T ) ⊆ N (λI − T ∗ ) is referred to as a normal eigenvalue.

48

2. Spectrum of an Operator

Proposition 2.A. Let H = {0} be a complex Hilbert space and let T denote the unit circle about the origin of the complex plane. (a) If H ∈ B[H] is hyponormal, then σP (H)∗ ⊆ σP (H ∗ ) and σR (H ∗ ) = ∅. (b) If N ∈ B[H] is normal, then σP (N ∗ ) = σP (N )∗ and σR (N ) = ∅. (c) If U ∈ B[H] is unitary, then σ(U ) ⊆ T . (d) If A ∈ B[H] is self-adjoint, then σ(A) ⊂ R . (e) If Q ∈ B[H] is nonnegative, then σ(Q) ⊂ [ 0, ∞). (f) If R ∈ B[H] is strictly positive, then σ(R) ⊂ [ α, ∞) for some α > 0. (g) If E ∈ B[H] is a nontrivial projection, then σ(E) = σP (E) = {0, 1}. Proposition 2.B. Similarity preserves the spectrum and its parts, and so it preserves the spectral radius. That is, let X and Y be nonzero complex Banach spaces. For every T ∈ B[X ] and W ∈ G[X , Y ], (a) σP (T ) = σP (W T W −1 ), (b) σR (T ) = σR (W T W −1 ), (c) σC (T ) = σC (W T W −1 ). So σ(T ) = σ(W T W −1), ρ(T ) = ρ(W T W −1), and r(T ) = r(W T W −1). If W is an isometric isomorphism (in particular, a unitary operator ), then the norm is also preserved : if W ∈ G[X , Y ] is an isometry, then T  = W T W −1. Proposition 2.C. σ(S T )\{0} = σ(T S)\{0} for every S, T ∈ B[X ]. 1

1

Proposition 2.D. If Q ∈ B[H] is nonnegative, then σ(Q 2 ) = σ(Q) 2 . Proposition 2.E. Take an arbitrary operator T ∈ B[X ] on a Banach space. Let d denote the usual distance in C . If λ ∈ ρ(T ), then r((λI − T )−1 ) = [d(λ, σ(T ))]−1 . If T ∈ B[H] on a Hilbert space is hyponormal and λ ∈ ρ(T ), then (λI − T )−1  = [d(λ, σ(T ))]−1 . Proposition 2.F. Let {Hk } be a collection of Hilbert spaces, let {Tk } be a (similarly indexed ) bounded family of operators k ] (i.e., supk Tk  <  Tk in B[H T in B[ ∞), and take the (orthogonal ) direct sum k k k Hk ]. Then 

(a) σP ( k Tk ) = k σP (Tk ), 

(b) σ( k Tk ) = k σ(Tk ) if the collection {Tk } is finite. In general (if the collection {Tk } is not finite), then  −  (c) ⊆ σ( k Tk ), and the inclusion may be proper. k σ(Tk )

2.7 Supplementary Propositions

49

However, if (λI − Tk )−1  = [d(λ, σ(Tk ))]−1 for each k and every λ ∈ ρ(Tk ),  −  (d) = σ( k Tk ), which happens whenever each Tk is hyponormal. k σ(Tk ) Proposition 2.G. Let T ∈ B[X ] be an operator on a complex Banach space. (a) T is normaloid if and only if there is a λ ∈ σ(T ) such that |λ| = T . (b) If T is compact and normaloid, then σP (T ) = ∅ and there is a λ ∈ σP (T ) such that |λ| = T . Proposition 2.H. Let H be a complex Hilbert space. (a) σR (T ) ⊆ {λ ∈ C : |λ| < T } for every operator T ∈ B[H], which is particularly relevant if T is normaloid . (b) σR (T ) ⊆ {λ ∈ C : |λ| < 1} if T ∈ B[H] is power bounded . Proposition 2.I. If H and K are complex Hilbert spaces and T ∈ B[H], then r(T ) =

inf

W ∈G[H,K]

W T W −1  =

inf

M ∈G[H]

M T M −1  =

inf

Q∈G[H]+

Q T Q−1 ,

+

where G[H] is the class of invertible nonnegative operators (cf. Section 1.5). An operator is uniformly stable if and only if it is similar to a strict contraction (by the spectral radius expression in the preceding proposition). Proposition 2.J. If T ∈ B[X ] is a nilpotent operator on a complex Banach space, then σ(T ) = σP (T ) = {0}. Proposition 2.K. An operator T ∈ B[H] on a complex Hilbert space is spectraloid if and only if w(T n ) = w(T )n for every n ≥ 0. An operator T ∈ B[H] a separable infinite-dimensional Hilbert space H on ∞ is diagonalizable if T x = k=0 αk x ; ek ek for every x ∈ H, for some orthonor∞ mal basis {ek }∞ k=0 for H and some bounded sequence {αk }k=0 of scalars. Proposition 2.L. If T ∈ B[H] is diagonalizable and H is complex, then   σP (T ) = λ ∈ C : λ = αk for some k ≥ 0 , σR (T ) = ∅, and   σC (T ) = λ ∈ C : inf k |λ − αk | = 0 and λ = αk for all k ≥ 0 . An operator S+ ∈ B[K+ ] on a Hilbert space K+ is a unilateral shift, and an operator S ∈ B[K] on a Hilbert space K is a bilateral shift, if there exists ∞ an infinite sequence {Hk }∞ k=0 and an infinite family {Hk }k=−∞ ∞of nonzero pairwise orthogonal subspaces of K + and K such that K+ = k=0 Hk and ∞ K = k=−∞ Hk (cf. Section 1.3), and both S+ and S map each Hk isometrically onto Hk+1 , so that each transformation U+ (k+1) = S+ |Hk : Hk → Hk+1 ,

50

2. Spectrum of an Operator

and each transformation Uk+1 = S|Hk : Hk → Hk+1 , is unitary, and therefore is the multiplicity of S+ and dim Hk+1 = dim Hk . Such a common dimension  ∞ 2 2 = H for all k, then K If H S. + = + (H) = k k=0 H and K =  (H) = ∞ k=−∞ H are the direct orthogonal sums of countably infinite copies of a single nonzero Hilbert space H, indexed either by the nonnegative integers or by all integers, which are precisely the Hilbert spaces consisting of all squaresummable H-valued sequences {xk }∞ k=0 and of all square-summable H-valued families {xk }∞ k=−∞ . In this case (if Hk = H for all k), U+ (k+1) = S+ |H = U+ and Uk+1 = S|H = U for all k, where U+ and U are any unitary operators on H. In particular, if U+ = U = I, the identity on H, then S+ and S are referred 2 (H) and on  2 (H). The to as the canonical unilateral and bilateral shifts on + ∗ ∗ adjoint S+ ∈ B[K+ ] of S+ ∈ B[K+ ] and the adjoint of S ∈ B[K] of S ∈ B[K] are referred  to as a backward unilateral shift. a∞ backward bilateral ∞shift and as ∞ ∞ ∞ in x for {x , and {x in for } x H } Writing k k k k k k=0 k=−∞ k=0 k=0 k=−∞ ∞ ∗ k=−∞ Hk , it follows that S+ : K+ → K+ and S+ : K+ → K+ , and S : K → K and S ∗ : K → K, are given by the formulas ∞ ∞ S+ x = 0 ⊕ k=1 U+ (k) xk−1 and S+∗ x = k=0 U+∗(k+1) xk+1 ∞ ∞ for all x = k=0 xk in K+ = k=0 Hk , with 0 being the origin of H0 , where U+ (k+1) is any unitary transformation of Hk onto Hk+1 for each k ≥ 0, and ∞ ∞ ∗ Sx = k=−∞ Uk xk−1 and S ∗ x = k=∞ Uk+1 xk+1 ∞ ∞ for all x = k=−∞ xk in K = k=−∞ Hk , where, for each integer k, Uk+1 is any unitary transformation of Hk onto Hk+1 . The spectrum of a bilateral shift is simpler than that of a unilateral shift, for bilateral shifts are unitary operators (i.e., besides being isometries as unilateral shifts are, bilateral shifts are normal too). For a full treatment of shifts on Hilbert spaces, see [56]. Proposition 2.M. Let D and T = ∂ D denote the open unit disk and the unit circle about the origin of the complex plane, respectively. If S+ ∈ B[K+ ] is a unilateral shift and S ∈ B[K] is a bilateral shift on complex spaces, then (a) σP (S+ ) = σR (S+∗ ) = ∅,

σR (S+ ) = σP (S+∗ ) = D ,

σC (S+ ) = σC (S+∗ ) = T .

(b) σ(S) = σ(S ∗ ) = σC (S ∗ ) = σC (S) = T . 2 A unilateral weighted shift T+ = S+ D+ in B[+ (H)] isthe product of a ∞ canonical unilateral shift S+ and a diagonal operator D+ = k=0 αk I, both in 2 ∞ (H)], where {αk }k=0 is a bounded sequence of scalars. A bilateral weighted B[+ shift T = SD in B[ 2 (H)]is the product of a canonical bilateral shift S and ∞ a diagonal operator D = k=−∞ αk I, both in B[ 2 (H)], where {αk }∞ k=−∞ is a bounded family of scalars. 2 Proposition 2.N. Let T+ ∈ B[+ (H)] be a unilateral weighted shift and let 2 T ∈ B[ (H)] be a bilateral weighted shift where H is a complex Hilbert space.

2.7 Supplementary Propositions

51

(a) If αk → 0 as |k| → ∞, then T+ and T are compact and quasinilpotent. If , in addition, αk = 0 for all k, then (b) σ(T+) = σR (T+) = σR2(T+) = {0} and σ(T+∗ ) = σP (T+∗ ) = σP2(T+∗ ) = {0}, (c) σ(T ) = σC (T ) = σC (T ∗ ) = σ(T ∗ ) = {0}. Proposition 2.O. Let T ∈ B[H] be an operator on a complex Hilbert space and let D be the open unit disk about the origin of the complex plane. w (a) If T n −→ O, then σP (T ) ⊆ D . u (b) If T is compact and σP (T ) ⊆ D , then T n −→ O.

(c) The concepts of weak, strong, and uniform stability coincide for a compact operator on a complex Hilbert space. Proposition 2.P. Take an operator T ∈ B[H] on a complex Hilbert space and let T = ∂ D be the unit circle about the origin of the complex plane. u w (a) T n −→ O and σC (T ) ∩ T = ∅. O if and only if T n −→

(b) If the continuous spectrum does not intersect the unit circle, then the concepts of weak, strong, and uniform stability coincide. The concepts of resolvent set ρ(T ) and spectrum σ(T ) of an operator T in the unital complex Banach algebra B[X ] introduced in Section 2.1, viz., ρ(T ) = {λ ∈ C : λI − T has an inverse in B[X ]} and σ(T ) = C \ρ(T ), of course, hold in any unital complex Banach algebra A, as does the concept of spectral radius r(T ) = sup λ∈σ(T ) |λ|, in relation to which the Gelfand–Beurling formula of 1 Theorem 2.10, namely, r(T ) = limn T n  n , holds in any unital complex Banach algebra (the proof being essentially the same as that of Theorem 2.10). A component of a set in a topological space is any maximal (in the inclusion ordering) connected subset of it. A hole of a compact set is any bounded component of its complement. Thus the holes of the spectrum σ(T ) are the bounded components of the resolvent set ρ(T ) = C \σ(T ). If A is a closed unital subalgebra of a unital complex Banach algebra A (for instance, A = B[X ] where X is a Banach space), and if T ∈ A, then let ρ (T ) be the resolvent set of T with respect to A, let σ  (T ) = C \ρ (T ) be the spectrum of T with respect to A, and set r (T ) = supλ∈σ (T ) |λ|. A homomorphism (or an algebra homomorphism) between two algebras is a linear transformation between them which also preserves the product operation. Let A be a maximal (in the inclusion ordering) commutative subalgebra of a unital complex Banach algebra A (i.e., a commutative subalgebra of A not included in any other commutative subalgebra of A). In fact, A is trivially unital, and closed in A because the closure of a commutative subalgebra of a Banach algebra is again commutative since multiplication is continuous in A. Consider

52

2. Spectrum of an Operator

the (unital complex commutative) Banach algebra C (of all complex numbers). Let A = {Φ : A → C : Φ is a homomorphism} stand for the collection of all algebra homomorphisms of A onto C . Proposition 2.Q. Let A be any unital complex Banach algebra (for instance, A = B[X ]). If T ∈ A, where A is any closed unital subalgebra of A, then (a) ρ (T ) ⊆ ρ(T ) and r (T ) = r(T ) (invariance of the spectral radius). Hence ∂σ  (T ) ⊆ ∂σ(T ) and σ(T ) ⊆ σ  (T ). Thus σ  (T ) is obtained by adding to σ(T ) some holes of σ(T ). (b) If the unital subalgebra A of A is a commutative, then   σ  (T ) = Φ(T ) ∈ C : Φ ∈ A for every T ∈ A . Moreover, if A is a maximal commutative subalgebra of A, then σ  (T ) = σ(T ). Proposition 2.R. If A is a unital complex Banach algebra, and if S, T in A commute (i.e., if S, T ∈ A and S T = T S), then σ(S + T ) ⊆ σ(S) + σ(T )

and

σ(ST ) ⊆ σ(S) · σ(T ).

If M is an invariant subspace for T , then it may happen that σ(T |M ) ⊆ σ(T ). Sample: every unilateral shift is the restriction of a bilateral shift to an invariant subspace (see, e.g., [74, Lemma 2.14] ). However, if M reduces T , then σ(T |M ) ⊆ σ(T ) by Proposition 2.F(b). The full spectrum of T ∈ B[H] (notation: σ(T )# ) is the union of σ(T ) and all bounded components of ρ(T ) (i.e., σ(T )# is the union of σ(T ) and all holes of σ(T )). Proposition 2.S. If M is T -invariant, then σ(T |M ) ⊆ σ(T )# . Proposition 2.T. Let T ∈ B[H] and S ∈ B[K] be operators on Hilbert spaces H and K. If σ(T ) ∩ σ(S) = ∅, then for every bounded linear transformation Y ∈ B[H, K] there exists a unique bounded linear transformation X ∈ B[H, K] such that X T − SX = Y . In particular , σ(T ) ∩ σ(S) = ∅ and X T = SX

=⇒

X = O.

This is the Rosenblum Corollary, which will be used to prove the Fuglede Theorems in Chapter 3. Notes: Again, as in Chapter 1, these are basic results needed throughout the text. We will not point out original sources here, but will instead refer to wellknown secondary sources where proofs for these propositions can be found, as well as deeper discussions on them. Proposition 2.A holds independently of the forthcoming Spectral Theorem (Theorem 3.11) — see, for instance,

2.7 Supplementary Propositions

53

[78, Corollary 6.18]. A partial converse, however, needs the Spectral Theorem (see Proposition 3.D in Chapter 3). Proposition 2.B is a standard result (see, e.g., [58, Problem 75] and [78, Problem 6.10]), as is Proposition 2.C (see, e.g., [58, Problem 76]). Proposition 2.D also bypasses the Spectral Theorem of the next chapter: if Q is a nonnegative operator, then it has a unique 1 nonnegative square root Q 2 by the Square Root Theorem (Proposition 1.M), 1 1 σ(Q 2 )2 = σ((Q 2 )2 ) = σ(Q) by the Spectral Mapping Theorem (Theorem 2.7), and σ(Q) ⊆ [0, ∞) by Proposition 2.A(e). Proposition 2.E is a rather useful technical result (see, e.g., [78, Problem 6.14]). Proposition 2.F is a synthesis of some scattered results (cf. [31, Proposition I.5.1], [58, Solution 98], [64, Theorem 5.42], [78, Problem 6.37], and Proposition 2.E). Proposition 2.G(a) is straightforward: r(T ) ≤ T , σ(T ) is closed, and T is normaloid if and only if r(T) = T . Proposition 2.G(b) is its compact version (see, e.g., [78, Corollary 6.32]). For Proposition 2.H(a) see, for instance, [74, Section 0.2]. Proposition 2.H(b) has been raised in [74, Section 8.2]; for a proof see [94]. The spectral radius formula in Proposition 2.I (see, e.g., [48, p. 22] or [74, Corollary 6.4]) ensures that an operator is uniformly stable if and only if it is similar to a strict contraction (hint: use the equivalence between (a) and (b) in Corollary 2.11). For Propositions 2.J and 2.K see [78, Sections 6.3 and 6.4]. The examples of spectra in Propositions 2.L, 2.M, and 2.N are widely known (see, e.g., [78, Examples 6.C, 6.D, 6.E, 6.F, and 6.G]). Proposition 2.O deals with the equivalence between uniform and weak stabilities for compact operators, and it is extended to a wider class of operators in Proposition 2.P (see [75, Problems 8.8. and 8.9]). Proposition 2.Q(a) is readily verified, where the invariance of the spectral radius follows by the Gelfand–Beurling formula. Proposition 2.Q(b) is a key result for proving both Theorem 2.8 (the Spectral Mapping Theorem for normal operators) and also Lemma 5.43 of Chapter 5 (for the characterization of the Browder spectrum in Theorem 5.44) — see, e.g., [98, Theorems 0.3 and 0.4]. For Propositions 2.R, 2.S, and 2.T see [104, Theorem 11.22], [98, Theorem 0.8], and [98, Corollary 0.13], respectively. Proposition 2.T will play an important role in the proof of the Fuglede Theorems of the next chapter (specifically in Corollary 3.5 and Theorem 3.17).

Suggested Readings Arveson [7] Bachman and Narici [9] Berberian [18] Conway [30, 31, 33] Douglas [38] Dowson [39] Dunford and Schwartz [45] Fillmore [48] Gustafson and Rao [52]

Halmos [58] Herrero [64] Istrˇ a¸tescu [69] Kato [72] Kubrusly [74, 75, 78] Radjavi and Rosenthal [98] Reed and Simon [99] Rudin [104] Taylor and Lay [111]

3 The Spectral Theorem

The Spectral Theorem is a milestone in the theory of Hilbert-space operators, providing a full statement about the nature and structure of normal operators. “Most students of mathematics learn quite early and most mathematicians remember till quite late that every Hermitian matrix (and, in particular, every real symmetric matrix) may be put into diagonal form. A more precise statement of the result is that every Hermitian matrix is unitarily equivalent to a diagonal one. The spectral theorem is widely and correctly regarded as the generalization of this assertion to operators on Hilbert space.” Paul Halmos [57]

For compact normal operators the Spectral Theorem can be investigated without requiring any knowledge of measure theory, and this leads to the concept of diagonalization. However, the Spectral Theorem for plain normal operators (the general case) requires some (elementary) measure theory.

3.1 Spectral Theorem for Compact Operators Throughout this chapter H will denote a nonzero complex Hilbert space. A bounded weighted sum of projections was defined in the last paragraph of Section 1.4 as an operator T on H such that  λ γ Eγ x Tx = γ∈Γ

for every x ∈ H, where {Eγ }γ∈Γ is a resolution of the identity on H made up of nonzero projections (i.e., {E γ }γ∈Γ is an orthogonal family of nonzero orthogonal projections such that γ∈Γ Eγ x = x for every x ∈ H), and {λγ }γ∈Γ is a bounded family of scalars. Every bounded weighted sum of projections is normal (Corollary 1.9). The spectrum of a bounded weighted sum of projections is characterized next as the closure of the set {λγ }γ∈Γ , −  σ(T ) = λ ∈ C : λ = λγ for some λ ∈ Γ . So, since T is normaloid, T  = r(T ) = supγ∈Γ |λγ |. © Springer Nature Switzerland AG 2020 C. S. Kubrusly, Spectral Theory of Bounded Linear Operators, https://doi.org/10.1007/978-3-030-33149-8_3

55

56

3. The Spectral Theorem

Lemma 3.1. If T ∈ B[H] is a weighted sum of projections, then   and σR (T ) = ∅, σP (T ) = λ ∈ C : λ = λγ for some γ ∈ Γ ,   σC (T ) = λ ∈ C : inf γ∈Γ |λ − λγ | = 0 and λ = λγ for all γ ∈ Γ .  Proof. Let T = γ∈Γ λγ Eγ be a bounded weighted sum of projections where {Eγ }γ∈Γ is a resolution of the identity on H and {λγ }γ∈Γ is a bounded sequence of scalars (see Section 1.4). Take an arbitrary x in H. Since {Eγ }γ∈Γ is  a resolution of the identity on H, we get x = γ∈Γ Eγ x, and the general version of the Pythagorean Theorem(see the Orthogonal Structure Theorem of Section 1.3) ensures that x2 = γ∈Γ Eγ x2 . Take any scalar λ ∈ C . Thus   (λI − T )x = γ∈Γ (λ − λγ )Eγ x and so (λI − T )x2 = γ∈Γ |λ − λγ |2 Eγ x2 by the same argument. If N (λI  exists x = 0 in H such  − T ) = {0}, then there that (λI − T )x = 0. Hence γ∈Γ Eγ x2 = 0 and γ∈Γ |λ − λγ |2 Eγ x2 = 0, which implies Eα x = 0 for some α ∈ Γ and |λ − λα |Eα x = 0, and therefore λ = λα . Conversely, take an arbitrary α ∈ Γ and an arbitrary nonzero vecO for every γ ∈ Γ ). Since tor x in R(Eα ) (recall: R(Eγ ) = {0} because Eγ = α γ, R(E ) ⊥ = we get R(Eα ) ⊥ R(Eγ ) whenever α α=γ∈Γ R(Eγ ), and con sequently R(Eα ) ⊆ ( α=γ∈Γ R(Eγ ))⊥ = α=γ∈Γ R(Eγ )⊥ = α=γ∈Γ N (Eγ ) (see Lemma 1.4 and Theorem 1.8).  Thus x ∈ N (Eγ ) for every α = γ ∈ Γ, which implies (λα I − T )x2 = γ∈Γ |λα − λγ |2 Eγ x2 = 0, and therefore N (λα I − T ) = {0}. Summing up: N (λI − T ) = {0} if and only if λ = λα for some α ∈ Γ . Equivalently,   σP (T ) = λ ∈ C : λ = λγ for some γ ∈ Γ . that λ = λγ for all γ ∈ Γ . The bounded weighted Take an arbitrary λ ∈ C such  sum of projections λI − T = γ∈Γ (λ − λγ )Eγ has an inverse, and this inverse  is the weighted sum of projections (λI − T )−1 = γ∈Γ (λ − λγ )−1 Eγ since   (λ − λβ )Eβ x (λ − λα )−1 Eα β∈Γ α∈Γ    Eγ x = x (λ − λα )−1 (λ − λβ )Eα Eβ x = = α∈Γ

β∈Γ

γ∈Γ

for every x in H (because the resolution of the identity {Eγ }γ∈Γ is an orthogonal family of continuous projections, and the inverse is unique). Moreover, the weighted sum of projections of (λI − T )−1 is bounded if and only if supγ∈Γ |λ − λγ |−1 < ∞; that is, if and only if inf γ∈Γ |λ − λγ | > 0. Thus   ρ(T ) = λ ∈ C : inf γ∈Γ |λ − λγ | > 0 . Furthermore, since σR (T ) = ∅ (a weighted sum of projections is normal — cf. Corollary 1.9 and Proposition 2.A(b)), we get σC (T ) = (C \ρ(T ))\σP (T ).  Compare with Proposition 2.L. Actually, Proposition 2.L is a particular case of Lemma 3.1, since Ek x = x ; ek ek for every x ∈ H defines the orthogonal projection Ek onto the unidimensional space spanned by ek .

3.1 Spectral Theorem for Compact Operators

57

Lemma 3.2. A weighted sum of projections T ∈ B[H] is compact if and only if the following triple condition holds. (i) σ(T ) is countable, (ii) 0 is the only possible accumulation point of σ(T ), and (iii) dim R(Eγ ) < ∞ for every γ such that λγ = 0.  Proof. Let T = γ∈Γ λγ Eγ ∈ B[H] be a weighted sum of projections. Claim. R(Eγ ) ⊆ N (λγ I − T ) for every γ ∈ Γ . Proof. Take any α ∈ Γ . If x ∈ R(Eα ), then x = Eα x and so T x = T Eα x =  λ γ∈Γ γ Eγ Eα x = λα Eα x = λα x (because Eγ ⊥ Eα whenever α = γ). Hence x ∈ N (λα I − T ), which concludes the proof of the claimed inclusion. If T is compact, then σ(T ) is countable and 0 is the only possible accumulation point of σ(T ) (Corollary 2.20), and dim N (λI − T ) < ∞ whenever λ = 0 (Theorem 1.19). Then dim R(Eγ ) < ∞ for every γ for which λγ = 0, by the above claim. Conversely, if T = O, then T is trivially compact. Thus suppose T = O. Since T is normal (Corollary 1.9), r(T ) > 0 (because normal operators are normaloid, i.e., r(T ) = T ) and so there exists λ = 0 in σP (T ) by Theorem 2.18. If σ(T ) is countable, then let {λk } be any enumeration of the countable set σP (T )\{0} = σ(T )\{0}. Hence  Tx = λk Ek x for every x ∈ H k

(cf. Lemma 3.1), where {Ek } is included in a resolution of the identity on H (and is itself a resolution of the identity / σP (T )). If {λk } is finite, n on H if 0 ∈ n R(E } , then R(T ) = ). If dim R(Ek ) < ∞ for each say {λk } = {λ k k k=1 k=1 −  n < ∞. Thus T is a finite-rank operator, and so k, then dim k=1 R(Ek ) compact (cf. Proposition 1.X). Now suppose {λk } is countably infinite. Since σ(T ) is a compact set (Theorem 2.1), the infinite set {λk } has an accumulation point in σ(T ) — this in fact is the Bolzano–Weierstrass Property which characterizes compact sets in metric spaces. If 0 is the only possible accumulation point of σ(T ), then 0 is the unique accumulation point of {λk }. Therefore, for each integer n ≥ 1 consider the partition {λk } = {λk } ∪ {λk }, where 1 1 n ≤ |λk | and |λk | < n . Since {λk } is a finite subset of σ(T ) (it has no accumulation point), it follows that {λk } is an infinite subset of σ(T ). Set  Tn = λk Ek ∈ B[H] for each n ≥ 1.  k

As we have seen, dim R(Tn ) < ∞; equivalently, Tn is a finite-rank operator. However, since Ej ⊥ Ek whenever j = k we get  2    2  Ek  x ≤ sup |λk  | (T − Tn )x2 =  λ Ek x2 ≤ n12 x2 k   k

k

k

u for all x ∈ H, and so Tn −→ T . Then T is compact by Proposition 1.Z.



58

3. The Spectral Theorem

Thus every bounded weighted sum of projections is normal and, if it is compact, it has a countable set of distinct eigenvalues. The Spectral Theorem for compact operators ensures the converse: every compact and normal operator T is a (countable) weighted sum of projections, whose weights are precisely the eigenvalues of T . Theorem 3.3. (Spectral Theorem for Compact Operators). If an operator T ∈ B[H] on a (nonzero complex ) Hilbert space H is compact and normal, then there exists a unique countable resolution of the identity {Ek } on H and a bounded set of scalars {λk } for which  T = λ k Ek , k

where {λk } = σP (T ) is the (nonempty) set of all (distinct) eigenvalues of T and each Ek is the orthogonal projection onto the eigenspace N (λk I − T ) (i.e., R(Ek ) = N (λk I − T )). If the above countable weighted sum of projections is infinite, then it converges in the (uniform) topology of B[H]. Proof. If T ∈ B[H] is compact and normal, then it has a nonempty point spectrum (Corollary 2.21). In fact, the heart of the matter is: T has enough eigenvalues so that its eigenspaces span the Hilbert space H.  − Claim. = H. λ∈σP (T ) N (λI − T ) −  Proof. Set M = λ∈σP (T ) N (λI − T ) , which is a subspace of H. Suppose M = H and hence M⊥ = {0}. Consider the restriction T |M⊥ of T to M⊥. Since T is normal, M reduces T (Theorem 1.16). Thus M⊥ is T -invariant, and T |M⊥ ∈ B[M⊥ ] is normal (Proposition 1.Q). Since T is compact, T |M⊥ is compact (Proposition 1.V). Then T |M⊥ is a compact normal operator on the Hilbert space M⊥ = {0}, and so σP (T |M⊥) = ∅ by Corollary 2.21. Thus there exist λ ∈ C and 0 = x ∈ M⊥ for which T |M⊥ x = λ x and hence T x = λ x. Then λ ∈ σP (T ) and x ∈ N (λI − T ) ⊆ M. This leads to a contradiction: 0 = x ∈ M ∩ M⊥ = {0}. Outcome: M = H. Since T is compact, the nonempty set σP (T ) is countable (Corollaries 2.20 and 2.21) and bounded (because T ∈ B[H] — cf. Theorem 2.1). Thus write σP (T ) = {λk }, where {λk } is a finite or a countably infinite bounded subset of C consisting of all eigenvalues of T . Each N (λk I − T ) is a subspace of H (since each λk I − T is bounded). Moreover, since T is normal, Lemma 1.15(a) ensures that a sequence of pairN (λk I − T ) ⊥ N (λj I − T ) if k = j. Then {N (λk I − T )} is −  wise orthogonal subspaces of H spanning H; that is, H = k N (λk I − T ) by the above claim. So the sequence {Ek } of the orthogonal projections onto each N (λk I − T ) is a resolution of the identity  on H (see Section 1.4 — recall: − T ) is unique). Thus x = k Ek x. Since T is linear and conEk onto N (λk I tinuous, T x = k T Ek x for every x ∈ H. But Ek x ∈ R(Ek ) = N (λk I − T ), and so T Ek x = λk Ek x, for each k and every x ∈ H. Therefore

3.2 Diagonalizable Operators

Tx =

 k

59

λk Ek x for every x ∈ H,

and hence T is a countable weighted sum of projections. If it is a finite weighted sum of projections, then we are done. On the other hand, suppose it is an infinite weighted sum of projections. In this case the above identity says n s λk Ek −→ T as n → ∞. k=1   n So the sequence k=1 λk Ek converges strongly to T . However, since T is compact, thisconvergence actually takes place in the uniform topology (i.e., n the sequence k=1 λk Ek converges uniformly to T ). Indeed,  

 2  ∞ n    2 ∞ λ k Ek x  =  λ k Ek x  = |λk |2 Ek x2  T− k=n+1 k=1 k=n+1 ∞ ≤ sup |λk |2 Ek x2 ≤ sup |λk |2 x2 k=n+1

k≥n+1

k≥n+1

for ∞ R(Ej )2⊥ R(Ek ) whenever j = k, and x = ∞every integer n ≥ 12 (reason: = E x so that x k k=1 k=1 Ek x — see Sections 1.3 and 1.4). Hence   

  n n     0 ≤ T − λk Ek  = sup  T − λk Ek x ≤ sup |λk | k=1

k=1

x=1

k≥n+1

for all n ∈ N . As T is compact, the infinite sequence {λn } of distinct elements in σ(T ) must converge to zero (Corollary 2.20(a)). Since limn λn = 0 we get limn supk≥n+1 |λk | = lim supn |λn | = 0 and so n u λk Ek −→ T as n → ∞.  k=1

3.2 Diagonalizable Operators If T in B[H] is compact and normal, then the (orthogonal) family of orthogonal projections {Eλ }λ∈σP (T ) onto each eigenspace N (λI − T ) is a countable resolution of the identity on H, and T is a weighted sum of projections. This is what the Spectral Theorem for compact (normal) operators says. Thus we write  λ Eλ , T = λ∈σP (T )

which is a uniform limit, and so it implies pointwise convergence (i.e., T x =  λ∈σP (T ) λ Eλ x for every x ∈ H). This is naturally identified with (i.e., it is unitarily equivalent to) an orthogonal direct sum of scalar operators,  λ Iλ , T ∼ = λ∈σP (T )

with each Iλ denoting the identity on each eigenspace N (λI − T ). In fact, Iλ = Eλ |R(Eλ ) , where R(Eλ ) = N (λI − T ). Thus under such an identification it is usual to write (a slight abuse of notation)  λ Eλ . T = λ∈σP (T )

60

3. The Spectral Theorem

Each of these equivalent representations is referred to as the spectral decomposition of a compact normal operator T . The Spectral Theorem for compact (normal) operators can be restated in terms of an orthonormal basis for N (T )⊥ consisting of eigenvectors of T , as follows. Corollary 3.4. Let T ∈ B[H] be compact and normal . nλ (a) For each λ ∈ σP (T )\{0} there is a finite orthonormal basis {ej (λ)}j=1 for N (λI − T ) consisting entirely of eigenvectors of T .

nλ (b) The set {ek } = λ∈σP (T )\{0} {ej (λ)}j=1 is a countable orthonormal basis ⊥ for N (T ) made up of eigenvectors of T .  nλ (c) T x = λ∈σP (T )\{0} λ j=1 x ; ej (λ) ej (λ) for every x ∈ H.  (d) T x = k λk x ; ek  ek for every x ∈ H, where {λk } is a sequence containing all nonzero eigenvalues of T including multiplicity (i.e., finitely repeated according to the dimension of the respective eigenspace).

Proof. The point spectrum σP (T ) is nonempty and countable by Theorem 3.3, and σP (T ) = {0} if and only if T = O by Corollary 2.21 or, equivalently, if and only if N (T )⊥ = {0} (i.e., N (T ) = H). If T = O, then the above assertions hold trivially (σP (T )\{0} = ∅, {ek } = ∅, N (T )⊥ = {0} and T x = 0x = 0 for every x ∈ H because the empty sum is null). Thus suppose T = O (so that N (T )⊥ = {0}) and take an arbitrary λ = 0 in σP (T ). (a) By Theorem 1.19 dim N (λI − T ) is finite, say dim N (λI − T ) = nλ for a positive integer nλ . This ensures the existence of a finite orthonormal basis nλ for the Hilbert space N (λI − T ) = {0}, and so each ej (λ) is an {ej (λ)}j=1 eigenvector of T (since 0 = ej (λ) ∈ N (λI − T )), concluding the proof of (a).

nλ ⊥ Claim. λ∈σP (T )\{0} {ej (λ)}j=1 is an orthonormal basis for N (T ) . Proof. By Theorem 3.3, H=

 λ∈σP (T )

N (λI − T )

−

.

Since {N (λI − T )}λ∈σP (T ) is a nonempty family of orthogonal subspaces of H (by Lemma 1.15(a)) we get 

⊥ ! N (λI − T )⊥ = N (λI − T ) N (T ) = λ∈σP (T )\{0}

λ∈σP (T )\{0}

(see, e.g., [78, Problem 5.8(b,e)]). Hence (see Section 1.3) 

− N (λI − T ) . N (T )⊥ = λ∈σP (T )\{0}

Therefore, since {N (λI − T )}λ∈σP (T ) is a family of orthogonal subspaces, the claimed result follows from (a).

λ (b) Since {ek } = λ∈σP (T )\{0} {ej (λ)}nj=1 is a countable set (a countable union of countable sets), the result in (b) follows from (a) and the above claim.

3.2 Diagonalizable Operators

61

(c,d) Consider the decomposition H = N (T ) + N (T )⊥ (see Section 1.3) and take an arbitrary x ∈ H. Thus x = u + v with u ∈ N (T ) and v ∈ N (T )⊥ . Consider the Fourier series expansion of v in terms of the orthonormal basis λ for the Hilbert space N (T )⊥ = {0}, namely, {ek } = λ∈σP (T )\{0} {ej (λ)}nj=1  λ   v ; ej (λ) ej (λ). Since the operator T is v = k v ; ek ek = λ∈σP (T )\{0} nj=1 linear and continuous, and since T ej (λ) = λ ej (λ) for every j = 1, . . . , nλ and every λ ∈ σP (T )\{0}, it follows that  nλ Tx = Tu + Tv = Tv = v ; ej (λ) T ej (λ) j=1 λ∈σP (T )\{0}  nλ = λ v ; ej (λ) ej (λ). λ∈σP (T )\{0}

j=1

However, x ; ej (λ) = u ; ej (λ) + v ; ej (λ) = v ; ej (λ) because u ∈ N (T ) and ej (λ) ∈ N (T )⊥, concluding the proof of (c), which implies the result in (d).  Remark. If T ∈ B[H] is compact and normal, and if H is nonseparable, then N (T ) is nonseparable and 0 ∈ σP (T ). Indeed, suppose T = O (otherwise the above statement is trivial). Thus N (T )⊥ = {0} is separable (i.e., it has a countable orthonormal basis) by Corollary 3.4, and hence N (T ) is nonseparable whenever H = N (T ) + N (T )⊥ is nonseparable. Moreover, if N (T ) is nonseparable, then N (T ) = {0} tautologically, which means 0 ∈ σP (T ). The expression for a compact normal operator T in Corollary 3.4(c) (and, equivalently, the spectral decomposition of T ) says T is a diagonalizable operator in the following sense: T = T |N (T )⊥ ⊕ O (since N (T ) reduces T ), where T |N (T )⊥ in B[N (T )⊥ ] is a diagonal operator with respect to a countable orthonormal basis {ek } for the separable Hilbert space N (T )⊥ . Generalizing: an operator T in B[H] (not necessarily compact) on a Hilbert space H (not necessarily separable) is diagonalizable if there is a resolution of the identity {Eγ }γ∈Γ on H and a bounded family of scalars {λγ }γ∈Γ for which T u = λγ u whenever u ∈ R(Eγ ).  Take an = γ∈Γ Eγ x in H. Since T is linear and continuous,  arbitrary x  T x = γ∈Γ T Eγ x = γ∈Γ λγ Eγ x. Hence T is a weighted sum of projections (and consequently it is normal). Thus we write   λ γ Eγ . λγ Eγ or T = T = γ∈Γ

γ∈Γ

 Conversely, if T is a weighted  (i.e., T x = γ∈Γ λγ Eγ x for  sum of projections every x ∈ H), then T u = γ∈Γ λγ Eγ u = γ∈Γ λγ Eγ Eα u = λα u for every u ∈ R(Eα ) (since Eγ Eα = O for γ = α and u = Eα u for u ∈ R(Eα )), and so T is diagonalizable. Conclusion: An operator T on H is diagonalizable if and only if it is a weighted sum of projections for some bounded sequence of scalars {λγ }γ∈Γ and some resolution of the identity {Eγ }γ∈Γ on H.

62

3. The Spectral Theorem

In this case {Eγ }γ∈Γ is said to diagonalize T . (See Proposition 3.A and recall: the identity on any Hilbert space H is trivially diagonalizable — it is actually a diagonal if H is the collection Γ2 of all square-summable families {ζγ }γ∈Γ of scalars ζγ ∈ C — and thus it is normal but not compact; cf. Proposition 1.Y.) The implication (b) ⇒ (d) in Corollary 3.5 below is the compact version of the Fuglede Theorem (Theorem 3.17) which will be developed in Section 3.5. Corollary 3.5. Suppose T is a compact operator on a Hilbert space H. (a) T is normal if and only if it is diagonalizable. Let {Ek } be a resolution of the identity on H that diagonalizes a compact and normal operator T ∈ B[H] into its spectral decomposition, and take any operator S ∈ B[H]. The following assertions are pairwise equivalent. (b) S commutes with T . (c) S commutes with T ∗ . (d) S commutes with every Ek . (e) R(Ek ) reduces S for every k. Proof. Take a compact operator T on H. If T is normal, then it is diagonalizable by the Spectral Theorem. The converse is trivial because a diagonalizable operator is normal. This proves (a). From now on suppose T is compact and normal, and consider its spectral decomposition  T = λ k Ek k

as in Theorem 3.3, where {Ek } is a resolution of the identity on H and {λk } = σP (T ) is the set of all (distinct) eigenvalues of T . Observe that  T∗ = λ k Ek . k

The central result is the implication (b) ⇒ (e). The main idea behind the proof relies on Proposition 2.T (see also [98, Theorem 1.16] ). Take any operator S on H. Suppose S T = T S. Since Ek T = λ k E k = T E k  (recall: Ek T = j λj Ek Ej = λk Ek = j λj Ej Ek = T Ek ), each R(Ek ) = N (λk I − T ) reduces T (Proposition 1.I). Then 

Ek T |R(Ek ) = T Ek |R(Ek ) = T |R(Ek ) = λk I : R(Ek ) → R(Ek ). So on R(Ek ), (Ej SEk )T |R(Ek ) = (Ej SEk )(T Ek ) = Ej S T Ek = Ej T SEk = (T Ej )(Ej SEk ) = T |R(Ej ) (Ej SEk ) whenever j = k. Moreover, since σ(T |R(Ek ) ) = {λk } and λk = λj , we get

3.3 Spectral Measure

63

σ(T Ek ) ∩ σ(T Ej ) = ∅, and hence by Proposition 2.T for j = k Ej SEk = O. Thus (as each Ek is self-adjoint and {Ek } is a resolution of the identity) " #   Ej y = Ej SEk x ; y = Ek SEk x ; y SEk x ; y = SEk x ; j

j

for every x, y ∈ H. Then SEk = Ek SEk . Equivalently, R(Ek ) is S-invariant  −  (Proposition 1.I) for every k. Furthermore, R(E = ) R(Ek )⊥ = j j=k j=k R(Ej ) (because {Ek } is a resolution of the identity on H). Since R(Ej ) is S-invariant for every j, then R(Ek )⊥ is S-invariant as well. Therefore R(Ek ) reduces S for every k. This proves that (b) implies (e). But (e) is equivalent to (d) by Proposition 1.I. (c). Indeed, if SEk = Ek S for every k, then S T ∗ =  Also (d) implies  ∗ k λk SEk = k λk Ek S = T S (as S is linear and continuous). Thus (b)  implies (c), and therefore (c) implies (b) because T ∗∗ = T .

3.3 Spectral Measure If T is a compact normal operator, then its point spectrum is nonempty and countable, which may not hold for noncompact normal operators. But this is not the main role played by compact operators in the Spectral Theorem — we  can deal with an uncountable weighted sum of projections T x = γ∈Γ λγ Eγ x. What is actually special with a compact operator is that a compact normal operator not only has a nonempty point spectrum − but it has enough eigenspaces N (λI − T ) = H (see proof of Theorem 3.3). to span H; that is, λ∈σP (T ) This makes the difference, since a normal (noncompact) operator may have an empty point spectrum or it may have eigenspaces but not enough to span the whole space H. The Spectral Theorem, however, survives the lack of compactness if the point spectrum is replaced with the whole spectrum (which is never empty). Such an approach for the general case of the Spectral Theorem (i.e., for normal, not necessarily compact operators) requires measure theory. Let R = R ∪{−∞}∪{+∞} stand for the extended real line. Take a measure space (Ω, AΩ , μ) where AΩ is a σ-algebra of subsets of a nonempty set Ω and μ : AΩ → R is a nonnegative measure on AΩ , which means μ(Λ) ≥ 0 for every Λ ∈ AΩ . Sets in AΩ are called measurable (or AΩ -measurable). A nonnegative measure μ is positive if μ(Ω) > 0. It is finite if μ(Ω) < ∞ (i.e., μ : AΩ → R ), and σ-finite if Ω is a countable union of measurable sets of finite measure.

64

3. The Spectral Theorem

Let F denote either the real line R or the complex plane C . A Borel σ-algebra (or a σ-algebra of Borel sets) is the σ-algebra AF of subsets of F generated by the usual (metric) topology of F (i.e., the smallest σ-algebra of subsets of F including all open subsets of F ). Borel sets are precisely the sets in AF (i.e., the AF -measurable sets). Thus all open, and consequently all closed, subsets of F are Borel sets (i.e., lie in AF ). If Ω ⊆ F is a Borel subset of F (i.e., if Ω ∈ AF ), then set AΩ = ℘ (Ω) ∩ AF . This is the σ-algebra of Borel subsets of Ω, where ℘ (Ω) is the power set of Ω (the collection of all subsets of Ω). A measure μ on AΩ = ℘ (Ω) ∩ AF is said to be a Borel measure if μ(K) < ∞ for every compact set K ⊆ Ω (recall: compact subsets of Ω lie in AΩ ). Every finite measure is a Borel measure, and every Borel measure 

is σ-finite. The support of a measure Λ ∈ AΩ : Λ is open and μ(Λ) = 0 , a μ on AΩ is the set support(μ) = Ω\ is the largest open set of measure zero. Equivaclosed set whose complement  lently, support(μ) = Λ ∈ AΩ : Λ is closed and μ(Ω\Λ) = 0 , the smallest closed set whose complement has measure zero. There is a vast literature on measure theory. See, for instance, [12], [23], [53], [81], [102], or [103] (which simply reflects the author’s preference but by no means exhausts any list). Remark. Take an arbitrary subset Ω of F . A set K ⊆ Ω ⊆ F is compact if it is complete and totally bounded. (In Ω ⊆ F total boundedness coincides with boundedness and so compactness in Ω ⊆ F means complete and bounded.) Thus K ⊆ Ω is compact in the relative topology of Ω if and only if K ⊆ F is compact in the topology of F (where compact means closed and bounded). But open and closed in the preceding paragraph refer to the topology of Ω. Note: if Ω is closed (open) in F , then closed (open) in Ω and in F coincide.

The disjoint union γ∈Γ Ωγ of a collection {Ωγ }γ∈Γ of subsets of C is obtained by reindexing the elements of each set Ωγ as follows. For every index γ ∈ Γ write Ωγ = {λδ (γ)}δ∈Γγ so that if there is a point λ in Ωα ∩ Ωβ for α = β in Γ, then the same point λ is represented as λω (α) ∈ Ωα for some ω ∈ Γα and as λτ (β) ∈ Ωβ for some τ ∈ Γβ , and these representations are interpreted as distinct elements. Thus under this definition the sets Ωα and Ωβ become disjoint. (Regard the sets in {Ωγ }γ∈Γ as subsets of distinct copies of C so that Ωα and Ωβ are regarded as disjoint for distinct indices α, β ∈ Γ .) Unlike the ordinary union, the disjoint union can be viewed as the union of all Ωγ considering multiplicity of points in it disregarding possible overlappings. From now on suppose either (i) Ω is a nonempty Borel subset of the complex plane C and AΩ is the σ-algebra of all Borel subsets of Ω, or (ii) Ω is a nonempty set consisting of a disjoint union of Borel subsets Ωγ of C and AΩ is the σ-algebra of all disjoint unions included in Ω made up of Borel subsets of C , again referred to as the σ-algebra of Borel subsets of Ω. Definition 3.6. Let Ω be a nonempty set and let AΩ be a σ-algebra of Borel subsets of Ω as above. A (complex) spectral measure in a (complex) Hilbert space H is a mapping E : AΩ → B[H] satisfying the following axioms.

3.3 Spectral Measure

65

(a) E(Λ) is an orthogonal projection for every Λ ∈ AΩ , (b) E(∅) = O and E(Ω) = I, (c) E(Λ1 ∩ Λ2 ) = E(Λ1 )E(Λ2 ) for every Λ1 , Λ2 ∈ AΩ ,    (d) E k E(Λk ) (ordinary union) whenever {Λk } is a countable k Λk = collection of pairwise disjoint sets in AΩ (i.e., E is countably additive). If {Λk } is a countably infinite collection of pairwise disjoint sets in AΩ , then the identity in (d) means convergence in the strong topology:

 n s E(Λk ) −→ E Λk . k=1

k

Indeed, since Λj ∩ Λk = ∅ if j = k, it follows by properties (b) and (c) that E(Λj )E(Λk ) = E(Λj ∩ Λk ) = E(∅) = O for j = k, and so {E(Λk )} is an projections in B[H]. Then, according to orthogonal sequence  nof orthogonal strongly to the orthogonal projecProposition 1.K,  k=1 E(Λk ) converges −   = )) tion in B[H] onto  k R(E(Λ k k R(E(Λk )) . Thus what prop

with the orthogonal projection in B[H] coincides Λ erty (d) says is: E k k  R(E(Λ of resolution of the identhe concept )) generalizes . This onto k k tity on H. In fact, if {Λk } is a partition of Ω, then the orthogonal sequence of orthogonal projections {E(Λk )} is such that

 n s E(Λk ) −→ Λk = E(Ω) = I. E k=1

k

Let E : AΩ → B[H] be a spectral measure into B[H]. For each pair of vectors x, y ∈ H consider the map πx,y : AΩ → C defined for every Λ ∈ AΩ by πx,y (Λ) = E(Λ)x ; y = x ; E(Λ)y. (Recall: E(Λ) is self-adjoint.) The function πx,y is an ordinary complex-valued (countably additive) measure on AΩ . For each x, y ∈ H consider the measure space (Ω, AΩ , πx,y ). Let B(Ω) denote the algebra of all complex-valued $ bounded AΩ -measurable functions φ : Ω → C on Ω. The integral $φ dπx,y of φ in B(Ω) with respect$ to the measure π$x,y will also be denoted $ by φ(λ) dπx,y , $ or φ dEλ x ; y, or φ dx ; Eλ y, or φ(λ) dEλ x ; y, or φ(λ) dx ; Eλ y. Lemma 3.7. Let E : AΩ → B[H] be a spectral measure. Every function φ in B(Ω) is πx,y -integrable, and there is a unique operator F ∈ B[H] for which % F x ; y = φ(λ) dEλ x ; y for every x, y ∈ H. Proof. Let φ : Ω → C be a (bounded AΩ -measurable) function in B(Ω). Set φ∞ = supλ∈Ω |φ(λ)|. Since E(·)x ; x = E(·)x2 is a positive measure, %  %    φ(λ) dEλ x ; x ≤ φ∞ dEλ x ; x = φ∞ E(Ω)x ; x = φ∞ x2

66

3. The Spectral Theorem

for x ∈ H. Apply the polarization identity and the parallelogram law (Proposition 1.A (a1 ,b)) for the sesquilinear form E(Λ) · ; · : H×H → C to get %       φ(λ) dEλ x ; y ≤ φ∞ x2 + y2 for every x, y ∈ H. So φ is πx,y -integrable. Take the sesquilinear form f : H×H → C given by % f (x, y) = φ(λ) dEλ x ; y for every x, y ∈ H, which is bounded (supx≤1,y≤1 |f (x, y)| ≤ 2φ∞ < ∞), and so is the linear functional f ( · , y) : H → C for each y ∈ H. Thus, by the Riesz Representation Theorem in Hilbert Space (see, e.g., [78, Theorem 5.62] ), for each y ∈ H there is a unique zy ∈ H for which f (x, y) = x ; zy  for every x ∈ H. This establishes a mapping F #: H → H that assigns to each y ∈ H a unique zy ∈ H such that f (x, y) = x ; F # y for every x ∈ H. As it is easy to show, F # is unique and lies in B[H]. So the mapping F = (F # )∗ is the unique operator in B[H] such that F x ; y = f (x, y) for every x, y ∈ H.  $ Notation. The unique F ∈ B[H] for which F x ; y = φ(λ) dEλ x ; y for every x, y ∈ H as in Lemma 3.7 is usually denoted by % F = φ(λ) dEλ . function χΛ of a set Λ$∈ AΩ , then In particular, if φ is the $ characteristic $ $ E(Λ)x ; y = πx,y (Λ) = Λ dπx,y = χΛ dπx,y = χΛ(λ) d Eλ x ; y = Λ dEλ x ; y for every x, y ∈ H. So the orthogonal projection E(Λ) ∈ B[H] is denoted by % % E(Λ) = χΛ(λ) dEλ = dEλ . Λ

Lemma 3.8. Let E $ : AΩ → B[H] be$a spectral measure. If φ, ψ are functions in B(Ω) and F = φ(λ) dEλ , G = ψ(λ) dEλ are operators in B[H], then % % (a) F ∗ = φ(λ) dEλ and (b) F G = φ(λ) ψ(λ) dEλ . $ Proof. $ Given φ and ψ in B(Ω), consider the operators F = φ(λ) dEλ and G = ψ(λ) dEλ in B[H] (cf. Lemma 3.7). Take an arbitrary pair x, y ∈ H. (a) Recall: E(Λ) is self-adjoint for every Λ ∈ AΩ . Thus we get % % F ∗ x ; y = y ; F ∗ x = F y ; x = φ(λ) dEλ y ; x = φ(λ) dEλ x ; y. $ $ (b) Set π = πx,y : AΩ → C . Since F x ; y = φ dπ = φ(λ) dEλ x ; y, % F Gx ; y = φ(λ) dEλ Gx ; y. Consider the measure ν = νx,y : AΩ → C defined for every Λ ∈ AΩ by

3.3 Spectral Measure

67

ν(Λ) = E(Λ)Gx ; y. %

Thus F Gx ; y =

φ dν.

$ $ Since Gx ; y = ψ dπ = ψ(λ) dEλ x ; y, % ν(Λ) = Gx ; E(Λ)y = ψ(λ) dEλ x ; E(Λ)y. Therefore, since E(Δ)x ; E(Λ)y = E(Λ)E(Δ)x ; y = E(Δ ∩ Λ)x ; y = π(Δ ∩ Λ) = π|Λ (Δ ∩ Λ) for every Λ, Δ ∈ AΩ , where the measure π|Λ : AΛ → C is the restriction of π : AΩ → C to AΛ = ℘ (Λ) ∩ AΩ , we get % % % % % dν = ν(Λ) = ψ(λ) dEλ x ; E(Λ)y = ψ dπ|Λ = χΛ ψ dπ = ψ dπ Λ

Λ

dν for every Λ ∈ AΩ . This is usually written as dν = ψ dπ, meaning that $ψ = dπ is ym derivative$ of ν with $ respect to π. In fact, since Λ dν = $ the Radon–Nikod´ ψ dπ for every Λ ∈ A , then φ dν = φ ψ dπ for every φ ∈ B(Ω). Hence Ω Λ % % % F Gx ; y = φ dν = φ ψ dπ = φ(λ) ψ(λ) dEλ x ; y. 

Lemma 3.9. If E : AΩ → B[H] $ a spectral measure, and if φ is a function in B(Ω), then the operator F = φ(λ) dEλ is normal . Proof. By Lemma 3.8, F ∗F =

%

|φ(λ)|2 dEλ = F F ∗.



Lemma 3.10. Let E : AΩ →$B[H] be a spectral measure. If ∅ = Ω ⊂ C is compact, then the operator F = λ dEλ is well defined and normal in B[H]. Also % p(F, F ∗ ) = p(λ, λ) dEλ is again normal in B[H] for every polynomial p(· , ·): Ω ×Ω → C in λ and λ. Proof. Consider Lemmas 3.7, 3.8, 3.9. If Ω ⊂ C is compact, then the identity map ϕ: Ω → Ω in B(Ω) and F defined for ϕ(λ) = λ is normal. For any funclies n,m tion p(· , ·) = i,j=0 αi,j (·)i (·)j : Ω → C , which also lies in B(Ω), set p(F, F ∗ ) = n,m i ∗j in B[X ], so that p(F, F ∗ ) p(F, F ∗ )∗ = p(F, F ∗ )∗ p(F, F ∗ ).  i,j=0 αi,j F F

Remark. Lemma 3.10 still holds if Ω = γ∈Γ Ωγ is a disjoint union where each -- ∈ Ω, then λ -- = λ ∈ Ωβ for a unique β ∈ Γ. Ωγ ⊂ C is compact. Actually if λ --) = λ ∈ C for each λ -- ∈ Ω. In this case Thus$ let ϕ : Ω → C be given by ϕ(λ ( · ), ϕ( · )) : Ω → C in B(Ω). F = ϕ(λ --) dEλ and p(· , ·) is replaced by p(ϕ -The standard form of the Spectral Theorem (Theorem 3.15) states the converse of Lemma 3.9. A proof of it relies on Theorem 3.11. In fact, Theorems 3.11 and 3.15 can be thought of as equivalent versions of the Spectral Theorem.

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3. The Spectral Theorem

3.4 Spectral Theorem: The General Case Assume all Hilbert spaces are nonzero and complex. Consider the notation in the paragraph preceding Proposition 3.B. Take the multiplication operator Mφ in B[L2 (Ω, μ)], which is normal, where φ lies in L∞ (Ω, μ), Ω is a nonempty set, and μ is a positive measure on AΩ (cf. Proposition 3.B in Section 3.6): Mφ f = φ f

for every

f ∈ L2 (Ω, μ).

Theorem 3.11. (Spectral Theorem – first version). If T is a normal operator on a Hilbert space, then there is a positive measure μ on a σ-algebra AΩ of subsets of a set Ω = ∅ and a function ϕ in L∞ (Ω, μ) such that T is unitarily equivalent to the multiplication operator Mϕ on L2 (Ω, μ), and ϕ∞ = T . Proof. Let T be a normal operator on a Hilbert space H. We split the proof into two parts. In part (a) we prove the theorem for the case where there exists astar-cyclic vector x for T , which means there exists a vector x ∈ H for which n ∗m x} = H (each index m and n runs over all nonnegative integers; m,n {T T that is, (m, n) ∈ N 0 × N 0 ). In part (b) we consider the complementary case. (a) Fix a unit vector x ∈ H. Take a compact ∅ = Ω ⊂ C . Let P (Ω) be the set of all polynomials p(· , ·) : Ω × Ω → C in λ and λ (equivalently, p(· , ·) : Ω → C such that λ → p(λ, λ)). Let C(Ω) be the set of all complex-valued continuous functions on Ω. Thus P (Ω) ⊂ C(Ω). By the Stone–Weierstrass Theorem, if Ω is compact, then P (Ω) is dense in the Banach space (C(Ω),  · ∞ ): P (Ω)− = C(Ω) in the sup-norm topology. (For real and complex versions of the Stone–Weierstrass Theorem see, e.g., [45, Theorems IV.6.16,17], [35, Theorems 7.3.1,2], [86, Theorems III.1.1,4], or [30, Theorem V.8.1].) Consider the functional Φ : P (Ω) → C given by Φ(p) = p(T, T ∗ )x ; x for every p ∈ P (Ω), which is linear on the linear space P (Ω). Moreover, |Φ(p)| ≤ p(T, T ∗ )x2 = p(T, T ∗ ) = r(p(T, T ∗ )) by the Schwarz inequality since x = 1 and p(T, T ∗ ) is normal (thus normaloid). From now on set Ω = σ(T ). According to Theorem 2.8, and with P (σ(T )) viewed as a linear manifold of the Banach space (C(Ω),  · ∞ ), |Φ(p)| ≤ r(p(T, T ∗ )) =

sup λ∈σ(p(T,T ∗ ))

|λ| = sup |p(λ, λ)| = p∞ λ∈σ(T )

Then Φ : (P (σ(T )),  · ∞ ) → (C , | · |) is a linear contraction. Since P (σ(T )) is dense in C(σ(T )) and Φ is linear continuous, Φ has a unique linear continuous extension over C(σ(T )), denoted again by Φ : (C(σ(T )),  · ∞ ) → (C , | · |). Claim. Φ : C(σ(T )) → C is a positive functional in the following sense: 0 ≤ Φ(ψ) for every ψ ∈ C(σ(T )) such that 0 ≤ ψ(λ) for every λ ∈ σ(T ).

3.4 Spectral Theorem: The General Case

69

Proof. Let ψ ∈ C(σ(T )) be such that 0 ≤ ψ(λ) for every λ ∈ σ(T ). Take an arbitrary ε > 0. Since Φ is a linear contraction and P (σ(T ))− = C(σ(T )), there is a polynomial pε ∈ P (σ(T )) such that 0 ≤ pε (λ, λ) for every λ ∈ σ(T ) and |Φ(pε ) − Φ(ψ)| ≤ pε − ψ∞ < ε. The Stone–Weierstrass Theorem for real functions ensures the existence of a polynomial qε in the real variables α, β with real coefficients for which |qε (α, β)2 − pε (λ, λ)| < ε for every λ = α + iβ ∈ σ(T ). 1

(Note: 0 ≤ pε (λ, λ) and the above may be replaced by |qε (α, β) − pε (λ, λ) 2 |). Take the Cartesian decomposition T = A + iB of T, where A and B are commuting self-adjoint operators (Proposition 1.O). Then qε (A, B) is self-adjoint, and so qε (A, B)2 is a nonnegative operator commuting with T and T ∗. Also, |qε (A, B)2 x ; x− Φ(pε )| = |qε (A, B)2 x ; x−pε (T, T ∗ )x ; x| = |(qε (A, B)2 − pε (T, T ∗ ))x ; x| ≤ qε (A, B)2 − pε (T, T ∗ ) = r(qε (A, B)2 − pε (T, T ∗ )) since qε (A, B)2 − pε (T, T ∗ ) is normal, thus normaloid (reason: qε (A, B)2 is a self-adjoint operator commuting with the normal operator pε (T, T ∗ )). Hence |qε (A, B)2 x ; x − Φ(pε )| ≤

sup λ=α+iβ∈σ(T )

|qε (α, β)2 − pε (λ, λ)| ≤ ε

by Theorem 2.8 again. So |qε (A, B)2 x ; x − Φ(ψ)| ≤ |qε (A, B)2 x ; x − Φ(pε )| + |Φ(pε ) − Φ(ψ)| ≤ 2ε. But 0 ≤ pε (A, B)2 x ; x (since O ≤ qε (A, B)2 ). Then for every ε > 0, 0 ≤ pε (A, B)2 x ; x ≤ 2ε + Φ(ψ). Therefore 0 ≤ Φ(ψ). This completes the proof of the claimed results. Take the bounded, linear, positive functional Φ on C(σ(T )). The Riesz Representation Theorem in C(Ω) (see, e.g., [53, Theorem 56.D], [103, Theorem 2.14], [102, Theorem 13.23], or [81, Theorem 12.5] ) ensures the existence of a finite positive measure μ on the σ-algebra AΩ = Aσ(T ) of Borel subsets of the compact set Ω = σ(T ) (where continuous functions are measurable) for which % Φ(ψ) = ψ dμ for every ψ ∈ C(Ω). Let U : P (Ω) → H be a mapping defined by Up = p(T, T ∗ )x for every p ∈ P (Ω). Regard P (Ω) as a linear manifold of the Banach space L2 (Ω, μ) equipped with the L2 -norm  · 2 . So U is a linear transformation, and % % p22 = |p|2 dμ = pp dμ = Φ(pp) = p(T, T ∗ )p(T, T ∗ )x ; x = p(T, T ∗ )p(T, T ∗ )∗ x ; x = p(T, T ∗ )∗ x2 = p(T, T ∗ )x2 = Up2

70

3. The Spectral Theorem

for every p ∈ P (Ω) (by the definition of p(T, T ∗ ) in the proof of Theorem 2.8, which is normal as in Lemma 3.10). Then U is an isometry, thus injective, and hence it has an inverse on its range   R(U) = p(T, T ∗ )x : p ∈ P (Ω) ⊆ H, say U −1 : R(U) → P (Ω). For each polynomial p in P (Ω) let q be the polynomial in P (Ω) defined by q(λ, λ) = λp(λ, λ) for every λ ∈ Ω. Let ϕ : Ω → Ω be the identity map (i.e., ϕ(λ) = λ ⊂ Ω for every λ ∈ Ω = σ(T ) ⊂ C ), which is bounded (i.e., ϕ ∈ L∞ (Ω, μ) since Ω is bounded). In this case we get q = ϕp

and

q(T, T ∗ )x = T p(T, T ∗ )x.

Thus U −1 T Up = U −1 T p(T, T ∗ )x = U −1 q(T, T ∗ )x = U −1 Uq = q = ϕ p = Mϕ p for every p ∈ P (Ω). Since P (Ω) is dense in the Banach space (C(Ω),  · ∞ ), and since Ω is a compact set in C and μ is a finite positive measure on AΩ , then P (Ω) is dense in the normed space (C(Ω),  · 2 ) as well, which in turn can be regarded as a dense linear manifold of the Banach space (L2 (Ω, μ),  · 2 ), and so P (Ω) is dense in (L2 (Ω, μ),  · 2 ). That is, P (Ω)− = L2 (Ω, μ) in the L2 -norm topology. Moreover, since U is a linear isometry of P (Ω) onto R(U), which are linear manifolds of the Hilbert spaces L2 (Ω, μ) and H, then U extends by continuity to a unique linear isometry, also denoted by U, of the Hilbert space P (Ω)− = L2 (Ω, μ) onto the range of U, viz., R(U) ⊆ H. (In fact, it extends onto the closure of R(U), but when U acts on P (Ω)− = L2 (Ω, μ) its range is closed in H because every linear isometry of a Banach space into a normed space has a closed range; see, e.g., [78, Problem 4.41(d)].) So U is a unitary transformation (i.e., a surjective isometry) between the Hilbert  spaces L2 (Ω, μ) and R(U). If, however, x is a star-cyclic vector for T (i.e., if m,n {T n T ∗m x} = H), then  −  R(U) = p(T, T ∗ )x : p ∈ P (Ω) = {T n T ∗m x} = H. m,n

Hence U is a unitary transformation in B[L2 (Ω, μ), H] for which U −1 T U = Mϕ . Therefore the operator T on H is unitarily equivalent to the multiplication operator Mϕ on L2 (Ω, μ) for the identity map ϕ : Ω → Ω on Ω = σ(T ) (i.e., ϕ(λ) = λ for every λ ∈ σ(T )) which lies in L∞ (Ω, μ). Since T is normal (thus normaloid) and unitarily equivalent to Mϕ (thus with the same norm as Mϕ ) we get ϕ∞ = T . Indeed, according to Proposition 3.B, ϕ∞ = ess sup |ϕ| ≤ sup |λ| = r(T ) = T  = Mϕ  = ϕ∞ . λ∈σ(T )

(b) On the other hand, suppose there is no star-cyclic vector for T . Assume T and H are nonzero to avoid trivialities. Since T is normal, the nontrivial

3.4 Spectral Theorem: The General Case

71

 subspace Mx = m,n {T n T ∗m x} of H reduces T for an arbitrary unit vector x in H. Consider the set  = { Mx : unit x ∈ H} of all orthogonal direct sums of these subspaces, which is partially ordered (in the inclusion ordering).  is not empty (if a unit y ∈ Mx ⊥, then My = m,n {T n T ∗m y} ⊆ Mx ⊥ because Mx ⊥ reduces T and so Mx ⊥ My ). Moreover, every chain in  has an upper bound in  (the union of all orthogonal direct sums in a chain of orthogonal direct sums in  is again an orthogonal direct sum in ). Thus Zorn’s Lemma ensures that  has a maximal element, say M = Mx , which coincides with H (otherwise it would not be maximal since M ⊕ M⊥ = H). Summing up: There exists an indexed family {xγ }γ∈Γ of unit vectors in H generating }γ∈Γ of orthogonal nontrivial suba (similarly indexed) collection {Mγ spaces of H such that each Mγ = m,n {T n T ∗m xγ } reduces T , each xγ is star-cyclic for T |Mγ , and H = γ∈Γ Mγ . Each restriction T |Mγ is a normal operator on the Hilbert space Mγ (Proposition 1.Q). By part (a) there is a positive finite measure μγ on the σ-algebra AΩγ of Borel subsets of Ωγ = σ(T |Mγ ) ⊆ C such that T |Mγ in B[Mγ ] is unitarily equivalent to the multiplication operator Mϕγ in B[L2 (Ωγ , μγ )], where each ϕγ : Ωγ → Ωγ is the identity map (ϕγ (λ) = λ for every λ ∈ Ωγ ), which is bounded on Ωγ = σ(T |Mγ ) (i.e., ϕγ ∈ L∞ (Ωγ , μγ )). Thus there is a unitary orUγ in B[Mγ , L2 (Ωγ , μ γ )] for which Uγ T |Mγ = Mϕγ Uγ . Take the external  2 , μ (Ω unitary L ), and take the U = thogonal direct sum γ γ γ∈Γ γ∈Γ Uγ     2 , T | in B[ U ) ( M (Ω , μ that ( L )] such Mγ ) = γ γ∈Γ γ γ γ∈Γ γ∈Γ  γ∈Γ γ  = T | M U U = M ( ). Therefore ( ) U Mγ ϕγ γ ϕγ γ∈Γ γ γ γ∈Γ γ∈Γ γ  T |Mγ in B[H] = B[ γ∈Γ Mγ ] is   unitarily equivalent to γ∈Γ Mϕγ in B[ γ∈Γ L2 (Ωγ , μγ )]. T =



γ∈Γ

σ(T |Mγ) ⊆ Let Ω = γ∈Γ Ωγ be the disjoint union of {Ωγ }γ∈Γ . Note: Ωγ =

σ(T ) since each Mγ reduces T (Proposition 2.F(b)), and so Ω ⊆ γ∈Γ σ(T ). of each Ωγ ; that Take the collection AΩ of all disjoint unions of Borel subsets

is, the collection of all disjoint unions of the form Λ = γ∈Γ  Λγ for any subset Γ  of Γ where Λγ is a Borel subset of Ωγ (i.e., Λγ ∈ AΩγ ). This is the σ-algebra of Borel subsets of Ω. Let μγ : AΩγ → R be the finite measure on each AΩγ whose existence was ensured in part (a). Consider the Borel subsets

Λ = j∈J Λj of Ω made up of countable disjoint unions of Borel subsets  of each ). Let μ be the measure on A given by μ(Λ) = Ωj (i.e., Λj ∈ A Ω Ω j j∈J μj (Λj )

for every Λ = j∈J Λj in AΩ with {Λj }j∈J being an arbitrary countable collection of sets with each Λj ∈ AΩj for

any countable

subset J of Γ. (For a proper subset Γ  of Γ we may identify γ∈Γ  Λγ with γ∈Γ Λγ by setting Λγ = ∅ for every γ ∈ Γ \Γ .) The measure μ is positive since each μj is (if Λj = Ωj , then μj (Λj ) > 0 for at least one j in one J ). Consider the measure space (Ω, AΩ , μ).

72

3. The Spectral Theorem

Definition. Let ϕ : Ω → C be a function obtained from the

identity maps -- lies in the disjoint union Ω = γ∈Γ Ωγ , then ϕγ : Ωγ → Ωγ as follows. If λ λ -- = λ ∈ Ωβ ⊂ C for a unique index β ∈ Γ and a unique λ ∈ Ωβ . Thus for each --) = ϕβ (λ) = λ ∈ C . This can be viewed as a sort of identity map λ -- ∈ Ω set ϕ(λ including multiplicity, referred to as an identity map with multiplicity. Since Ωγ = σ(T |Mγ ) ⊆ σ(T ) for all γ ∈ Γ, ϕ is bounded. So ϕ ∈ L∞ (Ω, μ) and ϕ∞ = ess sup |ϕ| ≤ sup |ϕ(λ --)| ≤ sup |λ| = r(T ) ≤ T . - ∈Ω λ λ∈σ(T ) (Actually, r(T ) = T  because T is normaloid.) Now we show:   2 γ∈Γ Mϕγ in B[ γ L (Ωγ , μγ )] is

unitarily equivalent to Mϕ in B[L2 (Ω, μ)].  First observe that the Hilbert spaces L2 (Ω, μ) and γ∈Γ L2 (Ωγ , μγ ) are unitarily equivalent. In fact, direct sums and topological sums of families of externally orthogonal  Hilbert spaces are unitarily equivalent, and therefore  2 ∼ L2 (Ωγ , μγ ))−. So take a unitary transformation (Ω , μ L ) ( γ γ = γ∈Γ γ∈Γ   V : γ∈Γ L2 (Ωγ , μγ ) → ( γ∈Γ L2 (Ωγ , μγ ))−. Next consider a transformation 

 W : γ∈Γ L2 (Ωγ , μγ ) → L2 ( γ∈Γ Ωγ , μ) taking each function ψ = γ∈Γ ψγ in 

2 2 2 γ ∈ L (Ωγ , μγ ) to a function W ψ in L ( γ∈Γ Ωγ , μ) γ∈Γ L (Ωγ , μγ ) with ψ

-- = λ ∈ Ωβ for only one β ∈ Γ. Take defined as follows. If λ -- ∈ γ∈Γ Ωγ , then λ --) = ψβ (λ). Regard--) = (W ψ)|Ωβ (λ this unique λ ∈ Ωβ ⊂ C and set (W ψ)(λ a linear Ω, this defines union ing {Ωγ }γ∈Γ as a partition of the disjoint $ $  and surjective map for which W ψ2 = Ω |W ψ|2 dμ = β∈Γ Ωβ |ψβ |2 dμβ =  2 2 , and so it is an isometry as well. Thus W is a unitary γ∈Γ ψγ  = ψ

transformation of ( γ∈Γ L2 (Ωγ , μγ )) onto L2 ( γ∈Γ Ωγ , μ). Then it extends by continuity to a unitary transformation, also denoted by W, over

the whole

Hilbert space ( γ∈Γ L2 (Ωγ , μγ ))− onto the Hilbert space L2 ( γ∈Γ Ωγ , μ). 

That is, W : ( γ∈Γ L2 (Ωγ , μγ ))− → L2 ( γ∈Γ Ωγ , μ) = L2 (Ω, μ) is unitary. So 

( γ∈Γ L2 (Ωγ , μγ ))− ∼ = L2 ( γ∈Γ Ωγ , μ) = L2 (Ω, μ) and hence by transitivity    2 2 ∼ 2 γ∈Γ L (Ωγ , μγ ) = L γ∈Γ Ωγ , μ = L (Ω, μ).   -- ∈ Ω so that Now take any ψ = γ∈Γ ψγ in γ∈Γ L2 (Ωγ , μγ ). Also take any λ --) = (W ψ)|Ωβ (λ --) = --) = ϕβ (λ) and (W ψ)(λ λ -- ∈ Ωβ for one β ∈ Γ . Recall: ϕ|Ωβ (λ --) =  (ϕW ψ)(λ --) = ϕγ (λ) ψβ (λ) =(ϕβ ψβ )(λ). On the ψβ (λ). Thus (Mϕ W ψ)(λ −1 that ( γ∈Γ M ψ = other hand, sinceV −1 ψ = ϕγ )V γ∈Γ ψγ , it follows   −1 ψ and hence V ( M M ψ = ϕ )V ψ = ϕ ϕγ γ γ∈Γ γ∈Γ γ ψγ . γ∈Γ γ γ   γ∈Γ ϕγ Thus (W V ( γ∈Γ Mϕγ )V −1 ψ)(λ --) = (W γ∈Γ ϕγ ψγ )(λ --) = (ϕβ ψβ )(λ). Then   Mϕ W = W V ( γ∈Γ Mϕγ )V −1 on γ∈Γ L2 (Ωγ , μγ ), which extends by conti nuity over all ( γ∈Γ L2 (Ωγ , μγ ))−. Thus for the unitary transformation W V,   WV γ∈Γ Mϕγ = Mϕ W V,

3.4 Spectral Theorem: The General Case

73

  ∼ and so γ∈Γ Mϕγ is unitarily equivalent to Mϕ (i.e., γ M ϕγ = Mϕ ) through ∼ ∼ the unitary W V. Then T = Mϕ by transitivity since T = γ∈Γ Mϕγ . That is, T in B[H] is unitarily equivalent to Mϕ in B[L2 (Ω, μ)]. Finally, since ϕ∞ ≤ T  and T ∼ = Mϕ we get (Proposition 3.B) 

ϕ∞ ≤ T  = Mϕ  ≤ ϕ∞ .

Remark. The identity map with multiplicity ϕ : Ω → σ(T ) ⊂ C may be neither injective nor surjective, but it is bounded. It takes disjoint unions in AΩ into ordinary unions in Aσ(T ) so that ϕ(Ω)− = σ(T ) (see Proposition 2.F(d) since ˜ ⊆ Ω (e.g., T |Mγ is normal). Thus for each Λ˜ ⊆ ϕ(Ω) there is one Λ = ϕ−1 (Λ) −1 ˜ − − ˜ ˜ a maximal) for which Λ = ϕ(Λ) (since ϕ(ϕ (Λ)) = Λ for every Λ˜ ⊆ ϕ(Ω)), and so each set Λ˜ in Aσ(T ) is such that Λ˜− = ϕ(Λ)− for some set Λ ∈ AΩ . The following commutative diagram illustrates the proof of part (b). H ⏐ ⏐ U'



T

−−−→ 

L2 (Ωγ , μγ ) ⏐ ⏐ V' − L2 (Ωγ , μγ ) ⏐ ⏐ W'

γ∈Γ

γ∈Γ

 γ∈Γ

L2 (Ω, μ)

H ⏐ ⏐ 'U

 γ∈Γ

Mϕ

−−−→

L2 (Ωγ , μγ ) ⏐ ⏐ 'V L2 (Ωγ , μγ ) ⏐ ⏐ 'W

−

L2 (Ω, μ).

Definition 3.12. A vector x in H is a cyclic vector for an operator T in B[H] if H is the smallest invariant subspace for T that contains x. If T has a cyclic vector, then it is called a cyclic operator . A vector x in H is a star-cyclic vector for an operator T in B[H] if H is the smallest reducing subspace for T that contains x. If T has a star-cyclic vector, then it is called a star-cyclic operator .  T in B[H] if and only   an

operator Therefore a vector x inH is cyclic for n {T x} = {p(T )x ∈ H : p is a if n {T n x} = H, where n {T n x} = n polynomial}− (i.e., x ∈ H is cyclic for T ∈ B[H] if and only if {p(T )x ∈ H : p is a polynomial}− = H, which means {Sx ∈ H : S ∈ P(T )}− = H, where P(T ) is the algebra of all polynomials in T with complex coefficients). Similarly, x ∈ H is star-cyclic for T ∈ B[H] if and only if {Sx ∈ H : S ∈ C ∗ (T )}− = H, where C ∗ (T ) is the C*-algebra generated by T (i.e., the smallest C*-algebra of operators from B[H] containing T and the identity I — see, e.g., [30, Proposition IX.3.2] ). Recall: C ∗ (T ) = P(T, T ∗ )−, the closure in B[H] of the set P(T, T ∗ ) of all polynomials in two (noncommuting) variables T and T ∗ (in any order) with complex coefficients (see, e.g., [6, p. 1] —C*-algebra will be reviewed in Section   m ∗n x} ∪ {T ∗n T m x} . 4.1), and {Sx ∈ H : S ∈ P(T, T ∗ )}− = m,n {T T

74

3. The Spectral Theorem

Remark. Reducing subspaces are invariant. So a cyclic vector is star-cyclic, and a cyclic operator is star-cyclic. A normed space is separable if and only the e.g., [78, Proposition 4.9]). Hence if it is spanned by a countable set (see,   m ∗n x}∪{T ∗n T m x} is sepasubspace {Sx ∈ H: S ∈ P(T, T ∗ )}−= m,n{T T rable, and so is {Sx ∈ H: S ∈ C ∗ (T )}−= {Sx ∈ H: S ∈ P(T, T ∗ )− }−. Thus if T is star-cyclic, then H is separable (see also [30, Proposition IX.3.3]). Therefore T ∈ B[H] is cyclic =⇒ T ∈ B[H] is star-cyclic =⇒ H is separable.     n ∗m m ∗n If T is normal , then x} x} ∪ {T ∗n T m x} = m,n{T T m,n{T T is the smallest subspace of H that contains x and reduces  T . Thus x ∈ H is a star-cyclic vector for a normal operator T if and only if m,n {T n T ∗m x} = H (cf. proof of Theorem 3.11). An important result from [20] reads as follows. If T is a normal operator, then T is cyclic whenever it is star-cyclic. Hence T ∈ B[H] is normal and cyclic ⇐⇒ T ∈ B[H] is normal and star-cyclic. It can also be shown along this line that x is a cyclic vector for a normal operator T if and only if it is a cyclic vector for T ∗ [58, Problem 164]. Consider the proof of Theorem 3.11. Part (a) deals with the case where a normal operator T is star-cyclic, while part (b) deals with the case where T is not star-cyclic. In the former case H must be separable by the above remark. As we show next if H is separable, then the measure in Theorem 3.11 can be taken to be finite. Indeed, the positive measure μ : AΩ → R constructed in part (a) is finite. However, the positive measure μ : AΩ → R constructed in part (b) may fail to be even σ-finite (see, e.g., [58, p. 66]). But finiteness can be restored by separability. Corollary 3.13. If T is a normal operator on a separable Hilbert space H, then there is a finite positive measure μ on a σ-algebra AΩ of subsets of a nonempty set Ω and a function ϕ in L∞ (Ω, μ) such that T is unitarily equivalent to the multiplication operator Mϕ on L2 (Ω, μ). Proof. Consider the argument in part (b) of the proof of Theorem 3.11. Since the collection {Mγ } is made up of orthogonal subspaces, we may take one unit vector from each subspace by the Axiom of Choice to get an orthonormal set {eγ } (of unit vectors), which is in one-to-one correspondence with the set {Mγ }. Hence {Mγ } and {eγ } have the same cardinality. Since {eγ } is an orthonormal set, its cardinality is not greater than the cardinality of any orthonormal basis for H, and so the cardinality of {Mγ } does not surpass the (Hilbert) dimension of H. If H is separable, then {Mγ } is countable. In this case, for notational convenience replace the index γ in the index set Γ with an integer index n in a countable index set, say N , throughout the proof of part (b). Since the orthogonal collection {Mn } of subspaces is now countable, it follows that the family of sets {Ωn }, where each Ωn = σ(T |Mn ), is countable. Recall: each positive measure μn : AΩn → R inherited from part (a) is finite. Take

3.4 Spectral Theorem: The General Case

75

 the positive measure

μ on AΩ defined in part (b). Thus μ(Λ) = j∈J μj (Λj ) for every set Λ = j∈J Λj in AΩ Ωj ) where J is

(each Λj in AΩj and so Λj ⊆ and hence μ(Ω) = N Ω . But now Ω = any subset of n N n∈ n∈N μn (Ωn ) =  N ) = μ (Ω ) < ∞ for each n ∈ . we are iden(Again, n n n n∈N μ(Ωn ), where μ(Ω

tifying each Ωn with m∈N Ωm when Ωm = ∅ for every m = n.) Conclusion: if H is separable, then μ is σ-finite. However, we can actually get a finite measure. Indeed, construct the measure μby multiplying each original finite measure μn by a positive scalar so that n∈N μn (Ωn ) 0 (since Ωγ is the support of each positive measure 3.11 μγ on AΩγ ). Thus μ(Λ) > 0 (cf. definitions of μγ and μ — proof of Theorem $ part (b)). However, since χΛ ∈ L2 (Ω, μ) and since E (Δ)f ; f  = χΔ |f |2 dμ for each Δ ∈ AΩ and every f ∈ L2 (Ω, μ) (cf. definition of E  in part (b)), % %  2 2  E (Λ)χΛ  = E (Λ)χΛ ; χΛ  = χΛ |χΛ | dμ = dμ = μ(Λ) > 0. Λ

˜ = E(Λ) ˆ ˜ = O.  ˜ = E(ϕ ˆ −1 (Λ)) = E (Λ) = 0 and so E(Λ) Thus E(Λ) $ $ ˆλ- = λ dEλ in the above proof are also The representations T = ϕ(λ --) dE called spectral decompositions of the normal operator T (see Section 3.2).

3.5 Fuglede Theorems and Reducing Subspaces Let Aσ(T ) be the σ-algebra of Borel subsets of the spectrum σ(T ) of T . The following lemma sets up the essential features for proving the next theorem. $ Lemma 3.16. Let T = λ dEλ be the spectral decomposition of a normal operator T ∈ B[H] as in Theorem 3.15. If Λ ∈ Aσ(T ) , then (a) E(Λ) T = T E(Λ). Equivalently, R(E(Λ)) reduces T . (b) Moreover, if Λ = ∅, then σ(T |R(E(Λ)) ) ⊆ Λ− . $ Proof. Let T = λ dEλ be the spectral decomposition of T . Take Λ˜ ∈ Aσ(T ) . $ $ ˜ = χ dEλ and T = λ dEλ , we get by Lemma 3.8 (a) Since E(Λ) ˜ Λ % % ˜ ˜ E(Λ) T = λ χΛ˜ (λ) dEλ = χΛ˜ (λ) λ dEλ = T E(Λ), ˜ reduces T (by Proposition 1.I). which is equivalent to saying that R(E(Λ)) (b) Let AΩ be the σ-algebra of Borel subsets of Ω as in Theorem 3.11. Take an arbitrary nonempty Λ ∈ AΩ . Recall (cf. proof of Theorem 3.15): % % %  ˆ =T ∼ --) dE M = ϕ(λ --) dEλ λ dEλ = ϕ(λ = ϕ λ ---, ˆ : AΩ → B[H] are given by E (Λ) = M χ where E  : AΩ → B[L2 (Ω, μ)] and E Λ ∗  ˆ and E(Λ) = U E (Λ) U for the same unitary transformation U : H → L2 (Ω, μ) ˆ and E  (Λ) reduce T and Mϕ as in (a), such that T = U ∗Mϕ U. So, since E(Λ) ∼ T |R(E(Λ)) = Mϕ |R(M χΛ ) . ˆ ˆ (Λ)) = U ∗ R(E (Λ) U ), Indeed, since R(E = (U ∗Mϕ U )|U ∗ R(E (Λ)U ) = U ∗Mϕ |R(E (Λ)) U = U ∗Mϕ |R(M χΛ ) U. T |R(E(Λ)) ˆ

80

3. The Spectral Theorem

Let μ|Λ : AΛ → C be the restriction of μ : AΩ → C to the sub-σ-algebra AΛ = AΩ ∩ ℘(Λ) so that R(M χΛ ) is unitarily equivalent to L2 (Λ, μ|Λ ). Explicitly,   ∼ L2 (Λ, μ|Λ ). R(M χΛ ) = f ∈ L2 (Ω, μ) : f = χΛ g for some g ∈ L2 (Ω, μ) = Consider the following notation: set Mϕ χΛ = Mϕ |L2 (Λ,μ|Λ ) in B[L2 (Λ, μ|Λ )] (not in B[L2 (Ω, μ)]). Since R(E (Λ)) = R(M χΛ ) reduces Mϕ , Mϕ |R(M χ ) ∼ = Mϕ χ . Λ

Λ

∼ Hence T |R(E(Λ)) = Mϕ χΛ by transitivity, and therefore (cf. Proposition 2.B) ˆ σ(T |R(E(Λ)) ) = σ(Mϕ χΛ ). ˆ Since Λ = ∅, we get by Proposition 3.B (for ϕ : Ω → C such that ϕ(λ --) = λ) !  ϕ(Δ)− ∈ C : Δ ∈ AΛ and μ|Λ (Λ\Δ) = 0 σ(Mϕ χΛ ) ⊆ !  ϕ(Δ ∩ Λ)− ∈ C : Δ ∈ AΩ and μ(Λ\Δ) = 0 ⊆ ϕ(Λ)− . = Take ∅ = Λ˜ ∈ Aσ(T ) . Let E : Aσ(T ) → B[H] be the spectral measure for which ˜ = E(ϕ ˆ −1 (Λ)) ˜ if Λ˜ ∈ Aϕ(Ω) (cf. proof of Theorem 3.15 part (c) — also see E(Λ) the remark after Theorem 3.11). Since Λ˜− = ϕ(Λ)− for some Λ ∈ AΩ , we get ) = σ(Mϕ χ ) ⊆ ϕ(Λ)− = Λ˜−.  σ(T | ˜ ) = σ(T |R(E(ϕ(Λ))) ) = σ(T | ˆ R(E(Λ))

R(E(Λ))

Λ

The next proof uses the same argument used in the proof of Corollary 3.5. $ Theorem 3.17. (Fuglede Theorem). Let T = λ dEλ be the spectral decomposition of a normal operator in B[H]. If S ∈ B[H] commutes with T, then S commutes with E(Λ) for every Λ ∈ Aσ(T ) . Proof. Take any Λ in Aσ(T ) . Suppose ∅ = Λ = σ(T ); otherwise the result is trivially verified with E(Λ) = O or E(Λ) = I. Set Λ = σ(T )\Λ in Aσ(T ) . Let Δ and Δ be arbitrary nonempty measurable closed subsets of Λ and Λ . That is, ∅ = Δ ⊆ Λ and ∅ = Δ ⊆ Λ lie in Aσ(T ) and are closed in C , and so they are closed relative to the compact set σ(T ). By Lemma 3.16(a) we get E(Λ)T |R(E(Λ)) = T E(Λ)|R(E(Λ)) = T |R(E(Λ)) : R(E(Λ)) → R(E(Λ)) for every Λ ∈ Aσ(T ) . This holds in particular for Δ and Δ . Take any operator S on H such that S T = T S. Thus, on R(E(Δ)), (∗) (E(Δ )SE(Δ))T |R(E(Δ)) = (E(Δ )SE(Δ))(T E(Δ)) = E(Δ )S T E(Δ) = E(Δ ) T SE(Δ) = (T E(Δ ))(E(Δ )SE(Δ)) = T |R(E(Δ )) (E(Δ )SE(Δ)). Since the sets Δ and Δ are closed and nonempty, Lemma 3.16(b) says σ(T |R(E(Δ)) ) ⊆ Δ

and

σ(T |R(E(Δ ) ) ⊆ Δ ,

and hence, as Δ ⊆ Λ and Δ ⊆ Λ = σ(T )\Λ, (∗∗)

σ(T |R(E(Δ)) ) ∩ σ(T |R(E(Δ )) ) = ∅.

3.5 Fuglede Theorems and Reducing Subspaces

81

Therefore, according to Proposition 2.T, (∗) and (∗∗) imply E(Δ )SE(Δ) = O. This identity holds for all Δ ∈ C Λ and all Δ ∈ C Λ , where   C Λ = Δ ∈ Aσ(T ) : Δ is a closed subset of Λ . Now, according to Proposition 3.C (in the inclusion ordering), R(E(Λ)) = sup R(E(Δ)) Δ∈ C Λ

R(E(Λ )) = sup R(E(Δ ))

and

Δ  ∈ C Λ

and so (in the extension ordering — see, e.g., [78, Problem 1.17] ), E(Λ) = sup E(Δ) Δ∈ C Λ

Then

and

E(Λ ) = sup E(Δ ). Δ  ∈ C Λ

E(Λ )SE(Λ) = O.

Since E(Λ) + E(Λ ) = E(Λ ∪ Λ ) = I, this leads to SE(Λ) = E(Λ ∪ Λ )SE(Λ) = (E(Λ) + E(Λ ))SE(Λ) = E(Λ)SE(Λ). Thus R(E(Λ)) is S-invariant (cf. Proposition 1.I). Since this holds for every Λ in Aσ(T ) , it holds for Λ = σ(T )\Λ so that R(E(Λ )) is S-invariant. But R(E(Λ)) ⊥ R(E(Λ )) (i.e., E(Λ)E(Λ ) = O) and H = R(E(Λ)) + R(E(Λ )) (since H = R(E(Λ ∪ Λ )) = R(E(Λ) + E(Λ )) ⊆ R(E(Λ)) + R(E(Λ )) ⊆ H). Hence R(E(Λ )) = R(E(Λ))⊥, and so R(E(Λ))⊥ is S-invariant as well. Then R(E(Λ)) reduces S, which, by Proposition 1.I, is equivalent to SE(Λ) = E(Λ)S.



A subspace is a closed linear manifold of a normed space. Invariant subspaces were discussed in Section 1.1 (M is T -invariant if T (M) ⊆ M). A subspace M of X is nontrivial if {0} = M = X . Reducing subspaces (i.e., subspaces M for which both M and M⊥ are T -invariant or, equivalently, subspaces M that are invariant for both T and T ∗ ) and reducible operators (i.e., operators that have a nontrivial reducing subspace) were discussed in Section 1.5. Hyperinvariant subspaces were defined in Section 1.9 (subspaces that are invariant for every operator commuting with a given operator). An operator is nonscalar if it is not a (complex) multiple of the identity; otherwise it is a scalar operator (Section 1.7). Scalar operators are normal. A projection E is nontrivial if O = E = I. Equivalently, a projection is nontrivial if and only if it is nonscalar . (Indeed, if E 2 = E, then it follows at once that O = E = I if and only if E = αI for every α ∈ C .) Corollary 3.18. Consider operators acting on a complex Hilbert space of dimension greater than 1.

82

3. The Spectral Theorem

(a) Every normal operator has a nontrivial reducing subspace. (b) Every nonscalar normal operator has a nontrivial hyperinvariant subspace which reduces every operator that commutes with it. (c) An operator is reducible if and only if it commutes with a nonscalar normal operator . (d) An operator is reducible if and only if it commutes with a nontrivial orthogonal projection. (e) An operator is reducible if and only if it commutes with a nonscalar operator that also commutes with its adjoint. $ Proof. Take Λ ∈ Aσ(T ) . Consider the spectral decomposition T = λ dEλ of a normal operator T as in Theorem 3.15. By Theorem 3.17, if S T = T S, then SE(Λ) = E(Λ)S or, equivalently, each subspace R(E(Λ)) reduces S, which means {R(E(Λ))} is a family of reducing subspaces for every operator that commutes with T . If σ(T ) has just one point, say σ(T ) = {λ}, then T = λI (by uniqueness of the spectral measure). Thus T is a scalar operator and so every subspace of H reduces T . If dim H > 1, then in this case the scalar T has many nontrivial reducing subspaces. So if T is nonscalar, then σ(T ) has more than one point (and dim H > 1). If λ, λ0 lie in σ(T ) and λ = λ0 , then let   D λ = B 1 |λ−λ0 | (λ) = ζ ∈ C : |ζ − λ| < 12 |λ − λ0 | 2 be the open disk of radius 12 |λ − λ0 | centered at λ. Set Dλ = σ(T ) ∩ D λ

and

Dλ = σ(T )\D λ .

The pair of sets {Dλ , Dλ } is a measurable partition of σ(T ) — both Dλ and Dλ lie in Aσ(T ) . Since Dλ and σ(T )\D − λ are nonempty relatively open subsets  of σ(T ) and σ(T )\D − ⊆ D , it follows by Theorem 3.15 that E(Dλ ) = O and λ λ E(Dλ ) = O. Then I = E(σ(T )) = E(Dλ ∪ Dλ ) = E(Dλ ) + E(Dλ ), and so E(Dλ ) = I − E(Dλ ) = I. Therefore O = E(Dλ ) = I. Equivalently, R(E(Dλ )) is nontrivial: {0} = R(E(Dλ )) = H. This proves (a) (for the particular case of S = T ), and also proves (b) — the nontrivial subspace R(E(Dλ )) reduces every S that commutes with T . Thus according to (b), if T commutes with a nonscalar normal operator, then it is reducible. The converse follows by Proposition 1.I, since every nontrivial orthogonal projection is a nonscalar normal operator. This proves (c). Since a projection P is nontrivial if and only if {0} = R(P ) = H, an operator T is reducible if and only if it commutes with a nontrivial orthogonal projection (by

3.5 Fuglede Theorems and Reducing Subspaces

83

Proposition 1.I), which proves (d). Since every nontrivial orthogonal projection is self-adjoint and nonscalar, it follows by (d) that if T is reducible, then it commutes with a nonscalar operator that also commutes with its adjoint. Conversely, suppose an operator T is such that L T = T L and L T ∗ = T ∗ L (and so L∗ T = T L∗ ) for some nonscalar operator L. Then (L + L∗ ) T = T (L + L∗ ) where L + L∗ is self-adjoint (thus normal). If this self-adjoint L + L∗ is not scalar, then T commutes with the nonscalar normal L + L∗ . On the other hand, if this self-adjoint L + L∗ is scalar, then the nonscalar L must be normal. Indeed, if L + L∗ = αI, then L∗ L = L L∗ = αL − L2 and therefore L is normal. Again in this case, T commutes with the nonscalar normal L. Thus in any case, T commutes with a nonscalar normal operator and so T is reducible according to (c). This completes the proof of (e).  The equivalent assertions in Corollary 3.5 which hold for compact normal operators remain equivalent for the general case (for plain normal — not necessarily compact — operators). Again the implication (a) ⇒ (c) in Corollary 3.19 below is the Fuglede Theorem (Theorem 3.17). $ Corollary 3.19. Let T = λ dEλ be a normal operator in B[H] and take any operator S in B[H]. The following assertions are pairwise equivalent. (a) S commutes with T. (b) S commutes with T ∗. (c) S commutes with every E(Λ). (d) R(E(Λ)) reduces S for every Λ. $ Proof. Let T = λ dEλ be the spectral decomposition of a normal operator in B[H]. Take any Λ ∈ Aσ(T ) and an arbitrary S ∈ B[H]. First we show: ST = TS

⇐⇒

SE(Λ) = E(Λ)S

⇐⇒

S T ∗ = T ∗S.

If S T = T S, then SE(Λ) = E(Λ)S by Theorem 3.17. So, for every x, y ∈ H, πx,S ∗ y (Λ) = E(Λ)x ; S ∗ y = SE(Λ)x ; y = E(Λ)Sx ; y = πSx,y (Λ). Hence % ∗ ∗ ∗ ST x ; y = T x ; S y = λ dEλ x ; S ∗ y % % = λ dSEλ x ; y = λ dEλ Sx ; y = T ∗Sx ; y for every x, y ∈ H (cf. Lemmas 3.7 and 3.8(a)). Thus ST = TS

=⇒

SE(Λ) = E(Λ)S

=⇒

S T ∗ = T ∗S

⇐⇒

S∗T = T S∗.

Since these hold for every S ∈ B[H], and since S ∗∗ = S, we get S∗T = T S∗

=⇒

S T = T S.

Therefore (a), (b), and (c) are equivalent, and (d) is equivalent to (c) by Proposition 1.I (as in the final part of the proof of Theorem 3.17). 

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3. The Spectral Theorem

Corollary 3.20. (Fuglede–Putnam Theorem). Suppose T1 in B[H] and T2 in B[K] are normal operators. If X ∈ B[H, K] intertwines T1 to T2 , then X intertwines T1∗ to T2∗ (i.e., if X T1 = T2 X, then X T1∗ = T2∗ X). Proof. Take T1 ∈ B[H], T2 ∈ B[K], and X ∈ B[H, K]. Consider the operators ( ) ( ) T1 O O O T = T1 ⊕ T2 = and S= X O O T2 in B[H ⊕ K]. If T1 and T2 are normal, then T is normal. If X T1 = T2 X, then  S T = T S and so S T ∗ = T ∗S by Corollary 3.19. Hence X T1∗ = T2∗ X.

3.6 Supplementary Propositions Let Γ be an arbitrary (not necessarily countable) nonempty index set. We have defined a diagonalizable operator in Section 3.2 as follows. (1) An operator T on a Hilbert space H is diagonalizable if there is a resolution of the identity {Eγ }γ∈Γ on H and a bounded family of scalars {λγ }γ∈Γ such that T u = λγ u whenever u ∈ R(Eγ ). As we saw in Section 3.2, this can be equivalently stated as follows. (1 ) An operator T on a Hilbert space H is diagonalizable if it is a weighted sum of projections for some bounded family of scalars {λγ }γ∈Γ and some resolution of the identity {Eγ }γ∈Γ on H. Thus every diagonalizable operator is normal by Corollary 1.9, and what the Spectral Theorem for compact normal operators (Theorem 3.3) says is precisely the converse for compact normal operators: a compact operator is normal if and only if it is diagonalizable — cf. Corollary 3.5(a) — and so (being normal) a diagonalizable operator is such that r(T ) = T  = supγ∈Γ |λγ | (cf. Lemma 3.1). However, a diagonalizable operator on a separable Hilbert space was defined in Section 2.7 (Proposition 2.L), whose natural extension to arbitrary Hilbert space (not necessarily separable) in terms of summable families (see Section 1.3), rather than summable sequences, reads as follows. (2) An operator T on a Hilbert space H is diagonalizable if there is an orthonormal basis  {eγ }γ∈Γ for H and a bounded family {λγ }γ∈Γ of scalars such that T x = γ∈Γ λγ x ; eγ eγ for every x ∈ H. There is no ambiguity here. Both definitions coincide: the resolution of the identity {Eγ }γ∈Γ behind the statement in (2) is given by Eγ x = x ; eγ eγ for every  x ∈ H and every γ ∈ Γ according to the Fourier series expansion x = γ∈Γ x ; eγ eγ of each x ∈ H in terms of the orthonormal basis {eγ }γ∈Γ . Now consider the Banach space Γ∞ made up of all bounded families a = {αγ }γ∈Γ of complex numbers and also the Hilbert space Γ2 consisting of all square-summable families z = {ζγ }γ∈Γ of complex numbers indexed by Γ . The definition of a diagonal operator in B[Γ2 ] goes as follows.

3.6 Supplementary Propositions

85

(3) A diagonal operator in B[Γ2 ] is a mapping D : Γ2 → Γ2 such that Dz = {αγ ζγ }γ∈Γ for every z = {ζγ }γ∈Γ ∈ Γ2 , where a = {αγ }γ∈Γ ∈ Γ∞ is a (bounded) family of scalars. (Notation: D = diag{αγ }γ∈Γ .) A diagonal operator D is clearly diagonalizable by the resolution of the identity Eγ z = z ; fγ fγ = ζγ fγ given by the canonical orthonormal basis {fγ }γ∈Γ for Γ2 , viz., fγ = {δγ,β }β∈Γ with δγ,β = 1 if β = γ and δγ,β = 0 if β = γ. So D is normal with r(D) = D = a∞ = supγ∈Γ |αγ |. Consider the Fourier series expansion of an arbitrary  x in H in terms of any orthonormal basis {eγ }γ∈Γ for H, namely, x = γ∈Γ x ; eγ eγ . Thus we can identify x in H with the square-summable family {x ; eγ }γ∈Γ in Γ2 (i.e., x is the image of between the unitarily equivalent {x ; eγ }γ∈Γ under a unitary transformation  spaces H and Γ2 ). Therefore if T x = γ∈Γ λγ x ; eγ eγ , then we can identify T x with the square-summable family {λγ x ; eγ }γ∈Γ in Γ2 , and so we can identify T with a diagonal operator D that takes z = {x ; eγ }γ∈Γ into Dz = {λγ x ; eγ }γ∈Γ . This means: T and D are unitarily equivalent — there is a unitary transformation U ∈ B[H, Γ2 ] for which U T = D U. It is in this sense that T is said to be a diagonal operator with respect to the orthonormal basis {eγ }γ∈Γ . Thus definition (2) can be rephrased as follows. (2 ) An operator T on a Hilbert space H is diagonalizable if it is a diagonal operator with respect to some orthonormal basis for H. Proposition 3.A. Let T be an operator on a nonzero complex Hilbert space. The following assertions are equivalent. (a) T is diagonalizable in the sense of definition (1) above. (b) H has an orthogonal basis made up of eigenvectors of T . (c) T is diagonalizable in the sense of definition (2) above. (d) T is unitarily equivalent to a diagonal operator . Take a measure space (Ω, AΩ , μ) where AΩ is a σ-algebra of subsets of a nonempty set Ω and μ is a positive measure on AΩ . Let L∞ (Ω, C , μ) — short notation: L∞ (Ω, μ), L∞ (μ), or just L∞ — denote the Banach space of all essentially bounded AΩ -measurable complex-valued functions f : Ω → C on Ω equipped with the usual sup-norm (i.e., f ∞ = ess sup |f | for f ∈ L∞ ). Let L2 (Ω, C , μ) — short notation: L2 (Ω, μ), L2 (μ), or just L2 — denote the Hilbert space of all square-integrable AΩ -measurable complex-valued $ functions 1 |f |2 dμ 2 f : Ω → C on Ω equipped with the usual L2 -norm (i.e., f 2 = for f ∈ L2 ). Multiplication operators on L2 will be considered next (see, e.g., [58, Chapter 7] ). Multiplication operators are prototypes of normal operators according to the first version of the Spectral Theorem (Theorem 3.11). Proposition 3.B. Let φ : Ω → C and f : Ω → C be complex-valued functions on a nonempty set Ω. Let μ be a positive measure on a σ-algebra AΩ of

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3. The Spectral Theorem

subsets of Ω. Suppose φ ∈ L∞ . If f ∈ L2 , then φf ∈ L2 . Thus consider the mapping Mφ : L2 → L2 defined by Mφ f = φ f

for every

f ∈ L2 .

That is, (Mφ f )(λ) = φ(λ)f (λ) μ-a.e. for λ ∈ Ω. This is called the multiplication operator on L2, which is linear and bounded (i.e., Mφ ∈ B[L2 ] ). Moreover : ∗

(a) Mφ = Mφ

∗

(i.e., (Mφ f )(λ) = φ(λ) f (λ) μ-a.e. for λ ∈ Ω).

(b) Mφ is a normal operator . (c) Mφ  ≤ φ∞ = ess sup |φ| = inf Λ∈NΩ supλ∈ Ω\Λ |φ(λ)|, where NΩ denotes the collection of all Λ ∈ AΩ such that μ(Λ) = 0. (d) σ(Mφ ) ⊆ ess R(φ) = essential range of φ   = α ∈ C : μ({λ ∈ Ω : |φ(λ) − α| < ε}) > 0 for every ε > 0   = α ∈ C : μ(φ−1 (Vα )) > 0 for every Borel neighborhood Vα of α   = φ(Λ)− ∈ C : Λ ∈ AΩ and μ(Ω\Λ) = 0 ⊆ φ(Ω)− = R(φ)−. If in addition μ is σ-finite, then (e) Mφ  = φ∞ , (f) σ(Mφ ) = ess R(φ). Particular case: Suppose (i) Ω is a nonempty bounded subset of C and (ii) ϕ : Ω → Ω = R(ϕ) ⊆ C is the identity map on Ω (i.e., ϕ(λ) = λ for every λ ∈ Ω). So ess R(ϕ) = ess Ω = {Λ− ∈ C : Λ ∈ AΩ and μ(Ω\Λ) = 0} ⊆ Ω −. Let support (μ) = {Λ ∈ AΩ : Λ is closed in Ω and μ(Ω\Λ) = 0} be the support of μ. As Λ− ∩ Ω is closed in Ω, support (μ) = ess R(ϕ) ∩ Ω = ess Ω ∩ Ω. This is the smallest relatively closed subset of Ω with complement of measure zero (i.e., Ω\support (μ) is the largest relatively open subset of Ω of measure zero). If ∅ = Λ ∈ AΩ is relatively open and if Λ ⊆ support (μ), then μ(Λ) > 0. (Indeed, if ∅ = Λ ⊆ support(μ) is open in Ω and if μ(Λ) = 0, then the set [Ω\support (μ)] ∪ Λ is a relatively open subset of Ω larger than Ω\support (μ) and of measure zero, which is a contradiction.) Moreover, if in addition Ω is closed in C (i.e., if Ω = ∅ is compact), then ess R(ϕ) = ess Ω ⊆ Ω and hence support(μ) = ess R(ϕ) = ess Ω is compact. Therefore if Ω is nonempty and compact and if μ is σ-finite, then σ(Mϕ ) = ess R(ϕ) = ess Ω = support(μ). If Ω = N , the set of all positive integers, if AΩ = ℘ (N ), the power set of N , and if μ is the counting measure (assigning to each finite set the number of elements in it, which is σ-finite), then a multiplication operator is reduced to 2 , where the bounded ℘(N )-measa diagonal operator on L2 (N , C , μ) = N2 = + ∞ is the bounded sequence urable function φ : N → C in L∞ (N , C , μ) = N∞ = + {αk } with αk = φ(k) for each k ∈ N , so that Mφ  = φ∞ = supk |αk | and

3.6 Supplementary Propositions

87

σ(Mφ ) = R(φ)− = {αk }−. What the Spectral Theorem says is: a normal operator is unitarily equivalent to a multiplication operator (Theorem 3.11), which turns out to be a diagonal operator in the compact case (Theorem 3.3). Proposition 3.C. Let Ω be a nonempty compact subset of the complex plane C and let E : AΩ → B[H] be a spectral measure (cf. Definition 3.6). For each

Λ ∈ AΩ set C Λ = {Δ ∈ AΩ : Δ is a closed subset of Λ}. Then R(E(Λ)) is the smallest subspace of H such that Δ∈ C Λ R(E(Δ)) ⊆ R(E(Λ)). Proposition 3.D. Let H be a complex Hilbert space and let T be the unit circle about the origin of the complex plane. If T ∈ B[H] is normal, then (a) T is unitary if and only if σ(T ) ⊆ T , (b) T is self-adjoint if and only if σ(T ) ⊆ R , (c) T is nonnegative if and only if σ(T ) ⊆ [0, ∞), (d) T is strictly positive if and only if σ(T ) ⊆ [α, ∞) for some α > 0, (e) T is an orthogonal projection if and only if σ(T ) ⊆ {0, 1}.       If T = 10 00 and S = 21 11 in B[C 2 ], then S 2 − T 2 = 34 32 . Therefore O ≤ T ≤ S does not imply T 2 ≤ S 2 . However, the converse holds. Proposition 3.E. If T, S ∈ B[H] are nonnegative operators, then 1

1

T ≤ S implies T 2 ≤ S 2 . $ Proposition 3.F. If T = λ dEλ is the spectral decomposition of a nonneg$ 1 1 ative operator T on a complex Hilbert space, then T 2 = λ 2 dEλ . $ Proposition 3.G. Let T = λ dEλ be the spectral decomposition of a normal operator T ∈ B[H]. Take an arbitrary λ ∈ σ(T ). (a) N (λI − T ) = R(E({λ})). (b) λ ∈ σP (T ) if and only if E({λ}) = O. (c) If λ is an isolated point of σ(T ), then λ ∈ σP (T ). (Every isolated point of the spectrum of a normal operator is an eigenvalue.) (d) If H is separable, then σP (T ) is countable. Recall: X ∈ B[H, K] intertwines T ∈ B[H] to S ∈ B[K] if X T = SX. If an invertible X intertwines T to S, then T and S are similar . If, in addition, X is unitary, then T and S are unitarily equivalent (cf. Section 1.9). Proposition 3.H. Let T1 ∈ B[H] and T2 ∈ B[K] be normal operators. If X ∈ B[H, K] intertwines T1 to T2 (i.e., X T1 = T2 X), then

88

3. The Spectral Theorem

(a) N (X) reduces T1 and R(X)− reduces T2 so that T1 |N (X)⊥ ∈ B[N (X)⊥ ] and T2 |R(X)− ∈ B[R(X)− ]. Moreover , (b) T1 |N (X)⊥ and T2 |R(X)− are unitarily equivalent. A special case of Proposition 3.H: if a quasiinvertible (i.e., injective with dense range) bounded linear transformation intertwines two normal operators, then they are unitarily equivalent. This happens in particular if X is invertible. Proposition 3.I. Two similar normal operators are unitarily equivalent. A subset of a metric space is nowhere dense if its closure has empty interior. Thus the spectrum σ(T ) of T ∈ B[H] is nowhere dense if and only if its interior σ(T )◦ is empty. The full spectrum σ(T )# was defined in Section 2.7: the union of the spectrum and its holes. So σ(T ) = σ(T )# if and only if ρ(T ) is connected. An operator T ∈ B[H] is reductive if all its invariant subspaces are reducing (Section 1.5). In particular, a normal operator is reductive if and only if its restriction to every invariant subspace is normal (Proposition 1.Q). Normal reductive operators are also called completely normal . Proposition 3.J. Suppose T ∈ B[H] is a normal operator. If σ(T ) = σ(T )# and σ(T )◦ = ∅, then T is reductive. If T is reductive, then σ(T )◦ = ∅. There are normal reductive operators for which the spectrum is different from the full spectrum. Example: the unitary diagonal U in Proposition 3.L(j) below is reductive (so σ(U )◦ = ∅) with σ(U ) = T . On the other hand, there are normal nonreductive operators with the same property. Example: a bilateral shift S is a nonreductive unitary operator with σ(S) = T (so σ(S)◦ = ∅). In fact, S has an invariant subspace M for which the restriction S|M is a unilateral shift (see, e.g., [74, Proposition 2.13] ), which is not normal, and so M does not reduce S by Proposition 1.Q (see Proposition 3.L(b,c) below). Proposition 3.K. Let T be a diagonalizable operator on a nonzero complex Hilbert space. The following assertions are equivalent. (a) Every nontrivial invariant subspace for T contains an eigenvector . (b) Every nontrivial invariant subspace M for T is  spanned by the eigenvectors of T |M (i.e., if M is T -invariant, then M = λ∈σP (T |M ) N (λI − T |M ) ). (c) T is reductive. There are diagonalizable operators which are not reductive. In fact, since a subset of C is a separable metric space, it includes a countable dense subset. Let Λ be any countable dense subset of an arbitrary nonempty compact subset Ω of C . Let {λk } be an enumeration of Λ so that supk |λk | < ∞ (since Ω is 2 ]. Then bounded). Consider the diagonal operator D = diag{λk } in B[+ σ(D) = Λ− = Ω

3.6 Supplementary Propositions

89

according to Proposition 2.L. Thus every closed and bounded subset of the 2 . Hence if Ω ◦ = ∅, complex plane is the spectrum of a diagonal operator on + then the diagonal D is not reductive by Proposition 3.J.   Let μ and η be measures on a σ-algebra AΩ . If η(Λ) = 0 ⇒ μ(Λ) = 0 for Λ ∈ AΩ , then μ is absolutely   continuous with respect to η (notation: μ  η). If η({λ}) = 0 ⇒ μ({λ}) = 0 for every singleton {λ} ∈ AΩ , then μ is continuous with respect to η. If there is a measurable partition {Λ, Δ} of Ω with η(Λ) = μ(Δ) = 0, then η and μ are singular (notation: η ⊥ μ or μ ⊥ η). If μ ⊥ η and if the set Λ = {λk } ∈ AΩ in the above partition is countable (union of measurable singletons), then μ is discrete with respect to η. If μ and η are σ-finite, then μ = μa + μs = μa + μsc + μsd , where μa, μs, μsc, and μsd are absolutely continuous, singular, singular-continuous, and singular-discrete (i.e., discrete) with respect to a reference measure η. This is the Lebesgue Decomposition Theorem (see, e.g., [81, Chapter 7] ). Take the σ-algebra AT of Borel sets in the unit circle T and the normalized Lebesgue measure η on AT (η(T ) = 1). A unitary operator is absolutely continuous, continuous, singular, singular-continuous, or singular-discrete (i.e., discrete) if its scalar spectral measure is absolutely continuous, continuous, singular, singular-continuous, or singular-discrete with respect to η. By the Lebesgue Decomposition Theorem and by the Spectral Theorem, every unitary operator U on a Hilbert space H is uniquely decomposed as the direct sum U = Ua ⊕ Us = Ua ⊕ Usc ⊕ Usd of an absolutely continuous unitary Ua on Ha , a singular unitary Us on Hs , a singular-continuous unitary Usc on Hsd , and a discrete unitary Usd on Hsd , with H decomposed as H = Ha ⊕ Hs = Ha ⊕ Hsc ⊕ Hsd (orthogonal direct sums). Proposition 3.L. Let U ∈ B[H] be a unitary operator on a Hilbert space H (thus σ(U ) ⊆ T and so σ(U )◦ = ∅). Let ϕ be the identity map on T . (a) Direct sums of bilateral shifts are bilateral shifts. (b) Every bilateral shift S ∈ B[H] has an invariant subspace M for which the restriction S|M ∈ B[M] is a unilateral shift. (c) A bilateral shift is a unitary operator which is not reductive. (d) U = S ⊕ W where S is a bilateral shift and W is a reductive unitary operator. (e) U is reductive if and only if it has no bilateral shift S as a direct summand . (f) If σ(U ) = T , then U is reductive. (g) A unitary operator is absolutely continuous if and only if it is a direct summand of a bilateral shift (or a bilateral shift itself ). (h) A singular unitary operator is reductive.

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3. The Spectral Theorem

(i) U is continuous if and only if σP (U ) = ∅. If U is discrete, then σP (U ) = ∅. (j) If {λk }k∈N is an enumeration of Q ∩ [0, 1), then D = diag({e2πiλk }) on 2+ is a reductive singular-discrete unitary diagonal operator with σ(D) = T . (k) If Mϕ is the bilateral shift on L2 (T,η) shifting the orthonormal basis {λk}k∈Z and Υ ∈ AT with 0 < η(Υ) < 1, then L2 (Υ,η) reduces Mϕ and U = Mϕ |L2(Υ,η) is an absolutely continuous unitary with σ(U) = Υ = T, thus reductive. () The multiplication operator Mϕ on L2 (T, μ) is a singular-continuous unitary if μ is the Borel–Stieltjes measure on AT generated by the Cantor function on the Cantor set Γ ∈ AT with σ(Mϕ ) = Γ = T, thus reductive. Notes: Diagonalizable operators are considered in Proposition 3.A (see, e.g., [78, Problem 6.25]). Multiplication operators are considered in Proposition 3.B (see, e.g., [30, Example IX.2.6], [32, Proposition I.2.6], [58, Chapter 7], and [98, p. 13]). The technical result in Proposition 3.C is needed for proving the Fuglede Theorem — see [98, Proposition 1.4]. The next propositions depend on the Spectral Theorem. Proposition 3.D considers the converse of Proposition 2.A (see, e.g., [78, Problem 6.26]). For Proposition 3.E see, for instance, [75, Problem 8.12]. Proposition 3.F is a consequence of Lemma 3.10 and Proposition 1.M (see, e.g., [78, Problem 6.45]). Proposition 3.G is a classical result — see [104, Theorem 12.29 and Exercise 18(b) p. 325)]; also [98, Proposition 1.2], [78, Problem 6.28]. Propositions 3.H and 3.I follow from Corollary 3.20 (see, e.g., [30, Propositions IX.6.10 and IX.6.11]). Reductive normal operators are considered in Propositions 3.J and 3.K (see, e.g., [39, Theorems 13.2 and 13.33] and [98, Theorems 1.23 and 1.25]). Proposition 3.L is a collection of results on decomposition of unitary operators. For items (a) and (b) see [74, Propositions 2.11 and 2.13]; item (b) and Proposition 1.Q imply (c); for (d) see [48, 1.VI p. 18]; item (e) follows from (d) — see also [98, Proposition 1.11] and [39, Theorem 13.14]; Propositions 2.A and 3.J lead to item (f) — see also [98, Theorem 1.23] and [39, Theorem 13.2]; for item (g) see [48, Exercise 6.8 p. 56]; items (e) and (g) imply (h) — see also [39, Theorem 13.15]; item (i) is a consequence of Proposition 3.G; for item (j) see [39, Example 13.5]; item (k) follows from [98, Theorem 3.6] and (g); for item () see Proposition 3.B, [81, Problem 7.15(c)], and remark after Lemma 4.7 (next section).

Suggested Readings Arveson [6, 7] Bachman and Narici [9] Berberian [15] Conway [30, 32] Douglas [38] Dowson [39] Dunford and Schwartz [46]

Halmos [54, 58] Helmberg [63] Kubrusly [78] Radjavi and Rosenthal [98] Reed and Simon [99] Rudin [104] Weidmann [113]

4 Functional Calculi

Take a fixed operator T in the operator algebra B[X ]. Suppose another operator ψ(T ) in B[X ] can be associated to each function ψ : σ(T ) → C in a suitable function algebra F (σ(T )). This chapter explores the map ΦT : F (σ(T )) → B[X ] defined by ΦT (ψ) = ψ(T ) for every ψ in F (σ(T )) — a map ψ → ψ(T ) taking each function ψ in a function algebra F (σ(T )) to the operator ψ(T ) in the operator algebra B[X ]. If ΦT is linear and preserves the product operation (i.e., if it is an algebra homomorphism), then it is referred to as a functional calculus for T (also called operational calculus). Sections 4.1 and 4.2 deal with the case where X is a Hilbert space and T is a normal operator. Sections 4.3 and 4.4 deal with the case where X is a Banach space ψ is an analytic function.

4.1 Rudiments of Banach and C*-Algebras Let A and B be algebras over the same scalar field F . If F = C , then these are referred to as complex algebras. A linear transformation Φ : A → B (of the linear space A into the linear space B) that preserves products (i.e., such that Φ(x y) = Φ(x) Φ(y) for every x, y in A) is a homomorphism (or an algebra homomorphism) of A into B. A unital algebra (or an algebra with identity) is an algebra with an identity element (i.e., with a neutral element under multiplication). An element x in a unital algebra A is invertible if there is an x−1 ∈ A such that x−1 x = x x−1 = 1, where 1 ∈ A denotes the identity in A. A unital homomorphism between unital algebras is one that takes the identity of A to the identity of B. If Φ is an isomorphism (of the linear space A onto the linear space B) and also a homomorphism (of the algebra A onto the algebra B), then it is an algebra isomorphism of A onto B. In this case A and B are said to be isomorphic algebras. A normed algebra is an algebra A which is also a normed space whose norm satisfies the operator norm property, viz., x y ≤ x y for every x, y in A. The identity element of a unital normed algebra has norm 1 ∈ F . A Banach algebra is a complete normed algebra. The spectrum σ(x) of an element x ∈ A in a unital complex Banach algebra A is the complement of the set ρ(x) = {λ ∈ C : λ 1 − x has an inverse in A}, and its spectral radius is the number r(x) = supλ∈σ(x) |λ|. An involution ∗ : A → A on an algebra A is a map x → x∗ such that (x∗ )∗ = x, (x y)∗ = y ∗ x∗, and © Springer Nature Switzerland AG 2020 C. S. Kubrusly, Spectral Theory of Bounded Linear Operators, https://doi.org/10.1007/978-3-030-33149-8_4

91

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(αx + β y)∗ = αx∗ + β y ∗ for all scalars α, β and every x, y in A. A ∗-algebra (or an involutive algebra) is an algebra equipped with an involution. If A and B are ∗-algebras, then a ∗-homomorphism (a ∗-isomorphism) between them is an algebra homomorphism (an algebra isomorphism) Φ : A → B that preserves involution (i.e., Φ(x∗ ) = Φ(x)∗ in B for every x in A — we use the same notation for involutions on A and B). An element x in a ∗-algebra is Hermitian if x∗ = x and normal if x∗x = x x∗ . An element x in a unital ∗-algebra is unitary if x∗x = x x∗ = 1 ∈ A. A C*-algebra is a Banach ∗-algebra A for which x∗x = x2 for every x ∈ A. In a ∗-algebra (x∗ )∗ is usually denoted by x∗∗. The origin 0 in a ∗-algebra and the identity 1 in a unital ∗-algebra are Hermitian. Indeed, since (αx)∗ = αx∗ for every x and every scalar α, we get 0∗ = 0 by setting α = 0. Since 1∗ x = (1∗ x)∗∗ = (x∗ 1)∗ = (1x∗ )∗ = x 1∗ and (x∗ 1)∗ = x∗∗ = x, we also get 1∗ x = x 1∗ = x, and so (by uniqueness of the identity) 1∗ = 1. In a unital C*-algebra the expression 1 = 1 is a theorem (rather than an axiom) since 12 = 1∗ 1 = 12  = 1, hence 1 = 1 (because 1 = 0 as 1 = 0 in A). Elementary properties of Banach algebras, C*-algebras, and algebra homomorphisms needed in the sequel are brought together in the next lemma. Lemma 4.1. Let A and B be algebras over the same scalar field and let Φ : A → B be a homomorphism. Take an arbitrary x ∈ A. (a) If A is a complex ∗-algebra, then x = a + ib with a∗ = a and b∗ = b in A. (b) If A, B, and Φ are unital and x is invertible, then Φ(x)−1 = Φ(x−1 ). (c) If A is unital and normed, and x is invertible, then x−1 ≤ x−1 . (d) If A is a unital ∗-algebra and x is invertible, then (x−1 )∗ = (x∗ )−1 . (e) If A is a C*-algebra, then x∗  = x (so x∗x = x2 = x∗ 2 = xx∗ ). (f) If A is a unital C*-algebra and x∗ x = 1, then x = 1. Suppose A and B are unital complex Banach algebras. (g) If Φ is a unital homomorphism, then σ(Φ(x)) ⊆ σ(x). (h) If Φ is an injective unital homomorphism, then σ(Φ(x)) = σ(x). 1

(i) r(x) = limn xn  n . Suppose A and B are unital complex C*-algebras. (j) If x∗ = x, then r(x) = x. (k) If Φ is a unital ∗-homomorphism, then Φ(x) ≤ x. () If Φ is an injective unital ∗-homomorphism, then Φ(x) = x. Proof. Let Φ : A → B be a homomorphism and take an arbitrary x ∈ A.

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(a) This is the Cartesian decomposition (as in Proposition 1.O). If A is a ∗-algebra, then set a = 21 (x∗ + x) ∈ A and b = 2i (x∗ − x) ∈ A. Thus a∗ = 1 −i 1 ∗ ∗ ∗ ∗ ∗ 2 (x + x ) = a, b = 2 (x − x ) = b, and a + i b = 2 (x + x − x + x) = x. (b) If A and B are unital algebras, Φ is a unital homomorphism, and x is invertible, then 1 = Φ(1) = Φ(x−1 x) = Φ(x x−1 ) = Φ(x−1 ) Φ(x) = Φ(x) Φ(x−1 ). (c) If A is a unital normed algebra, then 1 = 1 = x−1 x ≤ x−1 x whenever x is invertible. (d) If A is a unital ∗-algebra and if x−1 x = x x−1 = 1, then x∗ (x−1 )∗ = (x−1 x)∗ = (x x−1 )∗ = (x−1 )∗ x∗ = 1∗ = 1, and so (x∗ )−1 = (x−1 )∗ . (e) Let A be a C*-algebra. The result is trivial for x = 0 because 0∗ = 0. Since x2 = x∗ x ≤ x∗ x, it follows that x ≤ x∗  if x = 0. Replacing x with x∗ and since x∗∗ = x we get x∗  ≤ x. (f) Particular case of (e) since 1 = 1. (g) Let A and B be unital complex Banach algebras, and let Φ be a unital homomorphism. If λ ∈ ρ(x), then λ 1 − x is invertible in A. Since Φ is unital, Φ(λ 1 − x) = λ 1 − Φ(x) is invertible in B according to (b). Hence λ ∈ ρ(Φ(x)). Thus ρ(x) ⊆ ρ(Φ(x)). Therefore C \ρ(Φ(x)) ⊆ C \ρ(x). (h) If Φ : A → B is injective, then it has an inverse Φ−1 : R(Φ) → B on its range R(Φ) = Φ(A) ⊆ B which, being the image of a unital algebra under a unital homomorphism, is again a unital algebra. Moreover, Φ−1 itself is a unital homomorphism. Indeed, if y = Φ(u) and z = Φ(v) are arbitrary elements in R(Φ), then Φ−1 (y z) = Φ−1 (Φ(u) Φ(v)) = Φ−1 (Φ(u v)) = u v = Φ−1 (y) Φ−1 (z) and so Φ−1 is a homomorphism (since inverses of linear transformations are linear) which is trivially unital (since Φ is unital). Now apply (b) so that Φ−1 (y) is invertible in A whenever y is invertible in R(Φ) ⊆ B. If λ ∈ ρ(Φ(x)), then y = λ1 − Φ(x) = Φ(λ1 − x) is invertible in R(Φ) and hence Φ−1 (y) = Φ−1 (Φ(λ1 − x)) = λ1 − x is invertible in A. Thus λ ∈ ρ(x). Therefore ρ(Φ(x)) ⊆ ρ(x). This implies ρ(x) = ρ(Φ(x)) according to (g). (i) The proof of the Gelfand–Beurling formula follows similarly to the proof of the particular case in the complex Banach algebra B[X ] as in Theorem 2.10. (j) If A is a C*-algebra and if x∗ = x, then x2 = x∗ x = x2  and so, by induction, x2n = x2n  for every integer n ≥ 1. If A is unital and complex, 1 1 1 then r(x) = limn xn  n = limn x2n  2n = limn (x2n ) 2n = x by (i). (k) Let A and B be complex unital C*-algebras and Φ a ∗-homomorphism. Then Φ(x)2 = Φ(x)∗ Φ(x) = Φ(x∗) Φ(x) = Φ(x∗x) for every x ∈ A. Since x∗x is Hermitian, Φ(x∗x) is Hermitian because Φ is a ∗-homomorphism. Hence Φ(x∗x) = r(Φ(x∗x)) ≤ r(x∗x) = x∗x = x2 according to (j) and (g). () Consider the setup and the argument of the preceding proof and suppose in addition that the unital ∗-homomorphism Φ : A → B is injective. Thus apply (h) instead of (g) to get () instead of (k). 

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Remark. A unital ∗-homomorphism Φ between unital complex C*-algebras is a contraction by Lemma 4.1(k), and so it is a contractive unital ∗-homomorphism. If it is injective, then Φ is an isometry by Lemma 4.1(), and so an isometric unital ∗-homomorphism. Since an isometry of a Banach space into a normed space has a closed range (see, e.g., [78, Problem 4.41(d)]), if a unital ∗-homomorphism Φ between unital complex C*-algebras A and B is injective, then the unital ∗-algebra R(Φ) = Φ(A) ⊆ B, being closed in the Banach space B, is itself a Banach space, and so R(Φ) is a unital complex C*algebra. Thus if Φ : A → B between unital complex C*-algebras is an injective unital ∗-homomorphism, then Φ : A → R(Φ) is an isometric isomorphism (i.e., an injective and surjective linear isometry), and therefore A and R(Φ) are isometrically isomorphic unital complex C*-algebras. These are the elementary results on Banach and C*-algebras required in this chapter. For a thorough treatment of Banach algebra see, e.g., [11], [26], [38], [104], and [106], and for C*-algebra see, e.g., [6], [30], [34], [49], and [96]. Throughout this and the next section H will stand for a nonzero complex Hilbert space and so B[H] is a (unital complex) C*-algebra, where involution is the adjoint operation. Let C Ω denote the collection of all functions from a nonempty set Ω to the complex plane C , and let F (Ω) ⊆ C Ω be an algebra of complex-valued functions on Ω where addition, scalar multiplication, and product are pointwise defined as usual. Take the map ∗ : F (Ω) → F (Ω) assigning to each function ψ in F (Ω) the function ψ ∗ in F (Ω) given by ψ ∗ (λ) = ψ(λ) for λ ∈ Ω. This (i.e., complex conjugation) defines an involution on F (Ω), and so F (Ω) is a commutative ∗-algebra. If in addition F (Ω) is endowed with a norm that makes it complete (thus a Banach algebra) and if ψ ∗ ψ = ψ2, then F (Ω) is a (complex) C*-algebra. (For instance, if F (Ω) is the algebra of all complex-valued bounded functions on Ω equipped with the sup-norm  · ∞ , since (ψ ∗ ψ)(λ) = ψ ∗ (λ)ψ(λ) = ψ(λ)ψ(λ) = |ψ(λ)|2 for every λ ∈ Ω). Take any operator T in B[H]. A first attempt towards a functional calculus for T alludes to polynomials n in T . To each polynomial p : σ(T ) → C with complex coefficients, p(λ) = i=0 αi λi for λ ∈ σ(T ), associate the operator n p(T ) = αi T i . i=0



Let P (σ(T )) be the unital algebra of all polynomials (in one variable with complex coefficients) on σ(T ). The map ΦT : P (σ(T )) → B[H] given by ΦT (p) = p(T )

for each

p ∈ P (σ(T ))

m is a unital homomorphism from P (σ(T )) to B[H]. In fact, if q(λ) = j=0 βj λj , n  m n m then (p q)(λ) = i=0 j=0 αi βj λi+j so that ΦT (p q) = i=0 j=0 αi βj T i+j = ΦT (p) ΦT (q), and also ΦT (1) = 1(T ) = T 0 = I. This can be extended from finite power series (i.e., n from polynomials) to infinite power series: if a sequence {pn } with pn (λ) = k=0 αk λk for each n converges in some sense on σ(T ) to a

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∞ ∞ limit ψ, denoted by ψ(λ) = k=0 αk λk , then we may set ψ(T ) = k=0 αk T k 1 if the series converges in some topology of B[H]. For instance, if αk = k! for each k ≥ 0, then {pn } converges pointwise to the exponential function ψ such that ψ(λ) = eλ for every λ ∈ σ(T ) (on any σ(T ) ⊆ C ), and so we define ∞ 1 k eT = ψ(T ) = k! T k=0

(where the series converges uniformly in B[H]). However, unlike in the preceding paragraph, complex conjugation does not define an involution on P (σ(T )) (and so P (σ(T )) is not a ∗-algebra under it). Indeed, p∗ (λ) = p(λ) is not a polynomial in λ — the continuous function p( · ) : σ(T ) → C given by p(λ) = n i α λ is not a polynomial in λ (for σ(T ) ⊆ R , even if σ(T )∗ = σ(T )). i=0 i The extension of the mapping ΦT to some larger algebras of functions is the focus of this chapter.

4.2 Functional Calculus for Normal Operators Let Aσ(T ) be the σ-algebra of Borel subsets of the spectrum σ(T ) of an operator T on a Hilbert space H. Let B(σ(T )) be the Banach space of all bounded Aσ(T ) -measurable complex-valued functions ψ : σ(T ) → C on σ(T ) equipped with the sup-norm. Let L∞ (σ(T ), μ) be the Banach space of all essentially bounded (i.e., μ-essentially bounded) Aσ(T ) -measurable complex-valued functions ψ : σ(T ) → C on σ(T ) for any positive measure μ on Aσ(T ) also equipped with the sup-norm  · ∞ (i.e., μ-essential sup-norm). In both cases we have unital commutative C*-algebras, where involution is defined as complex conjugation, and the identity element is the characteristic function 1 = χσ(T ) of σ(T ). (Note: we will also deal with the identity map on σ(T ) given by ϕ(λ) = λχσ(T ) .) If T is normal, then let E : Aσ(T ) → B[H] be the unique spectral mea$ sure of its spectral decomposition T = λ dEλ as in Theorem 3.15. Theorem 4.2. Let$T ∈ B[H] be a normal operator and consider its spectral decomposition T = λ dEλ . For every function ψ ∈ B(σ(T )) there is a unique operator ψ(T ) ∈ B[H] given by % ψ(T ) = ψ(λ) dEλ , which is normal. Moreover, the mapping ΦT : B(σ(T )) → B[H] such that ΦT (ψ) = ψ(T ) is a unital ∗-homomorphism. Proof. The operator ψ(T ) is unique in B[H] and normal for every ψ ∈ B(σ(T )) by Lemmas 3.7 and 3.9. The mapping ΦT : B(σ(T )) → B[H] is a unital ∗-homomorphism by Lemma 3.8 and by the linearity of the integral. Indeed, $ (a) ΦT (α ψ + β φ) = (α ψ + β φ)(T ) = (α ψ(λ) + β φ(λ)) dEλ = α ψ(T ) + β φ(T ) = α ΦT (ψ) + β ΦT (φ), $ (b) ΦT (ψφ) = (ψφ)(T ) = ψ(λ) φ(λ) dEλ = ψ(T ) φ(T ) = ΦT (ψ) ΦT (φ),

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$ (c) ΦT (1) = 1(T ) = dEλ = I, $ (d) ΦT (ψ ∗ ) = ψ ∗ (T ) = ψ(λ) dEλ = ψ(T )∗ = ΦT (ψ)∗ , for every ψ, φ ∈ B(σ(T )) and every α, β ∈ C , where the constant function 1 (i.e., 1(λ) = 1 for every λ ∈ σ(T )) is the identity in the algebra B(σ(T )).  Let P (σ(T )) be the set of all polynomials p(· , ·) : σ(T ) × σ(T ) → C in λ and λ (or p(· , ·) : σ(T ) → C such that λ → p(λ, λ)). Let C(σ(T )) be the set of all complex-valued continuous (thus Aσ(T ) -measurable) functions on σ(T ). So P (σ(T )) ⊂ P (σ(T )) ⊂ C(σ(T )) ⊂ B(σ(T )), where the last inclusion follows from the Weierstrass Theorem since σ(T ) is compact — cf. proof of Theorem 2.2. Except for P (σ(T )), these are all unital ∗-subalgebras of B(σ(T )) and the restriction of the mapping ΦT to any of them remains a unital ∗-homomorphism into B[H]. Thus Theorem 4.2 holds for all these unital ∗-subalgebras of the unital C*-algebra B(σ(T )). The algebra P (σ(T )) is a unital subalgebra of P (σ(T )), and the restriction of ΦT to it is a unital homomorphism into B[H], and so Theorem 4.2 also holds for P (σ(T )) by replacing ∗-homomorphism with nplain homomorphism. So if T is normal and p lies in P (σ(T )), say p(λ) = i=0 αi λi, then by Theorem 4.2 % % n n p(T ) = p(λ) dEλ = αi λi dEλ = αi T i, i=0

n,m

i=0

and for p in P (σ(T )), say p(λ, λ) = i,j=0 αi,j λi λj (cf. Lemma 3.10), we get % % n,m n,m ∗ αi,j λi λj dEλ = αi,j T i T ∗j. p(T, T ) = p(λ, λ) dEλ = i,j=0

i,j=0

The quotient space of a set modulo an equivalence relation is a partition of the set. So we may regard B(σ(T )) as consisting of equivalence classes from L∞ (σ(T ), μ) — bounded functions are essentially bounded with respect to any measure μ. On the other hand, shifting from restriction to extension, it is sensible to argue that the preceding theorem might be extended from B(σ(T )) to some L∞ (σ(T ), μ), where bounded would be replaced by essentially bounded. But essentially bounded with respect to what measure μ ? Recall from Section 3.6: a measure μ on a σ-algebra is absolutely continuous with respect to a measure ν on the same σ-algebra if ν(Λ) = 0 implies μ(Λ) = 0 for measurable sets Λ. Two measures on the same σ-algebra are equivalent if they are absolutely continuous with respect to each other, which means they have the same sets of measure zero. That is, μ and ν are equivalent if ν(Λ) = 0 if and only if μ(Λ) = 0 for every measurable set Λ. (Note: two positive equivalent measures are not necessarily both finite or both not finite.) Remark. Let E : Aσ(T ) → B[H] be the spectral measure of the spectral decomposition of a normal operator T and consider the statements of Lemmas 3.7, 3.8, and 3.9 for Ω = σ(T ). Lemma 3.7 remains true if B(σ(T )) is replaced by

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L∞ (σ(T ), μ) for every positive measure μ : Aσ(T ) → R equivalent to the spectral measure E. The proof is essentially the same as that of Lemma 3.7. In fact, the spectral integrals of functions in L∞ (σ(T ), μ) coincide whenever the functions are equal μ-almost everywhere, and so a function in B(σ(T )) has the same integral as any representative from its equivalence class in L∞ (σ(T ), μ). Moreover, since μ and E have exactly the same sets of measure zero, we get φ∞ = ess sup |φ| =

inf

sup

Λ∈Nσ(T ) λ∈ σ(T )\Λ

|φ(λ)|

for every φ ∈ L∞ (σ(T ), μ), where Nσ(T ) denotes the collection of all Λ ∈ Aσ(T ) for which μ(Λ) = 0 or, equivalently, for which E(Λ) = O. Thus the sup-norm φ∞ of φ : σ(T ) → C with respect to μ dominates the sup-norm with respect to πx,x for every x ∈ H. Then the proof of Lemma 3.7 remains unchanged if measure μ equivalent to B(σ(T )) is replaced by L∞ (σ(T ), μ) for any positive $ E. (Reason: the inequality |f (x, x)| ≤ φ∞ dπx,x = φ∞ x2 for x ∈ H still holds for the sesquilinear form f : H × H → C in the proof of Lemma 3.7 if φ ∈ L∞ (σ(T ), μ).) Hence for$ each φ ∈ L∞ (σ(T ), μ) there is a unique operator F ∈ B[H] such that F = φ(λ) dEλ as in Lemma 3.7. This ensures that Lemmas 3.8 and 3.9 also hold if B(σ(T )) is replaced by L∞ (σ(T ), μ). Theorem 4.3. Let $T ∈ B[H] be a normal operator and consider its spectral decomposition T = λ dEλ . If μ is a positive measure on Aσ(T ) equivalent to the spectral measure E : Aσ(T ) → B[H], then for every ψ ∈ L∞ (σ(T ), μ) there is a unique operator ψ(T ) ∈ B[H] given by % ψ(T ) = ψ(λ) dEλ , which is normal. Moreover, the mapping ΦT : L∞ (σ(T ), μ) → B[H] such that ΦT (ψ) = ψ(T ) is a unital ∗-homomorphism. Proof. If μ : Aσ(T ) → R and E : Aσ(T ) → B[H] are equivalent measures, then the proof of Theorem 4.2 still holds according to the above remark.  Sometimes it is convenient that a positive measure μ on Aσ(T ) equivalent to the spectral measure E be finite. For example, if ψ lies L∞ (σ(T ), μ) and μ is $ in 2 2 2 finite, then ψ also lies in L (σ(T ), μ) since ψ2 = |ψ| dμ ≤ ψ2∞ μ(σ(T )) < ∞. This is important if an essentially constant function is expected to be in L2 (σ(T ), μ) as will be the case in the proof of Lemma 4.7 below. Definition 4.4. A scalar spectral measure for a normal operator is a positive finite measure equivalent to the spectral measure of its spectral decomposition. ˆ is its unique spectral measure on AΩ as in Let T be a normal operator. If E the proof of Theorem 3.15 part (b), then a positive finite measure μ on AΩ is a ˆ = O}. scalar spectral measure for T if, for any Λ ∈ AΩ , {μ(Λ) = 0 ⇐⇒ E(Λ) This also applies to the unique spectral measure E on Aσ(T ) as in the proof of ˆ ˜ ˜ σ(T ) of Theorem 3.15 part (c). Take the collections {E(Λ)} Λ∈AΩ and {E(Λ)}Λ∈A

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ˆ and E. For each x, y ∈ H consider the scalar-valued (complex) all images of E measure of Section 3.3 acting on AΩ or on Aσ(T ), namely, ˆ π ˆx,y (Λ) = E(Λ)x ; y for Λ ∈ AΩ

and

˜ = E(Λ)x ˜ ; y for Λ˜ ∈ Aσ(T ) . πx,y (Λ)

Definition 4.5. A separating vector for a collection {Cγ } of operators in B[H] is a vector e ∈ H such that e ∈ N (C) for every O = C ∈ {Cγ }. Lemma 4.6. If T ∈ B[H] is normal, then e ∈ H is a separating vector for ˆ ˆe,e : AΩ → R is a scalar spectral measure for T . {E(Λ)} Λ∈AΩ if and only if π ˆ : AΩ → B[H] be the spectral measure for a normal operator T in Proof. Let E 2 ˆ for every Λ ∈ AΩ and every x ∈ H, the meaB[H]. Since π ˆx,x (Λ) = E(Λ)x ˆx,x : AΩ → R is positive and finite for every x ∈ H. Recall: e ∈ H is a sepsure π ˆ ˆ arating vector for the collection {E(Λ)} Λ∈AΩ if and only if E(Λ)e = 0 whenˆ ˆ ˆ ever E(Λ) = O or, equivalently, if and only if {E(Λ)e = 0 ⇐⇒ E(Λ) = O}. ˆ Conclusion: e ∈ H is a separating vector for the collection {E(Λ)}Λ∈AΩ if and ˆ ˆ and π = O}, which is to say if and only if E ˆe,e only if {ˆ πe,e (Λ) = 0 ⇐⇒ E(Λ) ˆe,e is positive and finite, then by Definition are equivalent measures. Since π  ˆe,e is a scalar spectral measure for T . 4.4 this equivalence means π Lemma 4.7. If T is a normal operator on a separable Hilbert space H, then ˆ there is a separating vector e ∈ H for {E(Λ)} Λ∈AΩ . (So there is a vector e in H such that π ˆe,e : AΩ → R is a scalar spectral measure for T by Lemma 4.6). Proof. Take the positive measure μ on AΩ of Theorem 3.11. If H is separable, then μ is finite (Corollary 3.13). Let Mϕ be the multiplication operator on L2 (Ω, μ) with ϕ ∈ L∞ (Ω, μ) being the identity map with multiplicity. Let χΛ ∈ L∞ (Ω, μ) be the characteristic function of each Λ ∈ AΩ . Set E (Λ) = MχΛ for every Λ ∈ AΩ so that E  : AΩ → B[L2 (Ω, μ)] is the spectral measure in L2 (Ω, μ) for the spectral decomposition (cf. proof of Theorem 3.15 part (a)) % Mϕ = ϕ(λ --) dEλof the normal operator Mϕ ∈ B[L2 (Ω, μ)]. Hence, for each f ∈ L2 (Ω, μ), % %   πf,f (Λ) = E (Λ)f ; f  = M χΛ f ; f  = χΛ f f dμ = |f |2 dμ Λ

for every Λ ∈ AΩ . In particular, the function 1 = χΩ (i.e., the constant function 1(λ) = 1 for all λ ∈ Ω — the characteristic function χΩ of Ω) lies in L2 (Ω, μ) because μ is finite. This is a separating vector for {E (Λ)}Λ∈AΩ since E (Λ)1 = M χΛ 1 = χΛ = 0 for every Λ = ∅ in AΩ . Thus π1,1 is a scalar spectral measure for Mϕ by Lemma 4.6. If T ∈ B[H] is normal, then T ∼ = Mϕ by Theorem 3.11. ˆ Let E : AΩ → B[H] be the spectral measure in H for T which is given by ˆ E(Λ) = U ∗E (Λ) U for each Λ ∈ AΩ , where U ∈ B[H, L2 (Ω, μ)] is unitary (cf. proof of Theorem 3.15 part (b)). Set e = U ∗ 1 ∈ H (with 1 = χΩ in L2 (Ω, μ)).

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∗ ∗  ˆ ˆ This e is a separating vector for {E(Λ)} Λ∈AΩ since E(Λ)e = U E (Λ) U U 1 = ∗  ∗   ˆ ˆ U E (Λ)1 = 0 if E(Λ) = O (because E (Λ)1 = 0 with E (Λ) = U E(Λ) U = O).  Thus π ˆe,e is a scalar spectral measure for T (Lemma 4.6).

ˆe,e are scalar spectral measures on AΩ for Remarks. (a) Claim: μ = π1,1 = π both Mϕ and T . Actually, by the above proof the scalar spectral measure π1,1 for Mϕ is precisely the measure μ of Theorem 3.11. In fact, for every Λ ∈ AΩ % dμ = χΛ 2 = M χΛ 12 = E (Λ)12 = π1,1 (Λ) = E (Λ)1 ; 1 μ(Λ) = Λ 2 ∗ ˆ ˆ ˆ = U E(Λ)U U e ; U e = E(Λ)e ; e = E(Λ)e =π ˆe,e (Λ).

˜ = E(ϕ ˆ −1 (Λ)) ˜ for Λ˜ (b) Take the spectral measure E : Aσ(T ) → B[H] with E(Λ) − in Aϕ(Ω) ⊆ Aσ(T ) , where ϕ(Ω) = σ(T ) (cf. proof of Theorem 3.15 part (c) and ˜ = O, then E(Λ)e ˜ = E(ϕ ˆ −1 (Λ))e ˜ = 0: the remark after Theorem 3.11). If E(Λ) ˜ ˜ ; e ˜ Λ∈A ˜ . So π ( Λ) = E( Λ)e the same e is a separating vector for {E(Λ)} σ(T ) e,e ˜ for every Λ ∈ Aσ(T ) defines a scalar spectral measure πe,e on Aσ(T ) for T . Theorem 4.8. Let T ∈ B[H] be a normal operator on$ a separable Hilbert space H and consider its spectral decomposition T = λ dEλ . Let μ be an arbitrary scalar spectral measure for T . For every ψ ∈ L∞ (σ(T ), μ) there is a unique operator ψ(T ) ∈ B[H] given by % ψ(T ) = ψ(λ) dEλ , which is normal. Moreover, the mapping ΦT : L∞ (σ(T ), μ) → B[H] such that ΦT (ψ) = ψ(T ) is an isometric unital ∗-homomorphism. Proof. All but the fact that ΦT is an isometry is a particular case of Theorem 4.3. We show that ΦT is an isometry. By Lemma 3.8, the uniqueness in Lemma 3.7, and the uniqueness of the expression for ψ(T ), we get for every x ∈ H % % ΦT (ψ)x2 = ψ(T )∗ ψ(T )x ; x = ψ(λ) ψ(λ) dEλ x ; x = |ψ(λ)|2 dπx,x . $ Suppose ΦT (ψ) = O. Then |ψ(λ)|2 dπx,x = 0 for every x ∈ H. By Lemma 4.7 (and the above remarks), there is a separating$vector e ∈ H for {E(Λ)}Λ∈Aσ(T ) . Take the scalar spectral measure πe,e . Thus |ψ(λ)|2 dπe,e = 0 so that ψ = 0 in L∞ (σ(T ), πe,e ), which means ψ = 0 in L∞ (σ(T ), μ) for every measure μ equivalent to πe,e . In particular, this holds for every scalar spectral measure μ for T . Then {0} = N (ΦT ) ⊆ L∞ (σ(T ), μ), and hence the linear transformation ΦT : L∞ (σ(T ), μ) → B[H] is injective, and is so isometric by Lemma 4.1().  If T is normal on a separable Hilbert space H, then ΦT : L∞ (σ(T ), μ) → B[H] is an isometric unital ∗-homomorphism between C*-algebras (Theorem 4.8). Thus ΦT : L∞ (σ(T ), μ) → R(ΦT ) is an isometric unital ∗-isomorphism between the C*-algebras L∞ (σ(T ), μ) and R(ΦT ) ⊆ B[H] (see remark after Lemma 4.1), which is the von Neumann algebra generated by T (see Proposition 4.A).

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Theorem 4.9. Let T ∈ B[H] be a normal operator on $a separable Hilbert space H and consider its spectral decomposition T = λ dEλ . For every ψ ∈ C(σ(T )) there is a unique operator ψ(T ) ∈ B[H] given by % ψ(T ) = ψ(λ) dEλ , which is normal. Moreover, the mapping ΦT : C(σ(T )) → R(ΦT ) ⊆ B[H] such that ΦT (ψ) = ψ(T ) is an isometric unital ∗-isomorphism, where R(ΦT ) is the C*-algebra generated by T and the identity I in B[H]. Proof. The set C(σ(T )) is a unital ∗-subalgebra of the C*-algebra B(σ(T )), which can be viewed as a subalgebra of the C*-algebra L∞ (σ(T ), μ) for every measure μ (in terms of equivalence classes in L∞ (σ(T ), μ) of each function in B(σ(T ))). So ΦT : C(σ(T )) → B[H], the restriction of ΦT : L∞ (σ(T ), μ) → B[H] to C(σ(T )), is an isometric unital ∗-homomorphism under the same assumptions as in Theorem 4.8. Hence ΦT : C(σ(T )) → R(ΦT ) is an isometric unital ∗-isomorphism (cf. paragraph preceding the theorem statement). Thus it remains to show that R(ΦT ) is the C*-algebra generated by T and the identity I. Indeed, take R(ΦT ) = ΦT (C(σ(T ))). Since ΦT (P (σ(T ))) = P(T, T ∗ ), which is the algebra of all polynomials in T and T ∗, since P (σ(T ))− = C(σ(T )) in the sup-norm topology by the Stone–Weierstrass Theorem (cf. proof of Theorem 3.11), and since ΦT : C(σ(T )) → B[H] is an isometry (thus with a closed range), we get ΦT (C(σ(T ))) = ΦT (P (σ(T ))− ) = ΦT (P (σ(T )))− = P(T, T ∗ )− = C ∗ (T ), where C ∗ (T ) stands for the C*-algebra generated by T and I (see the paragraph following Definition 3.12, where T is now normal).  The mapping ΦT of Theorems 4.2, 4.3, 4.8, and 4.9 such that ΦT (ψ) = ψ(T ) is referred to as the functional calculus for T , and the theorems themselves are referred to as the Functional Calculus for Normal Operators. Since σ(T ) is compact, for any positive measure μ on Aσ(T ) C(σ(T )) ⊂ B(σ(T )) ⊂ L∞ (σ(T ), μ). (Again, the last inclusion is interpreted in terms of the equivalence classes in L∞ (σ(T ), μ) of each representative in B(σ(T )) as in the remark preceding Theorem 4.3.) Thus ψ lies in L∞ (σ(T ), μ) as in Theorem 4.3 or 4.8 whenever ψ satisfies the assumptions of Theorems 4.2 and 4.9, and so the next result applies to the settings of the preceding Functional Calculus Theorems. It says that ψ(T ) commutes with every operator that commutes with T . $ Corollary 4.10. Let T = λ dEλ be a normal operator in B[H]. Take any ψ in L∞ (σ(T ), μ) where μ is any positive measure on Aσ(T ) equivalent to the spectral measure E. Consider the operator ψ(T ) in B[H] of Theorem 4.3, viz., % ψ(T ) = ψ(λ) dEλ . If S in B[H] commutes with T , then S commutes with ψ(T ).

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101

Proof. If S T = T S, then by Theorem 3.17 (cf. proof of Corollary 3.19), % S ψ(T )x ; y = ψ(T )x ; S ∗ y = ψ(λ) dEλ x ; S ∗ y % % = ψ(λ) dSEλ x ; y = ψ(λ) dEλ Sx ; y = ψ(T )Sx ; y for every x, y ∈ H (see Lemma 3.7), which means S ψ(T ) = ψ(T )S.



If T is a normal operator, then the Spectral Mapping Theorem for Normal Operators in Theorem 2.8 (which deals with polynomials) does not depend on the Spectral Theorem. Moreover, for a normal operator the Spectral Mapping Theorem for Polynomials in Theorem 2.7 is a particular case of Theorem 2.8. However, extensions from polynomials to bounded or continuous functions do depend on the Spectral Theorem via Theorems 4.8 or 4.9, as we will see next. The forthcoming Theorems 4.11 and 4.12 are also referred to as Spectral Mapping Theorems for bounded or continuous functions of normal operators. Theorem 4.11. If T is a normal operator on a separable Hilbert space, if μ is a scalar spectral measure for T , and if ψ lies in L∞ (σ(T ), μ), then   σ(ψ(T )) = ess R(ψ) = ess ψ(λ) ∈ C : λ ∈ σ(T ) = ess ψ(σ(T )). Proof. If the theorem holds for a scalar spectral measure, then it holds for any scalar spectral measure. Suppose T is a normal operator on a separable Hilbert space. Let μ : AΩ → R be the positive measure of Theorem 3.11, which is a scalar spectral measure for Mϕ ∼ = T (cf. remarks following the proof of Lemma 4.7). Also let πe,e : Aσ(T ) → R be a scalar spectral measure for T (cf. remarks following the proof of Theorem 4.7 again). Take any ψ in L∞ (σ(T ), πe,e ). By Theorem 4.8, ΦT : L∞ (σ(T ), πe,e ) → B[H] is an isometric (thus injective) unital ∗-isomorphism between C*-algebras and ΦT (ψ) = ψ(T ). Then σ(ψ) = σ(ΦT (ψ)) = σ(ψ(T )) by Lemma 4.1(h). Consider the multiplication operator Mϕ on L2 (Ω, μ) as in ∼ Mϕ , L2 (Ω, μ) is separable. Since ϕ(Ω)− = σ(T ) (cf. Theorem 3.11. Since T = --) = ψ(λ) remark after Theorem 3.11), set ψ  = ψ ◦ ϕ in L∞ (Ω, μ). Since ψ  (λ for λ -- ∈ Ω, σ(ψ) = σ(ψ  ) in both algebras and ess R(ψ  ) = ess R(ψ). Since μ is a scalar spectral measure for Mϕ (cf. remarks following the proof of Lemma 4.7 once again), we may apply Theorem 4.8 to Mϕ . So using the same argument, σ(ψ  ) = σ(ψ  (Mϕ )). Claim. ψ  (Mϕ ) = Mψ . Proof. Let E  be the spectral measure for Mϕ where E (Λ) = M χΛ with χΛ being the characteristic function of Λ in AΩ as in the proof of Lemma 4.7.   given by πf,g (Λ) = E (Λ)f ; g for every Take the scalar-valued measure πf,g 2 Λ in AΩ and each f, g ∈ L (Ω, μ). Recalling the remarks following the proof of Lemma 4.7 for the fourth time, we get μ = π1,1 . Note: for every Λ in AΩ ,

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%





%



%

d πf,g = πf,g (Λ) = E (Λ)f ; g = χΛ f ; g = Λ 

f g d π1,1 .

f gdμ = Λ

Λ

 So d πf,g = f g d π1,1 (f g is the Radon–Nikod´ ym derivative of πf,g with respect $     --) dEλ- (Theorem 4.8), to π1,1 ; see proof of Lemma 3.8). Since ψ (Mϕ ) = ψ (λ % %  ψ  (Mϕ )f ; g = ψ  d Eλ- f ; g = ψ  d πf,g % %   = ψ f g d π1,1 = ψ f g dμ = Mψ f ; g

for every f, g ∈ L2 (Ω, μ). Hence ψ  (Mϕ ) = Mψ, proving the claimed identity. Therefore (since μ is finite and so σ-finite — cf. Proposition 3.B(f)) we get σ(ψ(T )) = σ(ψ) = σ(ψ  ) = σ(ψ  (Mϕ )) = σ(Mψ ) = ess R(ψ  ) = ess R(ψ).  Theorem 4.12. If T is a normal operator on a separable Hilbert space, then σ(ψ(T )) = ψ(σ(T ))

for every

ψ ∈ C(σ(T )).

Proof. Consider the setup of the previous proof with πe,e on Aσ (T ) and μ = π ˆe,e on AΩ (as in Theorem 3.11) which are scalar spectral measures for T and for Mϕ ∼ = T , respectively. Thus all that has been said so far about μ holds for every finite positive measure equivalent to it; in particular, for every scalar spectral measure for T . Recall: C(σ(T )) ⊂ B(σ(T )) ⊂ L∞ (σ(T ), ν) for every measure ν on Aσ (T ). Take an arbitrary ψ ∈ C(σ(T )). Since σ(T ) is compact in C and ψ is continuous, R(ψ) = ψ(σ(T )) is compact (see, e.g., [78, Theorem 3.64]) and so R(ψ)− = R(ψ). From Proposition 3.B we get   ess R(ψ) = α ∈ C : μ ˜(ψ −1 (Bε (α))) > 0 for all ε > 0 ⊆ R(ψ), where Bε (α) denotes the open ball of radius ε centered at α. Also, πe,e (Λ) > 0 for every nonempty Λ ∈ Aσ(T ) open relative to σ(T ) = σ(Mϕ ) (cf. Theorem 3.15). If there is an α in R(ψ)\ess R(ψ), then there is an ε > 0 such that πe,e (ψ −1 (Bε (α))) = 0. Since ψ is continuous, the inverse image ψ −1 (Bε (α)) of the open set Bε (α) in C must be open in σ(T ), and hence ψ −1 (Bε (α)) = ∅. But this is a contradiction: ∅ = ψ −1 ({α}) ⊆ ψ −1 (Bε (α)) because α ∈ R(ψ). Thus R(ψ)\ess R(ψ) = ∅, and so R(ψ) = ess R(ψ). Then, by Theorem 4.11, σ(ψ(T )) = ess R(ψ) = R(ψ) = ψ(σ(T )).



4.3 Analytic Functional Calculus: Riesz Functional Calculus The approach here is rather different from Section 4.2. However, Proposition 4.J will show that the Functional Calculus for Normal Operators of the previous section coincides with the Riesz Functional Calculus of this section when restricted to a normal operator on a Hilbert space (for analytic functions).

4.3 Analytic Functional Calculus: Riesz Functional Calculus

103

Measure theory was the means for Chapter 3 and Section 4.2 where operators necessarily act on Hilbert spaces. In contrast, the tool for the present section is complex analysis where the integrals are Riemann integrals (unlike the measure-theoretic integrals we have been dealing with so far) and operators may act on Banach spaces. For standard results of complex analysis used in this section the reader is referred, for instance, to [2], [14], [23], [29], and [103]. An arc is a continuous function from a nondegenerate closed interval of the real line into the complex plane, say α : [0, 1] → C . Let Υ be the range of an arc, Υ = R(α) = {α(t) ∈ C : t ∈ [0, 1]}, which is connected and compact (continuous image of connected sets is connected, and of compact sets is compact). Suppose Υ ⊂ C is nowhere dense to avoid space-filling curves. In this case Υ is referred to as a curve generated by an arc α. Let G = {#X : ∅ = X ∈ ℘ ([0, 1])} be the set of all cardinal numbers of nonempty subsets of the interval [0, 1]. Given an arc α : [0, 1] → Υ , set m(t) = #{s ∈ [0, 1) : α(s) = α(t)} ∈ G for each t ∈ [0, 1). Thus m(t) ∈ G is, for each t ∈ [0, 1), the cardinality of the subset of the interval [0, 1] consisting of all points s ∈ [0, 1) for which α(s) = α(t). This m(t) is the multiplicity of the point α(t) ∈ Υ . Consider the function m : [0, 1) → G defined by t → m(t), which is referred to as the multiplicity of the arc α. A pair (Υ, m) consisting of the range Υ and the multiplicity m of an arc is called a directed pair if the direction in which the arc traverses the curve Υ, according to the natural order of the interval [0, 1], is taken into account. An oriented curve generated by an arc α:[0, 1] → Υ is the directed pair (Υ, m), and the arc α itself is referred to as a parameterization of the oriented curve (Υ, m) (which is not unique for (Υ, m)). If an oriented curve (Υ, m) is such that α(0) = α(1) for some parameterization α (and so for all parameterizations), then it is called a closed curve. A parameterization α of an oriented curve (Υ, m) is injective on [0, 1) if and only if m = 1 (i.e., m(t) = 1 for all t ∈ [0, 1)). If one parameterization of (Υ, 1) is injective, then so are all of them. An oriented curve (Υ, 1) parameterized by an injective arc α on [0, 1) is a simple curve (or a Jordan curve). Thus an oriented curve consists of a directed pair (Υ, m) where Υ ⊂ C is the (nowhere dense) range of a continuous function α : [0, 1] → C called a parameterization of (Υ, m), and m is a G -valued function that assigns to each point t of [0, 1) the multiplicity of the point α(t) in Υ (i.e., each m(t) says how often α traverses the point λ = α(t) of Υ ). For notational simplicity we refer to Υ itself as an oriented curve when the multiplicity m is either clear in the context or is immaterial. In this case we simply say Υ is an oriented curve. By a partition P of the interval [0, 1] we mean a finite sequence {tj }nj=0 of points in [0, 1] such that 0 = t0 , tj−1 < tj for 1 ≤ j ≤ n, and tn = 1. The total  variation of a function α : [0, 1] → C is the supremum of the set {υ ∈ R : n υ = j=1 |α(tj ) − α(tj−1 )|} taken over all partitions P of [0, 1]. A function α : [0, 1] → C is of bounded variation if its total variation is finite. An oriented curve Υ is rectifiable if some (and so any) parameterization α of it is of bounded variation, and its length (Υ ) is the total variation of α. An arc α : [0, 1] → C

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is continuously differentiable (or smooth) if it is differentiable on the open interval (0, 1), it has one-sided derivatives at the end points 0 and 1, and its derivative α : [0, 1] → C is continuous on [0, 1]. In this case the oriented curve Υ parameterized by α is also referred to as continuously differentiable (or smooth), and as is known from advanced calculus, Υ is rectifiable with length % 1 % 1 % |α (t)|dt. |dα(t)| = (Υ ) = |dλ| = 0

0

Υ

The linear space of all continuously differentiable functions from [0, 1] to C is denoted by C 1([0, 1]). If there is a partition of [0, 1] such that each restriction α|[tj−1 ,tj ] of the continuous α is continuously differentiable (i.e., α|[tj−1 ,tj ] is in C 1([tj−1 , tj ])), then α : [0, 1] → C is a piecewise continuously differentiable arc (or piecewise smooth arc), and the curve Υ parameterized by α is an oriented piecewise continuously differentiable curve (or an oriented piecewise smooth $  n $ tj  |α[t (t)|dt. curve) which is rectifiable with length (Υ) = Υ |dλ| = j=1 tj−1 j−1 ,tj ] Every continuous function ψ : Υ → C has a Riemann–Stieltjes integral associated with a parameterization α : [0, 1] → Υ of a rectifiable curve Υ, viz., $1 $ ψ(λ) dλ = 0 ψ(α(t)) dα(t). If in addition α is smooth, then Υ % 1 % 1 % ψ(λ) dλ = ψ(α(t)) dα(t) = ψ(α(t)) α (t) dt. 0

0

Υ

Now consider a bounded function f : Υ → Y of an oriented curve Υ to a Banach space Y. Let α : [0, 1] → Υ be a parameterization of Υ and let P = {tj }nj=0 be a partition of the interval [0, 1]. The norm of a partition P is the number P  = max 1≤j≤n (tj − tj−1 ). A Riemann–Stieltjes sum for the function f with respect to α based on a given partition P is a vector n f (α(τj )) (α(tj ) − α(tj−1 )) ∈ Y Σ(f, α, P ) = j=1

with τj ∈ [tj−1 , tj ]. Suppose there is a unique vector Σ ∈ Y with the following property: for every ε > 0 there is a δε > 0 such that if P is an arbitrary partition of [0,1] with P  ≤ δε , then Σ(f, α,P ) −Σ≤ ε for all Riemann–Stieltjes sums Σ(f, α, P ) for f with respect to α based on P. If such a vector Σ exists, it is called the Riemann–Stieltjes integral of f with respect to α, denoted by % 1 % f (α(t)) dα(t) ∈ Y. Σ = f (λ) dλ = Υ

0

If the Y-valued function f : Υ → Y is continuous and the curve Υ is rectifiable, then it can be verified (exactly as in the case of complex-valued functions; see, e.g., [23, Proposition 5.3 and p. 385]) the existence of the Riemann–Stieltjes integral of f with respect to α. If α1 and α2 are parameterizations of Υ, then for every δ > 0 there are partitions P1 and P2 with norm less than δ for which curve Υ and a continΣ(f, α1 , P1 ) = Σ(f, α2 , P2 ). Given a rectifiable oriented $ uous Y-valued function f on Υ, then the integral Υ f (λ) dλ does not depend on the parameterization of Υ. We refer to it as the integral of f over Υ.

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105

Let C[Υ, Y] and B[Υ, Y] be the linear spaces of all Y-valued continuous functions and of all bounded functions on Υ equipped with the sup-norm, which are Banach spaces because Υ is compact and Y$is complete. Since Υ is : C[Υ, Y] → Y assigncompact, then C[Υ, Y] ⊆ B[Υ, Y]. The transformation Υ $ f (λ) dλ in Y is linear and ing to each function f in C[Υ, Y] its integral Υ $ continuous (i.e., linear and bounded — Υ ∈ B[ C[Υ, Y], Y ] ). Indeed, % %    f (λ) dλ ≤ sup f (λ) |dλ(t)| = f ∞ (Υ ).  λ∈Υ

Υ

Υ

If Z is a Banach space and S ∈ B[Y, Z], then (recall: Sf is continuous) % % S f (λ) dλ = Sf (λ) dλ. Υ

Υ

The reverse (or the opposite) of an oriented curve Υ is an oriented curve denoted by −Υ obtained from Υ by reversing the orientation of Υ. To reverse the orientation of Υ means to reverse the order of the domain [0, 1] of its parameterization α or, equivalently, to replace α(t) with α(−t) for t running over [−1, 0]. Reversing the orientation of a curve Υ has the effect of replacing $ $0 $1 $ f (λ) dλ = 0 f (α(t)) dα(t) with 1 f (α(t)) dα(t) = −Υ f (λ) dλ, and so Υ % % f (λ) dλ = − f (λ) dλ. −Υ

Υ

The preceding arguments and results all remain valid if the continuous function f is defined on a subset Λ of C that includes the curve Υ (by simply replacing it with its restriction to Υ, which remains continuous). In particular, if Λ is a nonempty open subset of C and Υ is a rectifiable curve included in Λ, and if f : Λ → Y is a continuous $ function on the open set Λ, then f has an integral over Υ ⊂ Λ ⊆ C , viz., Υ f (λ) dλ. A topological space is disconnected if it is the union of two disjoint nonempty subsets that are both open and closed. Otherwise the topological space is said to be connected . A subset of a topological space is called a connected set (or a connected subset) if, as a topological subspace, it is a connected topological space itself. Take any nonempty subset Λ of C . A component of Λ is a maximal connected subset of Λ, which coincides with the union of all connected subsets of Λ containing a given point of Λ. Thus any two components of Λ are disjoint. Since the closure of a connected set is connected, any component of Λ is closed relative to Λ. By a region (or a domain) we mean a nonempty connected open subset of C . Every open subset of C is uniquely expressed as a countable union of disjoint regions which are the components of it (see, e.g., [23, Proposition 3.9]). The closure of a region is sometimes called a closed region. Different regions may have the same closure (sample: a punctured open disk). Let Υ be a closed rectifiable oriented curve in C . A classical and important result in complex analysis establishes the value of the integral

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4. Functional Calculi

wΥ (ζ) =

1 2πi

% Υ

1 dλ λ−ζ

for every

ζ ∈ C \Υ.

This integral has an integer value which is constant on each component of C \Υ and zero on the unbounded component of C \Υ. The number wΥ (ζ) is

referred to as the winding number of Υ about ζ. If a closed rectifiable oriented curve Υ is a simple curve, then C \Υ has only two components, just one of them is bounded, and Υ is their common boundary (this is the Jordan Curve Theorem). In this case (i.e., for a simple closed rectifiable oriented curve Υ ) the winding number wΥ (ζ) takes on only three values for each ζ ∈ C \Υ , either 0 or ±1. If wΥ (ζ) = 1 for every ζ in the bounded component of C \Υ , then Υ is said to be positively (i.e., counterclockwise) oriented ; otherwise if wΥ (ζ) = −1, then Υ is said to be negatively (i.e., clockwise) oriented . For every ζ in the unbounded component of C \Υ we get wΥ (ζ) = 0. If Υ is positively oriented, then the reverse curve −Υ is negatively oriented, and vice versa. These notions m can be extended as follows. A finite union Υ = j=1 Υj of disjoint closed rectifiable oriented simple curves Υj is called apath, and its winding number m wΥ (ζ) about ζ ∈ C \Υ is defined by wΥ (ζ) = j=1 wΥj (ζ). A path Υ is positively oriented if for every ζ ∈ C \Υ the winding number wΥ (ζ) is either 0 or 1, and negatively oriented if for every ζ ∈ C \Υ the winding number wΥ (ζ) is

m either 0 or −1. If a path Υ = j=1 Υj is positively oriented, then the reverse

m path −Υ = j=1 −Υj is negatively oriented. The inside (notation: ins Υ ) and the outside (notation: out Υ ) of a positively oriented path Υ are the sets     and out Υ = ζ ∈ C \Υ : wΥ (ζ) = 0 . ins Υ = ζ ∈ C \Υ : wΥ (ζ) = 1

m From now on all paths are positively oriented . If Υ = j=1 Υj is a path and if there is a finite subset {Υj  }nj =1 of Υ such that Υj  ⊂ ins Υj  +1 , then these nested (disjoint closed rectifiable oriented) simple curves {Υj  } are oppositely oriented, with Υn being positively oriented because wΥ (ζ) is either 0 or 1 for every ζ ∈ C \Υ. An open subset of C is a Cauchy domain (or a Jordan domain) if it has finitely many components whose closures are pairwise disjoint, and its boundary is a path. The closure of a Jordan domain is sometimes referred to as a Jordan closed region. If Υ is a path, then {Υ, ins Υ, out Υ } is a partition of C . Since a path Υ is a closed set in C (it is a finite union of closed sets), and since it is the common boundary of ins Υ and out Υ, it follows that ins Υ and out Υ are open sets in C , and their closures are given by the union with their common boundary, ( ins Υ )− = Υ ∪ ins Υ

and

( out Υ )− = Υ ∪ out Υ.

If Υ is a path, Y is a Banach space, and f : Υ → Y is a continuous function (and so f has an integral over each closed rectifiable oriented simple curve Υj ), then the integral of the Y-valued function f over the path Υ ⊂ C is defined by % m % f (λ) dλ = f (λ) dλ. Υ

j=1

Υj

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107

Again, if f : Λ → Y is a continuous function on a nonempty open subset Λ of

C that includes a path Υ, then we define the integral of f over the path Υ as the integral of the restriction of f to Υ over Υ ⊂ Λ ⊆ C ; that is, as the

integral of f |Υ : Υ → Y (which is continuous as well) over the path Υ : % % f (λ) dλ = f |Υ (λ) dλ. Υ

Υ

Therefore if we are given a nonempty open subset Λ of C , a Banach space Y, and a continuous function f : Λ → Y, then the above identity establishes how to define the integral of f over an arbitrary path Υ included in Λ. The Cauchy Integral Formula says: if ψ : Λ → C is an analytic function on a nonempty open subset Λ of C including a path Υ and its inside ins Υ (i.e., Υ ∪ ins Υ ⊂ Λ ⊆ C ), then % ψ(λ) 1 ψ(ζ) = 2πi dλ for every ζ ∈ ins Υ. λ −ζ Υ In fact this is the most common application of the general case of the Cauchy Integral Formula (see, e.g., [23, Problem 5.O] ). What the basic assumption Υ ∪ ins Υ ⊂ Λ ⊆ C (i.e., ( ins Υ )− ⊂ Λ ⊆ C ) says is simply this: Υ ⊂ Λ ⊆ C and C \Λ ⊂ out Υ (i.e., wΥ (ζ) = 0 for every ζ ∈ C \Λ), and the assumption ζ ∈ ins Υ ⊂ Λ is equivalent to saying ζ ∈ Λ\Υ and wΥ (ζ) = 1. Since the function ψ(·) [(·) − ζ]−1 : Υ → C for ζ ∈ ins Υ is continuous on the path Υ, the above integral may be generalized to a Y-valued function by considering the following special case in a complex Banach space Y. Moreover, throughout this chapter X will also denote a nonzero complex Banach space. Set Y = B[X ], the Banach algebra of all operators on X . Take an operator T in B[X ] and let RT : ρ(T ) → G[X ] be its resolvent function, defined by RT (λ) = (λI − T )−1 for every λ in the resolvent set ρ(T ), where ρ(T ) is an open and nonempty subset of C (cf. Section 2.1). Let ψ : Λ → C be an analytic function on a nonempty open subset Λ of C for which the intersection Λ ∩ ρ(T ) is again nonempty. Recall from the proof of Theorem 2.2 that RT : ρ(T ) → G[X ] is continuous, and so is the product function ψRT : Λ ∩ ρ(T ) → B[X ] defined by (ψRT )(λ) = ψ(λ)RT (λ) ∈ B[X ] for every λ in the open subset Λ ∩ ρ(T ) of C . Then we can define the integral of ψRT over any path Υ ⊂ Λ ∩ ρ(T ), % % ψ(λ) RT (λ) dλ = ψ(λ) (λI − T )−1 dλ ∈ B[X ]. Υ

Υ

Thus, for any complex number ζ ∈ ins Υ, the integral of the C -valued function ψ(·) [(·) − ζ]−1 : Λ\{ζ} → C over any path Υ such that ( ins Υ )− ⊂ Λ is generalized, for any operator T ∈ B[X ], to the integral of the B[X ]-valued function ψ(·) [(·)I − T ]−1 : Λ ∩ ρ(T ) → B[X ] over any path Υ such that Υ ⊂ Λ ∩ ρ(T ). As we will see next, the function ψRT : Λ ∩ ρ(T ) → B[X ] is analytic, and the definition of a Banach-space-valued analytic function on a nonempty open subset of C is exactly the same as that of a scalar-valued analytic function.

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If Ω is a compact set included in an open subset Λ of C (so Ω ⊂ Λ ⊆ C ), then there exists a path Υ ⊂ Λ for which Ω ⊂ ins Υ and C \Λ ⊂ out Υ (see, e.g., [30, p. 200] ). As we have already seen, this condition is equivalent to Ω ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ. Take an operator T ∈ B[X ]. Set Ω = σ(T ), which is compact. Now take an open subset Λ of C such that σ(T ) ⊂ Λ. Let Υ be any path for which Υ ⊂ Λ and σ(T ) ⊂ ins Υ ⊂ Λ. In other words, for a given open set Λ including σ(T ) let the path Υ satisfy the following assumption: σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ. Since ρ(T ) = C \σ(T ), we get Υ ⊂ Λ ∩ ρ(T ) = ∅ by the above inclusions. Lemma 4.13. If ψ : Λ → C is an analytic function on a nonempty open subset Λ of C , then the integral % % ψ(λ) RT (λ) dλ = ψ(λ) (λI − T )−1 dλ Υ

Υ

does not depend on the choice of any path Υ that satisfies the assumption σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ. Proof. Let X be a Banach space and take an operator T in the Banach algebra B[X ]. Let RT : ρ(T ) → G[X ] be its resolvent function and let ψ : Λ → C be an analytic function on an open set Λ ⊆ C properly including σ(T ). So there is a path Υ satisfying the above assumption. Suppose Υ  and Υ  are distinct positively oriented paths satisfying the above assumption. We show that % % ψ(λ) RT (λ) dλ. ψ(λ) RT (λ) dλ = Υ

Υ 

Claim 1. The product ψRT : Λ ∩ ρ(T ) → B[X ] is analytic on Λ ∩ ρ(T ). Proof. First recall: Λ ∩ ρ(T ) = ∅. By the resolvent identity of Section 2.1, if λ and ζ are distinct points in ρ(T ), then   RT (λ) − RT (ζ) + RT (ζ)2 = RT (ζ) − RT (λ) RT (ζ) λ−ζ (cf. proof of Theorem 2.2, Claim 2). Since RT : ρ(T ) → G[X ] is continuous,  R (λ) − R (ζ)     T    T + RT (ζ)2  ≤ lim RT (ζ) − RT (λ)RT (ζ) = 0 lim  λ→ζ λ→ζ λ−ζ for every ζ ∈ ρ(T ). Thus the resolvent function RT : ρ(T ) → G[X ] is analytic on ρ(T ) and so it is analytic on the nonempty open set Λ ∩ ρ(T ). Since the function ψ : Λ → C also is analytic on Λ ∩ ρ(T ), and since the product of analytic functions on the same domain is again analytic, the product function ψRT : Λ ∩ ρ(T ) → B[X ] given by (ψRT )(λ) = ψ(λ)RT (λ) ∈ B[X ] for every λ ∈ Λ ∩ ρ(T ) is analytic on Λ ∩ ρ(T ), concluding the proof of Claim 1.

4.3 Analytic Functional Calculus: Riesz Functional Calculus

109

The Cauchy Theorem says: if ψ : Λ → C is analytic on a nonempty open subset Λ of C that includes a path Υ and its inside ins Υ , then % ψ(λ) dλ = 0 Υ

(see, e.g., [23, Problem 5.O]). This can be extended from scalar-valued functions to Banach-space-valued functions. Let Y be a complex Banach space, let Λ of C including f : Λ → Y be an analytic function on a nonempty open subset $ a path Υ and its inside ins Υ, and consider the integral Υ f (λ) dλ. Claim 2. If f : Λ → Y is analytic on a given nonempty open subset Λ of C and if Υ is any path for which ( ins Υ )− ⊂ Λ, then % f (λ) dλ = 0. Υ

Proof. Take an arbitrary nonzero ξ in Y ∗ (i.e., take a nonzero bounded linear functional ξ : Y → C ), and consider the composition ξ ◦ f : Λ → C of ξ with an analytic function f : Λ → Y, which is again analytic on Λ. Indeed,  f (λ) − f (ζ)      ξ (f (λ)) − ξ (f (ζ))   − ξ (f  (ζ)) ≤ ξ  − f  (ζ)  λ−ζ λ−ζ for every pair of distinct points λ and ζ in Λ. Since both ξ and the integral are linear and continuous, %

m %

m %

ξ f (λ) dλ = ξ f (λ) dλ = ξ f (λ) dλ j=1

Υ

=

m % j=1

Υj

j=1

%

ξ (f (λ)) dλ = Υj

Υj

ξ (f (λ)) dλ. Υ

$ By the Cauchy Theorem for scalar-valued functions we get Υ ξ (f (λ)) dλ = 0 $ (because ξ ◦ f is analytic on Λ). Therefore ξ( Υ f (λ) dλ) = 0. Since this holds for every ξ ∈ Y ∗, the Hahn–Banach $ Theorem (see, e.g., [78, Corollary 4.64] ) ensures the claimed result, viz., Υ f (λ) dλ = 0. Consider the nonempty open subset ins Υ  ∩ ins Υ  of C , which includes σ(T ). Let Δ be the (finite) union of open components of ins Υ  ∩ ins Υ  including σ(T ). Let Υ ⊆ Υ  ∪ Υ  be the path consisting of the boundary of Δ positively oriented, and so ins Υ = Δ and Δ− = ( ins Υ )− = Δ ∪ Υ . Observe that σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊆ ( ins Υ  ∩ ins Υ  )− ⊂ Λ. * = ins Υ  \Δ. If Δ * = ∅, then orient the boundThus Υ ⊂ Δ− ∩ ρ(T ) = ∅. Set Δ * ary of Δ so as to make it a positively oriented path Υ* ⊆ Υ  ∪ (−Υ ). (Note: If * = ∅, which implies Υ* = ∅.) Since ins Υ  ⊂ ins Υ , then Δ = ins Υ  so that Δ   * * ins Υ = Δ ∪ Δ with Δ ∩ Δ = ∅ and Υ and Υ are positively oriented, and so

110

4. Functional Calculi

$ $ $ is Υ* if it is not empty, then Υ  f (λ) dλ = Υ f (λ) dλ + Υ* f (λ) dλ for every B[X ]-valued continuous function f whose domain includes ( ins Υ  )−. Hence % % % ψ(λ) RT (λ) dλ. ψ(λ) RT (λ) dλ = ψ(λ) RT (λ) dλ + * Υ Υ Υ *− * − But ψRT is analytic $ on Λ ∩ ρ(T ) by Claim 1 and so, since Δ = ( ins Υ ) ⊂ Λ ∩ ρ(T ), we get Υ* ψ(λ) RT (λ) dλ = 0 by Claim 2. Therefore % % ψ(λ) RT (λ) dλ = ψ(λ) RT (λ) dλ. Υ

Υ

* as ins Υ  \Δ), Similarly (by exactly the same argument, redefining Δ % % ψ(λ) RT (λ) dλ = ψ(λ) RT (λ) dλ. Υ 



Υ

Definition 4.14. Take an arbitrary T ∈ B[X ]. If ψ : Λ → C is analytic on an open set Λ ⊆ C including the spectrum σ(T ) of T , and if Υ is any path such that σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ, then define the operator ψ(T ) in B[X ] by % % 1 1 ψ(T ) = 2πi ψ(λ) RT (λ) dλ = 2πi ψ(λ) (λI − T )−1 dλ. Υ

Υ

After defining the integral of ψRT over any path Υ such that Υ ⊂ Λ ∩ ρ(T ), its invariance for any path Υ such that σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ was ensured in Lemma 4.13. Thus the above definition of the operator ψ(T ) is clearly motivated by the Cauchy Integral Formula. Let T be an arbitrary operator in B[X ]. A complex-valued function ψ is said to be analytic on σ(T ) (or analytic on a neighborhood of σ(T )) if it is analytic on an open set including σ(T ). By a path about σ(T ) we mean a path Υ whose inside (properly) includes σ(T ) (i.e., a path for which σ(T ) ⊂ ins Υ ). So what is behind Definition 4.14 is: if ψ is analytic on σ(T ), then the integral in Definition 4.14 exists as an operator in B[X ], and the integral does not depend on the path about σ(T ) whose closure of its inside is included in the open set upon which ψ is defined (i.e., it is included in the domain of ψ). Lemma 4.15. If φ and ψ are analytic on σ(T ) and if Υ is any path about σ(T ) such that σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ, where the nonempty open set Λ ⊆ C is the intersection of the domains of φ and ψ, then % 1 φ(T ) ψ(T ) = 2πi φ(λ) ψ(λ) RT (λ) dλ, Υ

and so φ(T ) ψ(T ) = (φ ψ)(T ) since φ ψ is analytic on σ(T ). Proof. Let φ and ψ be analytic on σ(T ). Thus φ ψ is analytic on the intersection Λ of their domains. Let Υ and Υ  be arbitrary paths such that

4.3 Analytic Functional Calculus: Riesz Functional Calculus

111

σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ ins Υ  ⊆ Λ. Thus by Definition 4.14 and by the resolvent identity (cf. Section 2.1), % %



1 1 φ(T )ψ(T ) = 2πi φ(λ) RT (λ) dλ 2πi ψ(ζ) RT (ζ) dζ Υ %Υ % φ(λ) ψ(ζ) RT (λ) RT (ζ) dζ dλ = − 4π1 2  %Υ %Υ   φ(λ) ψ(ζ) (ζ − λ)−1 RT (λ) − RT (ζ) dζ dλ = − 4π1 2  %Υ %Υ φ(λ) ψ(ζ) (ζ − λ)−1 RT (λ) dζ dλ = − 4π1 2  %Υ %Υ 1 φ(λ) ψ(ζ) (ζ − λ)−1 RT (ζ) dλ dζ + 4π2 Υ Υ % %

1 φ(λ) ψ(ζ)(ζ − λ)−1 dζ RT (λ) dλ = − 4π2 Υ %Υ %

1 ψ(ζ) φ(λ)(ζ − λ)−1 dλ RT (ζ) dζ. + 4π2 Υ

Υ

$ Since λ ∈ Υ we get λ ∈ ins Υ , and hence Υ  ψ(ζ)(ζ − λ)−1 dζ = (2πi)ψ(λ) by the Cauchy Integral Formula. Moreover, since ζ ∈ Υ  we get ζ ∈ ins Υ , and so the − (·)]−1 is analytic on ins Υ  which includes Υ. Therefore $ function φ(·)[ζ −1 φ(λ)(ζ − λ) dλ = 0 by the Cauchy Theorem. Thus Υ % 1 φ(T ) ψ(T ) = 2πi φ(λ) ψ(λ) RT (λ) dλ.  

Υ

Take an arbitrary operator T ∈ B[X ]. Let A(σ(T )) denote the set of all analytic functions on the spectrum σ(T ) of T. That is, ψ ∈ A(σ(T )) if ψ : Λ → C is analytic on an open set Λ ⊆ C including σ(T ). As is readily verified, A(σ(T )) is a unital algebra where the domain of the product of two functions in A(σ(T )) is the intersection of their domains, and the identity element 1 in A(σ(T )) is the constant function 1(λ) = 1 for all λ ∈ Λ. The identity map also lies in A(σ(T )) (i.e., if ϕ : Λ → Λ ⊆ C is such that ϕ(λ) = λ for every λ ∈ Λ, then ϕ is in A(σ(T )) ). The next theorem is the main result of this section: the Analytic (or Holomorphic) Riesz (or Riesz–Dunford) Functional Calculus. Theorem 4.16. Riesz–Dunford Functional Calculus. Let X be a nonzero complex Banach space. Take an operator T in B[X ]. For every function ψ in A(σ(T )), let the operator ψ(T ) in B[X ] be defined as in Definition 4.14, viz., % 1 ψ(λ) RT (λ) dλ. ψ(T ) = 2πi Υ

The mapping ΦT : A(σ(T )) → B[X ] such that ΦT (ψ) = ψ(T ) is a unital homomorphism, which is continuous when A(σ(T )) is equipped with the sup-norm. Moreover, (a) since every polynomial p is such that p ∈ A(C ) ⊃ A(σ(T )),

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4. Functional Calculi

ΦT (p) = p(T ) =

1 2πi

%  m Υ

j=0

αj λj RT (λ) dλ =

m j=0

αj T j,

with Υ being any circle about the origin such that σ(T ) ⊂ ins Υ  ⊂ ( ins Υ )− ⊂ Λ m where Λ is an open neighborhood of σ(T ), and with p(λ) = j=0 αj λj for m λ ∈ Λ where {αj }j=0 is a finite set of complex coefficients. In particular, % 1 RT (λ) dλ = T 0 = I, ΦT (1) = 1(T ) = 2πi Υ

where 1 ∈ A(σ(T )) is the constant function, 1(λ) = 1 for every λ, and so ΦT takes the identity element 1 ∈ A(σ(T )) to the identity element I ∈ B[X ]. Also % 1 ΦT (ϕ) = ϕ(T ) = 2πi λ RT (λ) dλ = T, Υ

where ϕ ∈ A(σ(T )) is the identity map (i.e., ϕ(λ) = λ for every λ). (b) If ∞ ψ(λ) = k=0 αk λk is a power series expansion for ϕ ∈ A(σ(T )) on an open neighborhood Λ of σ(T ) with radius of convergence greater than r(T ), then %  ∞ ∞ 1 ΦT (ψ) = ψ(T ) = 2πi αk λk RT (λ) dλ = αk T k, k=0 k=0 Υ ∞ where k=0 αk T k converges in B[X ]. (c) If {ψn } is an arbitrary A(σ(T ))-valued sequence converging uniformly on compact subsets of Λ to ψ ∈ A(σ(T )), then {ΦT (ψn )} converges in B[X ] to ΦT (ψ). $ Proof. The integral Υ ( · )RT dλ : A(σ(T )) → B[X ] is a linear transformation between the linear spaces A(σ(T )) and B[X ], and so is ΦT : A(σ(T )) → B[X ]: % 1 ΦT (α φ + β ψ) = (α φ + β ψ)(T ) = 2πi (α φ + β ψ)(λ) RT (λ) dλ Υ % % 1 1 φ(λ) RT (λ) dλ + β 2πi ψ(λ) RT (λ) dλ = α ΦT (φ) + β ΦT (ψ) = α 2πi Υ

Υ

for every α, β ∈ C and every φ, ψ ∈ A(σ(T )). Moreover, by Lemma 4.15, % 1 ΦT (φ ψ) = 2πi (φ ψ)(λ) RT (λ) dλ = ΦT (φ) ΦT (ψ) Υ

for every φ, ψ ∈ A(σ(T )). Thus Φ is a homomorphism. Those integrals do not depend on the path Υ such that σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ. (a) First suppose ψ ∈ A(σ(T )) is analytic on an open neighborhood Λ of σ(T ) including a circle Υ about the origin with radius greater than the spectral radius r(T ) — e.g., if ψ is a polynomial. Since r(T ) < |λ| for λ ∈ Υ , we get by Corollary 2.12(a) % % ∞ k 1 dλ (2πi) ψ(T ) = ψ(λ) RT (λ) dλ = ψ(λ) k+1 T k=0 λ Υ Υ % % ∞ 1 ψ(λ) λ1 dλ + T k ψ(λ) λk+1 dλ, =I Υ

k=1

Υ

∞ 1 because the integral is linear and continuous, where the series k=0 λk+1 Tk converges uniformly (i.e., converges in the operator norm topology of B[X ]). Consider two special cases of elementary polynomials. First let ψ = 1. So

4.3 Analytic Functional Calculus: Riesz Functional Calculus 1 1(T ) = I 2πi

% Υ

1 λ

dλ +

∞ k=1

Tk

1 2πi

% Υ

1 λk+1

113

dλ = I + O = I

$ 1 1 dλ = 1(0) = 1 by the Cauchy Integral for ψ(λ) = 1(λ) = 1 for all$ λ, since 2πi Υ λ 1 dλ = 0. Indeed, with α(θ) = ρ eiθ for θ ∈ [0, 2π] Formula and, for k ≥ 1, Υ λk+1 $ 2π $ and ρ being the radius of the circle Υ, we get Υ λ1 dλ = 0 α(θ)−1 α (θ) dθ = $ $ $ 2π −1 −iθ 2π 1 ρ e ρ i eiθ dθ = i 0 dθ = 2πi and, for k ≥ 1, we also get Υ λk+1 dλ = 0 $ $ $ 2π 2π −(k+1) −i(k+1)θ −(k+1)  iθ −k 2π −ikθ α(θ) α (θ) dθ = 0 ρ e i ρ e dθ = i ρ e dθ = 0. 0 0 Now set ψ = ϕ, the identity map ϕ(λ) = λ for every λ. Thus % % % ∞ k 1 1 1 1 1 ϕ(T ) = I 2πi dλ + T 2πi T 2πi dλ = O + T + O = T, λ dλ + λk Υ

$

k=2

Υ

Υ

$ $ 2π 1 (i.e., 2πi dλ = 0 α (θ) dθ = Υ were computed above. Then % % 1 1 T 2 = ϕ(T )2 = 2πi ϕ(λ)2 RT (λ) dλ = 2πi λ2 RT (λ) dλ

1 since 2πi dλ = 0 by the Cauchy Theorem $ 2π iθ Υ i ρ 0 e dθ = 0) and the other two integrals

Υ

Υ

by Lemma 4.15, and so a trivial induction ensures % 1 T j = 2πi λj RT (λ) dλ Υ

for every integer j ≥ 0. Therefore, by linearity of the integral, % m j 1 p(T ) = αj T = 2πi p(λ) RT (λ) dλ j=0

m

Υ

if p(λ) = j=0 αj λ . That is, if p ∈ A(σ(T )) is any polynomial. (b) Next we extend from polynomials (i.e., from finite power series) to infinite series. power ∞ k α A function ψ ∈ A(σ(T )) has a power series expansion ψ(λ) = k=0 k ∞ λ onk a neighborhood of σ(T ) if the radius of convergence n of the series k=0 αk λ is greater than r(T ). So the polynomials pn (λ) = k=0 αk λk make a sequence {pn }∞ n=0 converging uniformly to ψ on an open neighborhood Λ of σ(T ) including a circle Υ about the origin with radius greater than $ the spectral radius 1 R (λ) dλ = I) r(T ). Thus supλ∈Λ |ψ(λ) − pn (λ)| → 0 and so (since 2πi Υ T %      1  ΦT (pn ) − ΦT (ψ) = pn (T ) − ψ(T ) =  2πi pn (λ) − ψ(λ) RT (λ) dλ Υ %        1 ≤ sup pn (λ) − ψ(λ)  2πi RT (λ) dλ = sup pn (λ) − ψ(λ) → 0 j

λ∈Υ

as n → ∞, where (since

Υ

λ∈Υ

$

1 2πi Υ

λk RT (λ) dλ = T k ) % % 

∞ 1 1 ψ(λ) RT (λ) dλ = 2πi αk λk RT (λ) dλ ΦT (ψ) = ψ(T ) = 2πi k=0 Υ Υ %

∞ ∞ 1 αk 2πi λk RT (λ) dλ = αk T k, = k=0

Υ

k=0

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4. Functional Calculi

$ because the integral is linear and continuous (i.e., Υ ∈ B[ C[Υ, B[X ]], B[X ] ] ). ∞ T k ∈ B[X ] is the This means ΦT (pn ) → ΦT (ψ) in B[X ], where nψ(T ) = k k=0 αk n uniform limit of the B[X ]-valued sequence k=0 αk T (i.e., { k=0 αk T k }∞ n=0 converges in the operator

m norm topology). (c) Finally consider the general case where the path Υ = j=1 Υj ⊂ ρ(T ) is a finite union of disjoint closed rectifiable oriented simple curves Υj ⊂ ρ(T ) such that σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ for some open neighborhood Λ of σ(T ). Take an arbitrary A(σ(T ))-valued sequence {ψn }∞ n=0 converging to ψ ∈ A(σ(T )) uniformly on compact subsets of Λ (i.e., supλ∈K⊂Λ |ψn (λ) − ψ(λ)| → 0 as n → ∞ where supλ∈K⊂Λ means sup over all λ in every compact subset of Λ — called compact convergence). Thus m %     1   ΦT (ψn ) − ΦT (ψ) = 2πi ψn (λ) − ψ(λ) RT (λ) dλ  j=1

Υj

supλ∈Υ RT (λ) (Υ ) supλ∈K⊂Λ |ψn (λ) − ψ(λ)| → 0 as n → ∞, $ where supλ∈Υj RT (λ) ≤ supλ∈Υ RT (λ) < ∞ and Υj |dλ| = (Υj ) ≤ (Υ ) < ∞. That is, ΦT (ψn ) − ΦT (ψ) → 0 whenever supλ∈K⊂Λ |ψn (λ) − ψ(λ)| → 0. So ψn − ψ∞ = supλ∈Λ |ψn (λ) − ψ(λ)| → 0 implies ΦT (ψn ) − ΦT (ψ) → 0. There fore ΦT : (A(σ(T )),  · ∞ ) → (B[X ],  · ) is continuous. ≤

m 2πi

Corollary 4.17. Take any operator in T ∈ B[X ]. If ψ ∈ A(σ(T )), then ψ(T ) commutes with every operator that commutes with T . Proof. This is the counterpart of Corollary 4.10. Let S be an operator in B[X ]. Claim. If S T = T S, then RT (λ)S = SRT (λ) for every λ ∈ ρ(T ). Proof. S = S (λI − T ) (λI − T )−1 = (λI − T )S (λI − T )−1 if S T = T S, and so RT (λ)S = (λI − T )−1 (λI − T )S (λI − T )−1 = SRT (λ) for λ ∈ ρ(T ). Therefore according to Definition 4.14, since S lies in B[X ], % % 1 1 ψ(λ) RT (λ) dλ = 2πi ψ(λ) SRT (λ) dλ S ψ(T ) = S 2πi Υ Υ % 1 ψ(λ) RT (λ) S dλ = ψ(T ) S. = 2πi



Υ

Theorem 2.7 is the Spectral Mapping Theorem for polynomials, which holds for every Banach-space operator. For normal operators on a Hilbert space, the Spectral Mapping Theorem was extended to larger classes of functions in Theorems 2.8, 4.11, and 4.12. Returning now to the general case of arbitrary Banach-space operators, Theorem 2.7 can be extended to analytic functions, and this is again referred to as the Spectral Mapping Theorem. Theorem 4.18. Take an operator T ∈ B[X ] on a complex Banach space X . If ψ is analytic on the spectrum of T (i.e., if ψ ∈ A(σ(T ))), then σ(ψ(T )) = ψ(σ(T )).

4.4 Riesz Idempotents and the Riesz Decomposition Theorem

115

Proof. Take T ∈ B[X ] and ψ : Λ → C in A(σ(T )). For an arbitrary λ ∈ σ(T ) ⊂ Λ consider the functionφ = φλ : Λ → C defined by φ(ζ) = 0 if ζ = λ and φ(ζ) = ψ(λ)−ψ(ζ) if ζ ∈ Λ\{λ}. Since the function ψ(λ) − ψ(·) = (λ − · ) φ(·) : Λ → C λ−ζ lies in A(σ(T )), Lemma 4.15, Theorem 4.16, and Corollary 4.17 ensure % 1 ψ(λ)I − ψ(T ) = 2πi [ψ(λ) − ψ(ζ)] RT (ζ) dζ %Υ 1 = 2πi (λ − ζ) φ(ζ) RT (ζ) dζ Υ

= (λI − T ) φ(T ) = φ(T ) (λI − T ). Suppose ψ(λ) ∈ ρ(ψ(T )). Then ψ(λ)I − ψ(T ) has an inverse [ψ(λ)I − ψ(T )]−1 in B[X ]. Since ψ(λ)I − ψ(T ) = φ(T ) (λI − T ), we get [ψ(λ)I − ψ(T )]−1 φ(T ) (λI − T ) = [ψ(λ)I − ψ(T )]−1 [ψ(λ)I − ψ(T )] = I, (λI − T ) φ(T ) [ψ(λ)I − ψ(T )]−1 = [ψ(λ)I − ψ(T )] [ψ(λ)I − ψ(T )]−1 = I. Thus λI − T has a left and a right bounded inverse, and so it has an inverse in B[X ], which means λ ∈ ρ(T ). This is a contradiction. Therefore if λ ∈ σ(T ), then ψ(λ) ∈ σ(ψ(T )), and hence   ψ(σ(T )) = ψ(λ) ∈ C : λ ∈ σ(T ) ⊆ σ(ψ(T )). Conversely, take an arbitrary ζ ∈ ψ(σ(T )), which means ζ − ψ(λ) = 0 for every 1 : Λ → C , which lies λ ∈ σ(T ), and consider the function φ (·) = φζ (·) = ζ−ψ(·) in A(σ(T )) with Λ ⊆ Λ. Take Υ such that σ(T ) ⊂ ins Υ ⊆ ( ins Υ )− ⊂ Λ as in Definition 4.14. Then by Lemma 4.15 and Theorem 4.16 % 1 φ (T ) [ζI − ψ(T )] = [ζI − ψ(T )] φ (T ) = 2πi RT (λ) dλ = I. Υ



Thus ζI − ψ(T ) has a bounded inverse, φ (T ) ∈ B[X ], and so ζ ∈ ρ(ψ(T )). Equivalently, if ζ ∈ σ(ψ(T )), then ζ ∈ ψ(σ(T )). That is, σ(ψ(T )) ⊆ ψ(σ(T )).



4.4 Riesz Idempotents and the Riesz Decomposition Theorem A clopen set in a topological space is a set that is both open and closed in it. If a topological space has a nontrivial (i.e., proper and nonempty) clopen set, then (by definition) it is disconnected . Consider the spectrum σ(T ) of an operator T ∈ B[X ] on a complex Banach space X . There are some different definitions of spectral set.We stick to the classical one [45, Definition VII.3.17]: A spectral set for T is a clopen set in σ(T ) (i.e., a subset of σ(T ) ⊂ C which is

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both open and closed relative to σ(T ) — also called a spectral subset of σ(T )). For each spectral set Δ ⊆ σ(T ) there is a function ψΔ ∈ A(σ(T )) (analytic on a neighborhood of σ(T )) such that ψ∅ = 0, ψσ(T ) = 1 and, if ∅ = Δ = σ(T ), + 1, λ ∈ Δ, ψΔ (λ) = 0, λ ∈ σ(T )\Δ. It does not matter which value ψΔ (λ) takes for those λ in a neighborhood of σ(T ) but not in σ(T ). If Δ is nontrivial (i.e., if ∅ = Δ = σ(T )), then σ(T ) must be disconnected (since ψΔ ∈ A(σ(T )) ). Set EΔ = EΔ (T ) = ψΔ (T ). Thus % % 1 1 EΔ = ψΔ (T ) = 2πi ψΔ (λ) RT (λ) dλ = 2πi RT (λ) dλ Υ

ΥΔ

for any path Υ such that σ(T ) ⊂ ins Υ and ( ins Υ )− is included in an open neighborhood of σ(T ) as in Definition 4.14, and ΥΔ is any path for which Δ ⊆ ins ΥΔ and σ(T )\Δ ⊆ out ΥΔ . The operator EΔ ∈ B[X ] is referred to as the Riesz idempotent associated with Δ. In particular, the Riesz idempotent associated with an isolated point λ0 of the spectrum of T , namely E{λ0 } = E{λ0 } (T ) = ψ{λ0 } (T ), will be denoted by % % 1 1 Eλ0 = 2πi RT (λ) dλ = 2πi (λI − T )−1 dλ, Υλ0

Υλ0

where Υλ0 is any simple closed rectifiable positively oriented curve (e.g., any positively oriented circle) enclosing λ0 but no other point of σ(T ). Take an arbitrary operator T in B[X ]. The collection of all spectral subsets of σ(T ) forms a (Boolean) algebra of subsets of σ(T ). Indeed, the empty set and whole set σ(T ) are spectral sets, complements relative to σ(T ) of spectral sets remain spectral sets, and finite unions of spectral sets are spectral sets, as are differences and intersections of spectral sets. Lemma 4.19. Let T be any operator in B[X ] and let EΔ ∈ B[X ] be the Riesz idempotent associated with an arbitrary spectral subset Δ of σ(T ). (a) EΔ is a projection with the following properties: (a1 ) E∅ = O,

Eσ(T ) = I

and

Eσ(T )\Δ = I − EΔ .

If Δ1 and Δ2 are spectral subsets of σ(T ), then (a2 ) EΔ1 ∩Δ2 = EΔ1 EΔ2

and

EΔ1 ∪Δ2 = EΔ1 + EΔ2 − EΔ1 EΔ2 .

(b) EΔ commutes with every operator that commutes with T . (c) The range R(EΔ ) of EΔ is a subspace of X which is S-invariant for every operator S ∈ B[X ] that commutes with T . Proof. (a) According to Lemma 4.15 a Riesz idempotent deserves its name: 2 and so EΔ ∈ B[X ] is a projection. By the definition of the function E Δ = EΔ ψΔ for an arbitrary spectral subset Δ of σ(T ) we get

4.4 Riesz Idempotents and the Riesz Decomposition Theorem

(i)

ψ∅ = 0,

ψσ(T ) = 1,

117

ψσ(T )\Δ = 1 − ψΔ ,

and ψΔ1 ∪Δ2 = ψΔ1 + ψΔ2 − ψΔ1 ψΔ2 . $ 1 Since EΔ = ψΔ (T ) = 2πi ψ (λ) RT (λ) dλ for every spectral subset Δ of Υ Δ σ(T ), where Υ is any path as in Definition 4.14, the identities in (a1 ) and (a2 ) follow from the identities in (i) and (ii), respectively (using the linearity of the integral, Lemma 4.15, and Theorem 4.16). (ii)

ψΔ1 ∩Δ2 = ψΔ1 ψΔ2

(b) Take any S ∈ B[X ]. According to Corollary 4.17, ST = TS

implies

SEΔ = EΔ S.

(c) Thus R(EΔ ) is S-invariant. Moreover, R(EΔ ) = N (I − EΔ ) because EΔ is a projection. Since EΔ is bounded, so is I − EΔ . Thus N (I − EΔ ) is closed. Then R(EΔ ) is a subspace (i.e., R(EΔ ) is a closed linear manifold) of X .  Theorem 4.20. Take T ∈ B[X ]. If Δ is any spectral subset of σ(T ), then σ(T |R(EΔ ) ) = Δ. Moreover, the map Δ → EΔ that assigns to each spectral subset of σ(T ) the associated Riesz idempotent EΔ ∈ B[X ] is injective. Proof. Consider the results in Lemma 4.19(a1 ). If Δ = σ(T ), then EΔ = I so that R(EΔ ) = X , and hence σ(T |R(EΔ ) ) = σ(T ). On the other hand, if Δ = ∅, then EΔ = O so that R(EΔ ) = {0}, and hence T |R(EΔ ) : {0} → {0} is the null operator on the zero space, which implies σ(T |R(EΔ ) ) = ∅. In both cases the identity σ(T |R(EΔ ) ) = Δ holds trivially. (Recall: the spectrum of a bounded linear operator is nonempty on every nonzero complex Banach space.) Thus suppose ∅ = Δ = σ(T ). Let ψΔ ∈ A(σ(T )) be the function defining the Riesz idempotent EΔ associated with Δ, + 1, λ ∈ Δ, ψΔ (λ) = 0, λ ∈ σ(T )\Δ, and so EΔ = ψΔ (T ) =

1 2πi

% ψΔ (λ) RT (λ) dλ Υ

for any path Υ as in Definition 4.14. If ϕ ∈ A(σ(T )) is the identity map, ϕ(λ) = λ for every λ ∈ σ(T ), then (cf. Theorem 4.16) T = ϕ(T ) = Claim 1. σ(T EΔ ) = Δ ∪ {0}.

1 2πi

% ϕ(λ) RT (λ) dλ. Υ

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Proof. Since T EΔ = ϕ(T ) ψΔ (T ) and % 1 ϕ(T ) ψΔ (T ) = 2πi ϕ(λ) ψΔ (λ) RT (λ) dλ = Υ

1 2πi

% (ϕ ψΔ )(λ) RT (λ) dλ, Υ

and since ϕ ψΔ ∈ A(σ(T )), the Spectral Mapping Theorem leads to σ(T EΔ ) = (ϕ ψΔ )(σ(T )) = ϕ(σ(T )) ψΔ (σ(T )) = σ(T ) ψΔ (σ(T )) = Δ ∪ {0} (cf. Lemma 4.15 and Theorems 4.16 and 4.18). Claim 2. σ(T EΔ ) = σ(T |R(EΔ ) ) ∪ {0}. Proof. Since EΔ is a projection, R(EΔ ) and N (EΔ ) are algebraic complements, which means R(EΔ ) ∩ N (EΔ ) = ∅ and X = R(EΔ ) ⊕ N (EΔ ) — plain direct sum (no orthogonality in a pure Banach space setup). Thus, with respect to the complementary decomposition X = R(EΔ ) ⊕ N (EΔ ), we get    O Δ) EΔ = OI O = I ⊕ O, and so T EΔ = T |R(E = T |R(EΔ ) ⊕ O beO O O cause EΔ T = T EΔ . Then λI − T EΔ = (λI − T |R(EΔ ) ) ⊕ λI for every λ ∈ C , and hence λ ∈ ρ(EΔ T ) if and only if λ ∈ ρ(T |R(EΔ ) ) and λ = 0. Equivalently, λ ∈ σ(EΔ T ) if and only if λ ∈ σ(T |R(EΔ ) ) or λ = 0. Claim 3. σ(T |R(EΔ ) ) ∪ {0} = Δ ∪ {0}. Proof. Claims 1 and 2. Claim 4. 0 ∈ Δ ⇐⇒ 0 ∈ σ(T |R(EΔ ) ). Proof. Suppose 0 ∈ Δ. Thus there exists a function φ0 ∈ A(σ(T )) for which + 0, λ ∈ Δ, φ0 (λ) = −1 λ , λ ∈ σ(T )\Δ. Observe that ϕ φ0 lies in A(σ(T )) and ϕ φ 0 = 1 − ψΔ

on

σ(T ),

and hence, since ϕ(T ) = T and φ0 (T ) ∈ B[X ], T φ0 (T ) = φ0 (T ) T = I − ψΔ (T ) = I − EΔ . If 0 ∈ σ(T |R(EΔ ) ), then T |R(EΔ ) has a bounded inverse on the Banach space R(EΔ ). Therefore there exists an operator [T |R(EΔ ) ]−1 on R(EΔ ) such that T |R(EΔ ) [T |R(EΔ ) ]−1 = [T |R(EΔ ) ]−1 T |R(EΔ ) = I. Now consider the operator [T |R(EΔ ) ]−1 EΔ in B[X ]. Since T EΔ = ϕ(T )ψΔ (T ) = ψΔ (T )ϕ(T ) = EΔ T ,



T [T |R(EΔ ) ]−1 EΔ + φ0 (T ) = [T |R(EΔ ) ]−1 EΔ + φ0 (T ) T = I ∈ B[X ]. Then T has a bounded inverse on X , namely [T |R(EΔ ) ]−1 EΔ + φ0 (T ), and so 0 ∈ ρ(T ), which is a contradiction (since 0 ∈ Δ ⊆ σ(T )). Thus

4.4 Riesz Idempotents and the Riesz Decomposition Theorem

0∈Δ

implies

119

0 ∈ σ(T |R(EΔ ) ).

Conversely, suppose 0 ∈ Δ. Then there is a function φ1 ∈ A(σ(T )) for which + λ−1 , λ ∈ Δ, φ1 (λ) = 0, λ ∈ σ(T )\Δ. Again, observe that ϕ φ1 lies in A(σ(T )) and ϕ φ 1 = ψΔ

on

σ(T ),

and hence, since ϕ(T ) = T and φ1 (T ) ∈ B[X ], T φ1 (T ) = φ1 (T ) T = ψΔ (T ) = EΔ . Since φ1 (T ) ψΔ (T ) = ψΔ (T ) φ1 (T ) by Lemma 4.15, the operator φ1 (T ) is R(EΔ )-invariant. Hence φ1 (T )|R(EΔ ) lies in B[R(EΔ )] and is such that T |R(EΔ ) φ1 (T )|R(EΔ ) = φ1 (T )|R(EΔ ) T |R(EΔ ) = EΔ |R(EΔ ) = I ∈ B[R(EΔ )]. Then T |R(EΔ ) has a bounded inverse on R(EΔ ), namely φ1 (T )|R(EΔ ) , and so 0 ∈ ρ(T |R(EΔ ) ). Thus 0 ∈ Δ implies 0 ∈ σ(T |R(EΔ ) ). Equivalently, 0 ∈ σ(T |R(EΔ ) )

implies

0 ∈ Δ.

This concludes the proof of Claim 4. Claims 3 and 4 ensure the identity σ(T |R(EΔ ) ) = Δ. Finally, consider the map Δ → EΔ assigning to each spectral subset Δ of σ(T ) the associated Riesz idempotent EΔ ∈ B[X ]. We have already seen at the beginning of this proof that if EΔ = O, then σ(T |R(EΔ ) ) = ∅. Hence EΔ = O

=⇒

Δ=∅

by the above displayed identity (which holds for all spectral subsets of σ(T )). Take arbitrary spectral subsets Δ1 and Δ2 of σ(T ). From Lemma 4.19(a), EΔ1 \Δ2 = EΔ1 ∩(σ(T )\Δ2 ) = EΔ1 (I − EΔ2 ) = EΔ1 − EΔ1 EΔ2 , where EΔ1 and EΔ2 commute since EΔ1 EΔ2 = EΔ1 ∩Δ2 = EΔ2 EΔ1 , and so (EΔ1 − EΔ2 )2 = EΔ1 + EΔ2 − 2EΔ1 EΔ2 = EΔ1 \Δ2 + EΔ2 \Δ1 = EΔ1Δ2 , where Δ1Δ2 = (Δ1 \Δ2 ) ∪ (Δ2 \Δ1 ) is the symmetric difference. Thus E Δ 1 = EΔ 2

=⇒

EΔ1Δ2 = O

=⇒

Δ1Δ2 = ∅

=⇒

Δ1 = Δ2 ,

since Δ1Δ2 = ∅ if and only if Δ1 = Δ2 , which proves injectivity.



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Remark. Let T be an operator in B[X ] and suppose σ(T ) is disconnected, so there are nontrivial sets in the algebra of all spectral subsets of σ(T ). We show that to each partition of σ(T ) into spectral sets there is a corresponding decomposition of the resolvent function RT . Indeed, consider a finite partition {Δj }nj=1 of nontrivial spectral subsets of σ(T ); that is, ∅ = Δj = σ(T ) are clopen subsets of σ(T ) for each j = 1, . . . , n

such that n j=1

Δj = σ(T )

Δj ∩ Δk = ∅ whenever j = k.

with

Consider the Riesz idempotent associated with each Δj , namely % 1 EΔj = 2πi RT (λ) dλ. ΥΔj

The results of Lemma 4.19 are readily extended for any integer n ≥ 2. Thus each EΔj is a nontrivial projection (O = EΔj = EΔj2 = I) and n j=1

EΔ j = I

with

EΔj EΔk = O whenever j = k.

Take an arbitrary λ ∈ ρ(T ). For each j = 1, . . . , n set T j = T EΔ j

and

Rj (λ) = RT (λ)EΔj

in B[X ], and so n j=1

Tj = T

and

n j=1

Rj (λ) = RT (λ).

These operators commute. Indeed, T RT (λ) = RT (λ) T and T EΔj = EΔj T , and hence RT (λ)EΔj = EΔj RT (λ) (cf. the claim in the proof of Corollary 4.17 and Lemma 4.19(b)). Therefore, if j = k, then Tj Tk = Rj (λ)Rk (λ) = Tk Rj (λ) = Rj (λ)Tk = O. On the other hand, for every j = 1, . . . , n, Tj Rj (λ) = Rj (λ)Tj = T RT (λ)EΔj and, since (λI − T )RT (λ) = RT (λ)(λI − T ) = I,     λEΔj − Tj Rj (λ) = Rj (λ) λEΔj − Tj = EΔj . Moreover, by Claim 1 in the proof of Theorem 4.20, σ(Tj ) = Δj ∪ {0}.

4.4 Riesz Idempotents and the Riesz Decomposition Theorem

121

Clearly, the spectral radii are such that r(Tj ) ≤ r(T ). Since Tjk = T k EΔj for every nonnegative integer k, we get (cf. Corollary 2.12)  ∞ k  ∞ T j k T 1 Rj (λ) = (λI − T )−1 EΔj = λ1 E = Δ j λ λ λ k=0

k=0

if λ in ρ(T ) is such that r(T ) < |λ| and, again by Corollary 2.12,  ∞ T E Δ j k ∞ RTj (λ) = (λI − T EΔj )−1 = λ1 = λ1 λ k=0

k=0

Tj λ

k

if λ in ρ(Tj ) is such that r(Tj ) < |λ|, where the above series converge uniformly in B[X ]. Hence if r(T ) < |λ|, then Rj (λ) = RT (λ)EΔj = RT EΔj (λ) = RTj (λ) where each Rj can actually be extended to be analytic in ρ(Tj ). The properties of Riesz idempotents in Lemma 4.19 and Theorem 4.20 lead to a major result in operator theory which reads as follows. Operators whose spectra are disconnected have nontrivial hyperinvariant subspaces. (For the original statement of the Riesz Decomposition Theorem see [100, p. 421].) Corollary 4.21. (Riesz Decomposition Theorem). Let T ∈ B[X ] be an operator on a complex Banach space X . If σ(T ) = Δ1 ∪ Δ2 , where Δ1 and Δ2 are disjoint nonempty closed sets, then T has a complementary pair {M1 , M2 } of nontrivial hyperinvariant subspaces, viz., M1 = R(EΔ1 ) and M2 = R(EΔ2 ), for which σ(T |M1 ) = Δ1 and σ(T |M2 ) = Δ2 . Proof. If Δ1 and Δ2 are disjoint closed sets in C and σ(T ) = Δ1 ∪ Δ2 , then they are both clopen subsets of σ(T ) (i.e., spectral subsets of σ(T )), and hence the Riesz idempotents EΔ1 and EΔ2 associated with them are such that their ranges M1 = R(EΔ1 ) and M2 = R(EΔ2 ) are subspaces of X which are hyperinvariant for T by Lemma 4.19(c). Since Δ1 and Δ2 are also nontrivial (i.e., ∅ = Δ1 = σ(T ) and ∅ = Δ2 = σ(T )), the spectrum σ(T ) is disconnected (because they are both clopen subsets of σ(T )) and the projections EΔ1 and EΔ2 are nontrivial (i.e., O = EΔ1 = I and O = EΔ2 = I by the injectivity of Theorem 4.20), and so the subspaces M1 and M2 are also nontrivial (i.e., {0} = R(EΔ1 ) = X and {0} = R(EΔ2 ) = X ). Thus by Lemma 4.19(a) we get EΔ1 + EΔ2 = I and EΔ1 EΔ2 = O, which means the operators EΔ1 and EΔ2 are complementary projections (not necessarily orthogonal even if X were a Hilbert space), and so their ranges are complementary subspaces (i.e., M1 + M2 = X and M1 ∩ M2 = {0}) as in Section 1.4. Finally, according to  Theorem 4.20, σ(T |M1 ) = Δ1 and σ(T |M2 ) = Δ2 . Remark. Since M1 = R(EΔ1 ) and M2 = R(EΔ2 ) are complementary subspaces (i.e., M1 + M2 = X and M1 ∩ M2 = {0}), X can be identified with the direct sum M1 ⊕ M2 (not necessarily an orthogonal direct sum even if X

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were a Hilbert space), which means there exists an isomorphism (the natural one) Ψ : X → M1 ⊕ M2 between the normed spaces X and M1 ⊕ M2 (see, e.g., [78, Theorem 2.14]). The normed space M1 ⊕ M2 is not necessarily complete and the invertible linear transformation Ψ is not necessarily bounded. In other words, X ∼ M1 ⊕ M2 , where ∼ stands for algebraic similarity (i.e., isomorphic equivalence) and ⊕ stands for direct (not necessarily orthogonal) sum. Thus, since M1 and M2 are both T -invariant (recall: both EΔ1 and EΔ2 commute with T by Lemma 4.19), it follows that T ∼ T |M1 ⊕ T |M2 , which means Ψ T Ψ −1 = T |M1 ⊕ T |M2 . (If X were Hilbert and the subspaces orthogonal, then we might say they would reduce T .) A pair Δ1 and Δ2 of subsets of σ(T ) are complementary spectral sets for T if, besides being spectral sets (i.e., subsets of σ(T ) which are both open and closed relative to σ(T )) they also form a nontrivial partition of σ(T ); that is, ∅ = Δ1 = σ(T ), ∅ = Δ2 = σ(T ), Δ1 ∪ Δ2 = σ(T ), and Δ1 ∩ Δ2 = ∅. Thus by the Riesz Decomposition Theorem, for every pair of complementary spectral sets, the ranges of their Riesz idempotents are complementary nontrivial hyperinvariant subspaces for T , and the spectra of the restrictions of T to those ranges coincide with themselves. Corollary 4.22. Let Δ1 and Δ2 be complementary spectral sets for an operator T ∈ B[X ], and let EΔ1 and EΔ2 be the Riesz idempotents associated with them. If λ ∈ Δ1 , then EΔ2 (N (λI − T )) = {0}, EΔ2 (R(λI − T )) = R(EΔ2 ), EΔ1 (N (λI − T )) = N (λI − T ) ⊆ R(EΔ1 ), EΔ1 (R(λI − T )) = R((λI − T )EΔ1 ) ⊆ R(EΔ1 ), so that R(λI − T ) = EΔ1 (R(λI − T )) + R(EΔ2 ). If dim R(EΔ1 ) < ∞, then dim N (λI − T ) < ∞ and R(λI − T ) is closed . Proof. Let T be an operator on a complex Banach space X . Claim 1. If Δ1 and Δ2 are complementary spectral sets and λ ∈ C , then (a)

N (λI − T ) = EΔ1 (N (λI − T )) + EΔ2 (N (λI − T )),

(b)

R(λI − T ) = EΔ1 (R(λI − T )) + EΔ2 (R(λI − T )).

Proof. Let EΔ1 and EΔ2 be the Riesz idempotents associated with Δ1 and Δ2 . By Lemma 4.19(a) R(EΔ1 ) and R(EΔ2 ) are complementary subspaces (i.e., X = R(EΔ1 ) + R(EΔ2 ) and R(EΔ1 ) ∩ R(EΔ2 ) = ∅, since Δ1 and Δ2 are complementary spectral sets, and so EΔ1 and EΔ2 are complementary projections). Thus (cf. Section 1.1) there is a unique decomposition

4.4 Riesz Idempotents and the Riesz Decomposition Theorem

123

x = u + v = EΔ 1 x + E Δ 2 x for every vector x ∈ X , with u ∈ R(EΔ1 ) and v ∈ R(EΔ2 ), where u = EΔ1 x and v = EΔ2 x (since EΔ1 (R(EΔ2 )) = {0} and EΔ2 (R(EΔ1 )) = {0} because EΔ1 EΔ2 = EΔ2 EΔ1 = O). So we get the decompositions in (a) and (b). Claim 2. If λ ∈ σ(T )\Δ for some spectral set Δ of σ(T ), then EΔ (N (λI − T )) = {0}

and

EΔ (R(λI − T )) = R(EΔ ).

Proof. Let Δ be any spectral set of σ(T ) and let EΔ be the Riesz idempotent associated with it. Since R(EΔ ) is an invariant subspace for T (Lemma 4.19), take the restriction T |R(EΔ ) ∈ B[R(EΔ )] of T to the Banach space R(EΔ ). If λ ∈ σ(T )\Δ, then λ ∈ ρ(T |R(EΔ ) ). So (λI − T )|R(EΔ ) = λI|R(EΔ ) − T |R(EΔ ) has a bounded inverse, where I|R(EΔ ) = EΔ |R(EΔ ) stands for the identity on R(EΔ ). Thus, since R(EΔ ) is T -invariant, RT |R(EΔ ) (λ) (λI − T )EΔ = (λI|R(EΔ ) − T |R(EΔ ) )−1 (λI|R(EΔ ) − T |R(EΔ ) )EΔ = I|R(EΔ ) EΔ = EΔ , where RT |R(EΔ ) : ρ(T |R(EΔ ) ) → G[R(EΔ )] is the resolvent function of T |R(EΔ ) . Now take an arbitrary x ∈ N (λI − T ). Since EΔ T = T EΔ , EΔ x = RT |R(EΔ ) (λ) (λI − T )EΔ x = RT |R(EΔ ) (λ) EΔ (λI − T )x = 0. Therefore, EΔ (N (λI − T )) = {0}. Moreover, since (λI − T )EΔ = λEΔ − T EΔ = λI|R(EΔ ) − T |R(EΔ ) , it follows that if λ ∈ σ(T )\Δ, then λ ∈ ρ(T |R(EΔ ) ) by Theorem 4.20 (where T |R(EΔ ) ∈ B[R(EΔ )]), and so (see diagram of Section 2.2) R((λI − T )EΔ ) = R(λI|R(EΔ ) − T |R(EΔ ) ) = R(EΔ ). Since T EΔ = EΔ T , and A(R(B)) = R(AB) for all operators A and B, EΔ (R(λI − T )) = R(EΔ (λI − T )) = R((λI − T )EΔ ) = R(EΔ ). This concludes the proof of Claim 2. From now on suppose λ ∈ Δ1 . In this case EΔ2 (N (λI − T )) = {0} according to Claim 2, and so we get by Claim 1(a)

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N (λI − T ) = EΔ1 (N (λI − T )) ⊆ R(EΔ1 ). Hence dim R(EΔ1 ) < ∞

=⇒

dim N (λI − T ) < ∞.

Again, according to Claim 2 we get EΔ2 (R(λI − T )) = R(EΔ2 ). Thus, from Claim 1(b), R(λI − T ) = EΔ1 (R(λI − T )) + R(EΔ2 ), where (since T EΔ = EΔ T and A(R(B)) = R(AB) for all A and B), EΔ1 (R(λI − T )) = R(EΔ1 (λI − T )) = R((λI − T )EΔ1 ) ⊆ R(EΔ1 ). But R(EΔ2 ) is a subspace (i.e., a closed linear manifold) of X (Lemma 4.19). If R(EΔ1 ) is finite-dimensional, then EΔ1 (R(λI − T )) is finite-dimensional, and so R(λI − T ) is the sum of a finite-dimensional linear manifold and a closed linear manifold, which is closed (cf. Proposition 1.C). Hence, dim R(EΔ1 ) < ∞

=⇒

R(λI − T ) is closed.



Remark. Let λ0 be an isolated point of the spectrum of an operator T in B[X ]. A particular case of the preceding corollary for Δ1 = {λ0 } says: if dim R(Eλ0 ) < ∞, then dim N (λ0 I − T ) < ∞ and R(λ0 I − T ) is closed. We will prove the converse in Theorem 5.19 (next chapter): dim R(Eλ0 ) < ∞ ⇐⇒ dim N (λ0 I − T ) < ∞ and R(λ0 I − T ) is closed. Moreover, λ0 I − T |R(Eλ0 ) is a quasinilpotent operator on R(Eλ0 ). Indeed, σ(T |R(Eλ0 ) ) = {λ0 } according to Theorem 4.20 and so, by the Spectral Mapping Theorem (Theorem 2.7), σ(λ0 I − T |R(Eλ0 ) ) = {0}. Corollary 4.23. Let T be a compact operator on a complex Banach space X . If λ ∈ σ(T )\{0}, then λ is an isolated point of σ(T ) such that dim R(Eλ ) < ∞, where Eλ ∈ B[X ] is the Riesz idempotent associated with it, and {0} = N (λI − T ) ⊆ R(Eλ ) ⊆ N ((λI − T )n ) for some positive integer n such that R(Eλ ) ⊆ N ((λI − T )n−1 ).

4.4 Riesz Idempotents and the Riesz Decomposition Theorem

125

Proof. Take a compact operator T ∈ B∞[X ] on a nonzero complex Banach space X . Consider the results of Section 2.6, which were proved on a Hilbert space but, as we had mentioned there, still hold on a Banach space. Suppose λ ∈ σ(T )\{0}, and so λ is an isolated point of σ(T ) by Corollary 2.20. Take the restriction T |R(Eλ ) in B[R(Eλ )] of T to the Banach space R(Eλ ) (cf. Lemma 4.19), which is again compact (Proposition 1.V). According to Theorem 4.20, σ(T |R(Eλ ) ) = {λ} = {0}. Thus 0 ∈ ρ(T |R(Eλ ) ). Hence the compact operator T |R(Eλ ) in B∞ [R(Eλ )] is invertible, which implies (cf. Proposition 1.Y) dim R(Eλ ) < ∞. Set m = dim R(Eλ ). Note that m = 0 because Eλ = O by the injectivity of Theorem 4.20. Since σ(T |R(Eλ ) ) = {λ} we get σ(λI − T |R(Eλ ) ) = {0} by the Spectral Mapping Theorem (cf. Theorem 2.7). So the operator λI − T |R(Eλ ) is quasinilpotent on the m-dimensional space R(Eλ ), and therefore λI − T |R(Eλ ) in B[R(Eλ )] is a nilpotent operator for which (λI − T |R(Eλ ) )m = O. Indeed, this is a purely finite-dimensional algebraic result which is obtained together with the well-known Cayley–Hamilton Theorem (see, e.g., [55, Theorem 58.2]). Thus since Eλ is a projection commuting with T whose range is T -invariant (cf. Lemma 4.19(c) again), (λI − T )m Eλ = O. Moreover, since X = {0} we get O = (λI − T )0 = I, and so (since Eλ = O) (λI − T )0 Eλ = O. Thus there exists an integer n ∈ [1, m] such that (λI − T )n Eλ = O

and

(λI − T )n−1 Eλ = O.

In other words, R(Eλ ) ⊆ N ((λI − T )n )

and

R(Eλ ) ⊆ N (λI − T )n−1 .

(Recall: AB = O if and only if R(B) ⊆ N (A), for all operators A and B.) The remaining assertions are readily verified. In fact, {0} = N (λI − T ) since λ ∈ σ(T )\{0} is an eigenvalue of the compact operator T (by the Fredholm Alternative of Theorem 2.18), and N (λI − T ) ⊆ R(Eλ ) by Corollary 4.22.  Remark. According to Corollary 4.23 and Proposition 4.F, if T ∈ B∞[X ] (i.e., if T is compact) and λ ∈ σ(T )\{0}, then λ is a pole of the resolvent function RT : ρ(T ) → G[X ], and the integer n in the statement of the preceding result is the order of the pole λ. Moreover, it follows from Proposition 4.G that R(Eλ ) = N ((λI − T )n ).

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4.5 Supplementary Propositions If H is a nonzero complex Hilbert space, then B[H] is a unital C*-algebra where the involution ∗ is the adjoint operation (see Section 4.1). Every T ∈ B[H] determines a C*-subalgebra of B[H], denoted by C ∗ (T ) and referred to as the C*-algebra generated by T , which is the smallest C*-algebra of operators from B[H] containing T and the identity I, and which coincides with the closure P (T, T ∗ )− in B[H] of all polynomials in T and T ∗ with complex coefficients (see, e.g., [6, p. 1] — cf. paragraph following Definition 3.12; see also proof of Theorem 4.9). The Gelfand–Naimark Theorem asserts the converse. Every unital C*-algebra is isometrically ∗-isomorphic to a C*-subalgebra of B[H] (see, e.g., [6, Theorem 1.7.3] — i.e., for every abstract unital C*-algebra A there exists an involution-preserving isometric algebra isomorphism of A onto a C*-subalgebra of B[H]). Consider a set S ⊆ B[H] of operators on a Hilbert space H. The commutant S  of S is the set S  = {T ∈ B[H] : T S = S T for every S ∈ S} of all operators that commute with every operator in S, which is a unital subalgebra of B[H]. The double commutant S  of S is the unital algebra S  = (S ). The Double Commutant Theorem reads as follows. If A is a unital C*-subalgebra of B[H], then A− = A, where A− stands for the weak closure of A in B[H] (which coincides with the strong closure — see [30, Theorem IX.6.4]). A von Neumann algebra A is a C*-subalgebra of B[H] such that A = A, which is unital and weakly (thus strongly) closed. Take a normal operator on a separable Hilbert space H. Let A∗ (T ) be the von Neumann algebra generated by T ∈ B[H], which is the intersection of all von Neumann algebras containing T , and coincides with the weak closure of P(T, T ∗ ) (compare with the above paragraph). Proposition 4.A. Consider the setup of Theorem 4.8. The range of ΦT coincides with the von Neumann algebra generated by T (i.e., R(ΦT ) = A∗ (T )). Proposition 4.B. Take T ∈ B[X ]. If ψ ∈ A(σ(T )) and φ ∈ A(σ(ψ(T ))), then the composition (φ ◦ ψ) ∈ A(σ(T )) and φ(ψ(T )) = (φ ◦ ψ)(T ). That is, % 1 φ(ψ(T )) = 2πi φ( ψ(λ)) RT (λ) dλ, Υ

where Υ is any path about σ(T ) for which σ(T ) ⊂ ins Υ ⊂ ( ins Υ )− ⊂ Λ, and the open set Λ ⊆ C is the domain of φ ◦ ψ. Proposition 4.C. Let EΔ ∈ B[X ] be the Riesz idempotent associated with a spectral subset Δ of σ(T ). The point, residual, and continuous spectra of the restriction T |R(EΔ ) ∈ B[R(EΔ )] of T ∈ B[X ] to R(EΔ ) are given by σP (T |R(EΔ )) = Δ∩σP (T ), σR (T |R(EΔ )) = Δ∩σR (T ), σC (T |R(EΔ )) = Δ∩σC (T ). Proposition 4.D. If λ0 is an isolated point of the spectrum σ(T ) of an operator T ∈ B[X ], then the range of the Riesz idempotent Eλ0 ∈ B[X ] associated

4.5 Supplementary Propositions

127

with λ0 is given by

  1 R(Eλ0 ) = x ∈ X : (λ0 I − T )n x n → 0 .

Proposition 4.E. If λ0 is an isolated point of the spectrum σ(T ) of an operator T ∈ B[X ], then (with convergence in B[X ]) ∞ (λ − λ0 )k Tk RT (λ) = (λI − T )−1 = k=−∞ ∞ ∞ = (λ − λ0 )−k T−k + T0 + (λ − λ0 )k Tk k=1

k=1

for every λ in the punctured disk Bδ0 (λ0 )\{λ0 } = {λ ∈ C : 0 < |λ − λ0 | < δ0 } ⊆ ρ(T ), where δ0 = d(λ0 , σ(T )\{λ0 }) is the distance from λ0 to the rest of the spectrum, and Tk ∈ B[X ] is such that % 1 Tk = 2πi (λ − λ0 )−(k+1) RT (λ) dλ Υλ0

for every k ∈ Z , where Υλ0 is any positively oriented circle centered at λ0 with radius less than δ0 . An isolated point of σ(T ) is a singularity (or an isolated singularity) of the resolvent function RT : ρ(T ) → G[X ]. The expansion of RT in Proposition 4.E is the Laurent $ expansion of RT about an isolated point of the spectrum. Since 1 RT (λ) dλ we get, for every positive integer m (i.e., for m ∈ N ), Eλ0 = 2πi Υλ 0

T−m = Eλ0 (T − λ0 I)m−1 = (T − λ0 I)m−1 Eλ0 , and so T−1 = Eλ0 , the Riesz idempotent associated with the isolated point λ0 . The notion of poles associated with isolated singularities of analytic functions is extended as follows. An isolated point of σ(T ) is a pole of order n of RT , for some n ∈ N , if T−n = O and Tk = O for every k < −n (i.e., the order of a pole is the largest positive integer |k| such that T−|k| = O). Otherwise, if the number of nonzero coefficients Tk with negative indices is infinite, then the isolated point λ0 of σ(T ) is said to be an essential singularity of RT . Proposition 4.F. Let λ0 be an isolated point of the spectrum σ(T ) of T ∈ B[X ], and let Eλ0 ∈ B[X ] be the Riesz idempotent associated with λ0 . The isolated point λ0 is a pole of order n of RT if and only if (a)

(λ0 I − T )n Eλ0 = O

and

(λ0 I − T )n−1 Eλ0 = O.

Therefore, if λ0 is a pole of order n of RT , then (b)

{0} = R((λ0 I − T )n−1 Eλ0 ) ⊆ N (λ0 I − T ),

and so λ0 is an eigenvalue of T (i.e., λ0 ∈ σP (T )). Actually (by item (a) and Corollary 4.22), λ0 is a pole of order 1 of RT if and only if

128

4. Functional Calculi

{0} = R(Eλ0 ) = N (λ0 I − T ).

(c)

Proposition 4.G. Take T ∈ B[X ]. If λ0 is a pole of order n of RT , then R(Eλ0 ) = N ((λ0 I − T )n )

and

R(EΔ ) = R((λ0 I − T )n ),

where Eλ0 denotes the Riesz idempotent associated with λ0 and EΔ denotes the Riesz idempotent associated with the spectral set Δ = σ(T )\{λ0 }. Proposition 4.H. Let EΔ be the Riesz idempotent associated with a spectral subset Δ of σ(T ). If dim R(EΔ ) < ∞, then Δ is a finite set of poles. Proposition 4.I. Take T ∈ B[X ]. Let Δ be a spectral subset of σ(T ). Consider the Riesz idempotent EΔ ∈ B[X ] associated with Δ. If ψ ∈ A(σ(T )), then ψ ∈ A(σ(T |R(EΔ ) )) and ψ(T )|R(EΔ ) = ψ(T |R(EΔ ) ). Thus % %   1 1 ψ(λ)R (λ) dλ = ψ(λ)RT |R(EΔ ) (λ) dλ.  T 2πi 2πi ΥΔ

R(EΔ )

ΥΔ

For a normal operator on a complex Hilbert space, the functional calculi of Sections 4.2 and 4.3 (cf. Theorems 4.2 and 4.16) coincide for every analytic function on a neighborhood of the spectrum. $ Proposition 4.J. If T = λ dEλ ∈ B[H] is the spectral decomposition of a normal operator T on a Hilbert space H and if ψ ∈ A(σ(T )), then % % 1 ψ(T ) = ψ(λ) dEλ = 2πi ψ(λ) RT (λ) dλ, Υ

with the first integral as in Lemma 3.7 and the second as in Definition 4.14. Consider Definition 3.6. The notion of spectral measure E : AΩ → B[X ] (carrying the same properties set forth in Definition 3.6) can be extended to a (complex) Banach space X where the projections E(Λ) for Λ ∈ AΩ are not orthogonal but are bounded. If an operator T ∈ B[X ] is such that E(Λ) T = T E(Λ) and σ(T |R(E(Λ)) ) ⊆ Λ− (see Lemma 3.16), then T is called a spectral operator (cf. [47, Definition XV.2.5] ). Take ψ ∈ B(Ω), a bounded AΩ -meas$ urable complex-valued function on Ω. An integral ψ(λ) dEλ can $ be defined in such a Banach-space setting as the uniform limit of integrals φn (λ) dEλ = n n α E(Λ ) of measurable simple functions φ = α χ (i.e., of finite i i n i Λi i=1 i=1 linear combinations of characteristic functions χΛi of measurable sets Λi ∈ AΩ ) — see, $ e.g., [46, pp. 891–893]. If a spectral operator T ∈ B[X ] is such that T = λ dEλ , then it is said to be of scalar type (cf. [47, Definition XV.4.1]). Proposition 4.J holds for Banach-space spectral operators of scalar type. In $ $ function ϕ(λ) = this case, the preceding integral λ dEλ = ϕ(λ) dEλ (of the n λ ∈ C for every λ ∈ Ω, defined as the limit of a sequence i=1 βi E(Λi ) of integrals of simple functions) coincides with the integral of Theorem 3.15 if X

4.5 Supplementary Propositions

129

is a Hilbert space, and if the spectral measure E : AΩ → B[X ], with Ω = σ(T ), takes on orthogonal projections only. $ Proposition 4.K. If T = λ dEλ ∈ B[H] is the spectral decomposition of a normal operator T on a Hilbert space H, then the orthogonal projection E(Λ) coincides with the Riesz idempotent EΛ for every clopen Λ in Aσ(T ) : % % 1 E(Λ) = dEλ = 2πi RT (λ) dλ = EΛ , Λ

ΥΛ

where ΥΔ is any path for which Δ ⊆ ins ΥΔ and σ(T )\Δ ⊆ out ΥΔ .  In particular, let T = k λk Ek be the spectral decomposition of a compact normal operator T on a Hilbert space, with {λk } = σP (T ) and {Ek } being a countable resolution of the identity where each orthogonal projection Ek is such that R(Ek ) = N (λk I − T ) (Theorem 3.3). Take any nonzero eigenvalue λk in σ(T )\{0} = σP (T )\{0} = {λk }\{0} so that dim N (λk I − T ) < ∞ (cf. Theorems 1.19 and 2.18 and Corollary 2.20). Each orthogonal projection Ek coincides with the Riesz idempotent Eλk associated with each isolated point 0 = λk of σ(T ) according to Proposition 4.K. Now we come back to a Banachspace setting: according to Corollary 4.23 and Proposition 4.G, N (λk I − T ) = N ((λk I − T )nk ) = R(Eλk ) where the integer nk > 0 is the order of the pole λk and the integer 0 < dim R(Eλk ) = dim N (λk I − T ) < ∞ is the finite multiplicity of the eigenvalue λk . In general, dim R(Eλk ) is sometimes referred to as the algebraic multiplicity of the isolated point λk of the spectrum, while dim N (λk I − T ) — the multiplicity of the eigenvalue λk — is sometimes called the geometric multiplicity of λk . In the preceding case (i.e., if T is compact and normal on a Hilbert space) these multiplicities are finite and coincide. Proposition 4.L. Every isolated point of the spectrum of a hyponormal operator is an eigenvalue. In fact, if λ0 is an isolated point of σ(T ), then {0} = R(Eλ0 ) ⊆ N (λ0 I − T ) for every hyponormal operator T ∈ B[H], where Eλ0 is the Riesz idempotent associated with the isolated point λ0 of the spectrum of T . Notes: For Proposition 4.A see, for instance, [30, Theorem IX.8.10]. Proposition 4.B is the composition counterpart of the product-preserving result of Lemma 4.15 (see, e.g., [66, Theorem 5.3.2] or [45, Theorem VII.3.12]). For Proposition 4.C see [45, Exercise VII.5.18] or [111, 1st edn. Theorem 5.7-B]. Proposition 4.D follows from Theorem 4.20 since, according to Theorem 2.7, 1 λ0 I − T |R(Eλ0 ) is quasinilpotent and so limn (λI − T )n x n = 0 whenever x lies in R(Eλ ). For the converse see [111, 1st edn. Lemma 5.8-C]. The Laurent expansion in Proposition 4.E and also Proposition 4.F are standard results (see [30, Lemma VII.6.11, Proposition 6.12, and Corollary 6.13]). Proposition 4.G refines the results of Proposition 4.F — see, e.g., [45, Theorem VII.3.24].

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4. Functional Calculi

For Proposition 4.H see [45, Exercise VII.5.34]. Proposition 4.I is a consequence of Theorem 4.20 (see, e.g., [45, Theorem VII.3.20]). Proposition 4.J also holds for Banach-space spectral operators of the scalar type (cf. [47, Theorem XV.5.1]), and Proposition 4.K is a particular case of it for the characteristic function in a neighborhood of a clopen set in the σ-algebra of Borel subsets of σ(T ). Proposition 4.L is the extension of Proposition 3.G from normal to hyponormal operators, which is obtained by the Riesz Decomposition Theorem (Corollary 4.21) — see, for instance, [78, Problem 6.28].

Suggested Readings Arveson [7] Brown and Pearcy [23] Conway [30, 32, 33] Dowson [39] Dunford and Schwartz [45, 46, 47]

Hille and Phillips [66] Radjavi and Rosenthal [98] Riesz and Sz.-Nagy [100] Sz.-Nagy and Foia¸s [110] Taylor and Lay [111]

5 Fredholm Theory in Hilbert Space

The central theme of this chapter is the investigation of compact perturbations. Of particular concern will be the properties of the spectrum of an operator which are invariant under compact perturbations. In other words, the properties of the spectrum of T which are also possessed by the spectrum of T + K for every compact operator K. As in Sections 1.8 and 2.6, when dealing with compact operators, we assume in this chapter that all operators lie in B[H] where H is a Hilbert space. Although the theory is amenable to being developed on a Banach space as well, the approaches are quite different. Hilbert spaces are endowed with a notably rich structure which allows significant simplifications. The Banach-space case is considered in Appendix A, where the differences between the two approaches are examined, and an account is given of the origins and consequences of such differences.

5.1 Fredholm and Weyl Operators — Fredholm Index Let H be a nonzero complex Hilbert space. Let B∞[H] be the (two-sided) ideal of compact operators from B[H]. An operator T ∈ B[H] is left semi-Fredholm if there exist A ∈ B[H] and K ∈ B∞[H] for which A T = I + K, and right semiFredholm if there exist A ∈ B[H] and K ∈ B∞[H] for which T A = I + K. It is semi-Fredholm if it is either left or right semi-Fredholm, and Fredholm if it is both left and right semi-Fredholm. Let F [H] and Fr [H] be the classes of all left semi-Fredholm operators and of all right semi-Fredholm operators:   F [H] = T ∈ B[H] : A T = I + K for some A ∈ B[H] and some K ∈ B∞[H] ,   Fr [H] = T ∈ B[H] : T A = I + K for some A ∈ B[H] and some K ∈ B∞[H] . The classes of all semi-Fredholm and Fredholm operators from B[H] will be denoted by SF[H] and F[H], respectively: SF[H] = F [H] ∪ Fr [H]

and

F[H] = F [H] ∩ Fr [H].

According to Proposition 1.W, K ∈ B∞[H] if and only if K ∗ ∈ B∞[H]. Thus © Springer Nature Switzerland AG 2020 C. S. Kubrusly, Spectral Theory of Bounded Linear Operators, https://doi.org/10.1007/978-3-030-33149-8_5

131

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5. Fredholm Theory in Hilbert Space

T ∈ F [H]

if and only if

T ∗ ∈ Fr [H].

T ∈ SF[H]

if and only if

T ∗ ∈ SF[H],

T ∈ F[H]

if and only if

T ∗ ∈ F[H].

Therefore

The following results of kernel and range of a Hilbert-space operator and its adjoint (Lemmas 1.4 and 1.5) will often be used throughout this chapter: N (T ∗ ) = H  R(T )− = R(T )⊥ , R(T ∗ ) is closed if and only if R(T ) is closed. Theorem 5.1. (a) An operator T ∈ B[H] is left semi-Fredholm if and only if R(T ) is closed and N (T ) is finite-dimensional . (b) Hence

  F [H] = T ∈ B[H] : R(T ) is closed and dim N (T ) < ∞ ,   Fr [H] = T ∈ B[H] : R(T ) is closed and dim N (T ∗ ) < ∞ .

Proof. Let A, T , and K be operators on H, where K is compact. (a1 ) If T is left semi-Fredholm, then there are operators A and K such that A T = I + K, and so N (T ) ⊆ N (A T ) = N (I + K) and R(A T ) = R(I + K). According to the Fredholm Alternative (Corollary 1.20), dim N (I + K) < ∞ and R(I + K) is closed. Therefore (i)

dim N (T ) < ∞,

(ii)

dim T (N (A T )) < ∞,

(iii)

R(A T ) is closed.

This implies (iv)

T (N (A T )⊥ ) is closed.

Indeed, the restriction (A T )|N (A T )⊥ : N (A T )⊥ → H is bounded below by (iii) since (A T )|N (A T )⊥ is injective with range R(A T ) (Theorem 1.2). Thus there exists an α > 0 such that αv ≤ A T v ≤ A T v for every v ∈ N (A T )⊥. Then T |N (A T )⊥ : N (A T )⊥ → H is bounded below, and so T |N (A T )⊥ has a closed range (Theorem 1.2 again), proving (iv). But (ii) and (iv) imply (v)

R(T ) is closed.

In fact, since H = N (A T ) + N (A T )⊥ by the Projection Theorem, it follows that R(T ) = T (H) = T (N (A T ) + N (A T )⊥ ) = T (N (A T )) + T (N (A T )⊥ ). Thus, by assertions (ii) and (iv), R(T ) is closed (since the sum of a closed linear

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133

manifold and a finite-dimensional linear manifold is closed — see Proposition 1.C). This concludes the proof of (v) which, together with (i), concludes half of the claimed result: T ∈ F [H] =⇒ R(T )− = R(T ) and dim N (T ) < ∞. (a2 ) Conversely, suppose dim N (T ) is finite and R(T ) is closed. Since R(T ) is closed, the restriction T |N (T )⊥ : N (T )⊥ → H (which is injective because N (T |N (T )⊥) = {0}) has a closed range R(T |N (T )⊥) = R(T ). Hence it has a bounded inverse on its range (Theorem 1.2). Let E ∈ B[H] be the orthogonal projection onto R(E) = R(T |N (T )⊥ ) = R(T ), and define A ∈ B[H] as follows: A = (T |N (T )⊥ )−1 E. If u ∈ N (T ), then A T u = 0 trivially. On the other hand, if v ∈ N (T )⊥, then A T v = (T |N (T )⊥ )−1E T |N (T )⊥ v = (T |N (T )⊥ )−1 T |N (T )⊥ v = v. Thus for every x = u + v in H = N (T ) + N (T )⊥ we get A T x = A T u + A T v = v = E  x, where E  ∈ B[H] is the orthogonal projection onto N (T )⊥ . Therefore A T = I + K, where −K = I − E  ∈ B[H] is the complementary orthogonal projection onto the finite-dimensional space N (T ). Hence K is a finite-rank operator, and so compact (Proposition 1.X). Thus T is left semi-Fredholm. (b) Since T ∈ F [H] if and only if T ∗ ∈ Fr [H], and since R(T ∗ ) is closed if and only if R(T ) is closed (Lemma 1.5), item (b) follows from item (a).  Corollary 5.2. Take an operator T ∈ B[H]. (a) T is semi-Fredholm if and only if R(T ) is closed and N (T ) or N (T ∗ ) is finite-dimensional :   SF[H] = T ∈ B[H] : R(T ) is closed, dim N (T ) < ∞ or dim N (T ∗ ) < ∞ . (b) T is Fredholm if and only if R(T ) is closed and both N (T ) and N (T ∗ ) are finite-dimensional :   F[H] = T ∈ B[H] : R(T ) is closed, dim N (T ) < ∞ and dim N (T ∗ ) < ∞ . Proof. SF[H] = F [H] ∪ Fr [H] and F[H] = F [H] ∩ Fr [H]. Use Theorem 5.1.  Let Z be the set of all integers and set Z = Z ∪ {−∞} ∪ {+∞}, the extended integers. Take any T in SF[H] so that N (T ) or N (T ∗ ) is finite-dimensional. The Fredholm index ind (T ) of T in SF[H] is defined in Z by ind (T ) = dim N (T ) − dim N (T ∗ ). It is usual to write α(T ) = dim N (T ) and β(T ) = dim N (T ∗ ), and hence ind (T ) = α(T ) − β(T ). Since T ∗ and T lie in SF[H] together, we get ind (T ∗ ) = −ind (T ).

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Note: β(T ) = α(T ∗ ) = dim R(T )⊥ = dim(H  R(T )− ) = dim H/R(T )−, where the last identity is discussed in the forthcoming Remark 5.3(a). According to Theorem 5.1, T ∈ F [H] implies dim N (T ) < ∞ and, dually, T ∈ Fr [H] implies dim N (T ∗ ) < ∞. Therefore T ∈ F [H]

implies

ind (T ) = +∞,

T ∈ Fr [H]

implies

ind (T ) = −∞.

In other words, if T ∈ SF[H] = F [H] ∪ Fr [H], then ind (T ) = +∞

implies

T ∈ Fr [H]\F [H] = Fr [H]\F[H],

ind (T ) = −∞

implies

T ∈ F [H]\Fr [H] = F [H]\F[H].

Remark 5.3. (a) Banach Space. If T is an operator on a Hilbert space H, then N (T ∗ ) = R(T )⊥ = H  R(T )− by Lemma 1.4. Thus Theorem 5.1 and Corollary 5.2 can be restated with N (T ∗ ) replaced by H  R(T )−. However, consider the quotient space H/R(T )− of H modulo R(T )− consisting of all cosets x + R(T )− for each x ∈ H. The natural mapping of H/R(T )− onto H  R(T )− (viz., x + R(T )− → Ex where E is the orthogonal projection onto R(T )⊥) is an isomorphism. Then dim H/R(T )− = dim(H  R(T )− ). Therefore we get the following restatement of Theorem 5.1 and Corollary 5.2. T is right semi-Fredholm if and only if R(T ) is closed and H/R(T ) is finite-dimensional. It is semi-Fredholm if and only if R(T ) is closed and N (T ) or H/R(T ) is finite-dimensional. It is Fredholm if and only if R(T ) is closed and both N (T ) and H/R(T ) are finite-dimensional. This is how the theory advances in a Banach space. Many properties among those that will be developed in this chapter work smoothly in any Banach space. Some of them, in being translated literally into a Banach-space setting, will behave well if the Banach space is complemented. A normed space X is complemented if every subspace of it has a complementary subspace (i.e., if there is a (closed) subspace N of X such that M + N = X and M ∩ N = {0} for every (closed) subspace M of X ). For instance, Proposition 1.D (on complementary subspaces and continuous projections) is an example of a result that acquires its full strength only on complemented Banach spaces. However, if a Banach space is complemented, then it is isomorphic to a Hilbert space [91] (i.e., topologically isomorphic, by the Inverse Mapping Theorem). So complemented Banach spaces are identified with Hilbert spaces: only Hilbert spaces (up to an isomorphism) are complemented . We stick to Hilbert spaces in this chapter. The Banach-space counterpart will be considered in Appendix A. (b) Closed Range. As will be verified in Appendix A (cf. Remark A.1 and Corollary A.9), if codim R(T ) < ∞, then R(T ) is closed — codimension means dimension of any algebraic complement, which coincides with dimension of the

5.1 Fredholm and Weyl Operators — Fredholm Index

135

quotient space. So, in a Hilbert space, dim H/R(T ) < ∞ ⇐⇒ codim R(T ) < ∞ =⇒ codim R(T )− < ∞ ⇐⇒ dim H/R(T )− < ∞ ⇐⇒ codim N (T ∗ )⊥ < ∞ ⇐⇒ dim N (T ∗ ) < ∞ (see also Lemmas 1.4 and 1.5 and the Projection Theorem). Thus, since dim H/R(T ) < ∞ =⇒ R(T ) = R(T )−, then (cf. Theorem 5.1)   R(T ) = R(T )− and dim N (T ∗ ) < ∞ ⇐⇒ dim H/R(T ) < ∞. (c) Finite Rank. The following statement was in fact proved in Theorem 5.1: T is left semi-Fredholm if and only if there exists an operator A ∈ B[H] and a finite-rank operator K ∈ B[H] for which A T = I + K. Since T is right semi-Fredholm if and only if T ∗ is left semi-Fredholm, and so there exists a finite-rank operator K such that A T ∗ = I + K or, equivalently, such that T A∗ = I + K ∗ , and since K is a finite-rank operator if and only if K ∗ is (see, e.g., [78, Problem 5.40]), we also get T is right semi-Fredholm if and only if there exists an operator A ∈ B[H] and a finite-rank operator K ∈ B[H] for which T A = I + K. So the definitions of left semi-Fredholm, right semi-Fredholm, semi-Fredholm, and Fredholm operators are equivalently stated if “compact” is replaced with “finite-rank”. For Fredholm operators we can even have the same A when stating that I − A T and I − T A are of finite rank (cf. Proposition 5.C). (d) Finite Dimension. Take any operator T ∈ B[H]. The rank and nullity identity of linear algebra says dim H = dim N (T ) + dim R(T ) (cf. proof of Corollary 2.19). Since H = R(T )− ⊕ R(T )⊥ (Projection Theorem), then dim H = dim R(T ) + dim R(T )⊥. Thus if H is finite-dimensional (and so N (T ) and N (T ∗ ) are finite-dimensional), then ind (T ) = 0. Indeed, if dim H < ∞, then dim N (T ) − dim N (T ∗ ) = dim N (T ) − dim R(T )⊥ = dim N (T ) + dim R(T ) − dim H = dim H − dim H = 0. Since linear manifolds of finite-dimensional spaces are closed, then on a finitedimensional space every operator is Fredholm with a null index (Corollary 5.2):   dim H < ∞ =⇒ T ∈ F[H] : ind (T ) = 0 = B[H]. (e) Fredholm Alternative. If K ∈ B∞[H] and λ = 0, then R(λI − K) is closed and dim N (λI − K) = dim N (λI − K ∗ ) < ∞. This is the Fredholm Alternative for compact operators of Corollary 1.20, which can be restated in terms of Fredholm indices, according to Corollary 5.2, as follows. If K ∈ B∞[H] and λ = 0, then λI − K is Fredholm with ind (λI − K) = 0. However, ind (λI − K) = 0 means dim N (λI − K) = dim R(λI − K)⊥ (since dim N (λI − K ∗ ) = dim R(λI − K)⊥ by Lemma 1.5), and this implies that

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N (λI − K) = {0} if and only if R(λI − K)− = H (equivalently, if and only if R(λI − K)⊥ = {0}). Since R(λI − K) is closed if λI − K is Fredholm, we get still another form of the Fredholm Alternative (cf. Theorems 1.18 and 2.18). If K ∈ B∞[H] and λ = 0, then N (λI − K) = {0} ⇐⇒ R(λI − K) = H (i.e., if K is a compact operator and λ is a nonzero scalar, then λI − K is injective if and only if it is surjective). (f) Fredholm Index. Corollary 5.2 and some of its straightforward consequences can also be naturally rephrased in terms of Fredholm indices. Indeed, dim N (T ) and dim N (T ∗ ) are both finite if and only if ind (T ) is finite (reason: ind (T ) was defined for semi-Fredholm operators only, and so if one of dim N (T ) or dim N (T ∗ ) is infinite, then the other must be finite). Thus T is Fredholm if and only if it is semi-Fredholm with a finite index, and hence   T ∈ SF[H] =⇒ T ∈ F[H] ⇐⇒ |ind (T )| < ∞ . Moreover, since F [H]\Fr [H] = F [H]\F[H] and Fr [H]\F [H] = Fr [H]\F[H],   T ∈ Fr [H]\F [H] ⇐⇒ ind (T ) = +∞ , T ∈ SF[H] =⇒   T ∈ SF[H] =⇒ T ∈ F [H]\Fr [H] ⇐⇒ ind (T ) = −∞ . (g) Product. Still from Corollary 5.2, a nonzero scalar operator is Fredholm with a null index. This is readily generalized: If T ∈ F[H], then γ T ∈ F[H] and ind (γ T ) = ind (T ) for every γ ∈ C \{0}. A further generalization leads to the most important property of the index, namely its logarithmic additivity: ind (S T ) = ind (S) + ind (T ) whenever such an addition makes sense. Theorem 5.4. Take S, T ∈ B[H]. (a) If S, T ∈ F [H], then S T ∈ F [H]. If S, T ∈ Fr [H], then S T ∈ Fr [H]. (Therefore, if S, T ∈ F[H], then S T ∈ F[H].) (b) If S, T ∈ F [H] or S, T ∈ Fr [H] (in particular, if S, T ∈ F[H]), then ind (S T ) = ind (S) + ind (T ). Proof. (a) If S and T are left semi-Fredholm, then there are A, B, K, L in B[H], with K, L being compact, such that B S = I + L and A T = I + K. Then (A B)(S T ) = A(I + L)T = A T + A L T = I + (K + A L T ). But K + A L T is compact (because B∞[H] is an ideal of B[H]). Thus S T is left semi-Fredholm. Summing up: S, T ∈ F [H] implies S T ∈ F [H]. Dually,

5.1 Fredholm and Weyl Operators — Fredholm Index

137

S, T ∈ Fr [H] if and only if S ∗, T ∗ ∈ F [H], which (as we saw above) implies T ∗ S ∗ ∈ F [H], which means ST = (T ∗ S ∗ )∗ ∈ Fr [H]. (b) We split the proof of ind (S T ) = ind (S) + ind (T ) into three parts. (b1 ) Take S, T ∈ F [H]. First suppose ind (S) and ind (T ) are both finite or, equivalently, suppose S, T ∈ F[H] = F [H] ∩ Fr [H]. Consider the surjective transformation L : N (S T ) → R(L) ⊆ H defined by Lx = T x for every x ∈ N (S T ), which is clearly linear with N (L) = N (T ) ∩ N (S T ) by the definition of L. Since N (T ) ⊆ N (S T ), N (L) = N (T ). If y ∈ R(L) = L(N (S T )), then y = Lx = T x for some vector x ∈ H such that S T x = 0, and so Sy = 0, which implies y ∈ R(T ) ∩ N (S). Therefore we get R(L) ⊆ R(T ) ∩ N (S). Conversely, if y ∈ R(T ) ∩ N (S), then y = T x for some vector x ∈ H and Sy = 0, and so S T x = 0, which implies x ∈ N (S T ) and y = T x = Lx ∈ R(L). Thus R(T ) ∩ N (S) ⊆ R(L). Hence R(L) = R(T ) ∩ N (S). Recalling again the rank and nullity identity of linear algebra, dim X = dim N (L) + dim R(L) for every linear transformation L on a linear space X . Thus, with X = N (S T ) and since dim N (S) < ∞, dim N (S T ) = dim N (T ) + dim (R(T ) ∩ N (S)) = dim N (T ) + dim N (S) + dim (R(T ) ∩ N (S)) − dim N (S). Moreover, since R(T ) is closed, R(T ) ∩ N (S) is a subspace of N (S). So (by the Projection Theorem) N (S) = R(T ) ∩ N (S) ⊕ (N (S)  (R(T ) ∩ N (S))), and hence dim (N (S)  (R(T ) ∩ N (S))) = dim N (S) − dim (R(T ) ∩ N (S)) (because dim N (S) < ∞). Thus dim N (T ) + dim N (S) = dim N (S T ) + dim (N (S)  (R(T ) ∩ N (S))). Swapping T with S ∗ and S with T ∗ (which have finite-dimensional kernels and closed ranges), it follows by Lemma 1.4 and Proposition 1.H(a,b), dim N (S ∗ ) + dim N (T ∗ ) = dim N (T ∗ S ∗ ) + dim(N (T ∗ )  (R(S ∗ ) ∩ N (T ∗ ))) = dim N ((S T )∗ ) + dim(R(T )⊥  (N (S)⊥ ∩ R(T )⊥)) = dim N ((S T )∗ ) + dim(R(T )⊥  (R(T ) + N (S))⊥ ) = dim N ((S T )∗ ) + dim(N (S)  (R(T ) ∩ N (S))). Therefore dim N (S) − dim N (S ∗ ) + dim N (T ) − dim N (T ∗ ) = dim N (S T ) − dim N ((S T )∗ ),

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5. Fredholm Theory in Hilbert Space

and so ind (S T ) = ind (S) + ind (T ). (b2 ) If S, T ∈ F [H], then ind (S), ind (T ) = +∞. Suppose S, T ∈ F [H] and one of ind (S) or ind (T ) is not finite; that is, ind (S) = −∞ or ind (T ) = −∞. If ind (S) = −∞, then dim N (S ∗ ) = ∞, and so dim R(S)⊥ = dim N (S ∗ ) = ∞. Since R(S T ) ⊆ R(S), we get R(S)⊥ ⊆ R(S T )⊥, and hence dim N ((T S)∗ ) = dim R(S T )⊥ = ∞. Since T S ∈ F [H] by item (a), then dim N (T S) < ∞ by Theorem 5.1. Thus ind (T S) = dim N (T S) − dim N ((T S)∗ ) = −∞. On the other hand, if ind (S) = −∞, then ind (T ) = −∞, and the same argument leads to ind (S T ) = −∞. So, in both cases, ind (S T ) = ind (T S) and ind (S T ) = −∞ = ind (S) + ind (T ). (b3 ) Suppose S, T ∈ Fr [H]. Thus S ∗, T ∗ ∈ F [H], which (as we saw in items (b1 ) and (b2 )) implies that ind (T ∗ S ∗ ) = ind (S ∗ T ∗ ) = ind (S ∗ ) + ind (T ∗ ). But ind (S ∗ ) = −ind (S), ind (T ∗ ) = −ind (T ), ind (T ∗ S ∗ ) = ind ((S T )∗ ) = −ind (S T ), and ind (S ∗ T ∗ ) = ind ((T S)∗ ) = −ind (T S). Thus, if S, T ∈ Fr [H], ind (S T ) = ind (S) + ind (T ).



If S, T ∈ SF[H]\F[H] (and so ind (S) and ind (T ) are both not finite), then the expression ind (S) + ind (T ) makes sense as an extended integer in Z if and only if either ind (S) = ind (T ) = +∞ or ind (S) = ind (T ) = −∞. But this is equivalent to saying that either S, T ∈ Fr [H]\F[H] = Fr [H]\F [H] or S, T ∈ F [H]\F[H] = F [H]\Fr [H]. Therefore if S, T ∈ SF[H] with one of them in Fr [H]\F [H] and the other in F [H]\Fr [H], then the index expression of Theorem 5.4 does not make sense. Moreover, if one of them lies in Fr [H]\F [H] and the other lies in F [H]\Fr [H], then their product may not be an operator in SF[H]. For instance, let H be an infinite-dimensional Hilbert space, and let S+ be the canonical ∞ unilateral shift (of infinite multiplicity) 2 (H) = on the Hilbert space + k=0 H (cf. Section 2.7). As is readily veri∗ 2 dim N (S ) = 0, and R(S+ ) is closed in + (H). In fact, fied, dim N (S+ ) = ∞, + ∞ ∞ ∗ 2 N (S+ ) = + (H)  k=1 H ∼ = H, N (S+ ) = {0}, and R(S+ ) = {0} ⊕ k=1 H. ∗ Therefore S+ ∈ Fr [H]\F [H] and S+ ∈ F [H]\Fr [H] (according to Theorem 5.1). However, S+ S+∗ ∈ SF[H]. Indeed, S+ S+∗ = O ⊕ I (where ∞ O stands for the null operator on H and I for the identity operator on k=1 H) does not lie in SF[H] since it is self-adjoint and dim N (S+ S+∗ ) = ∞ (cf. Theorem 5.1). Corollary 5.5. Take an arbitrary nonnegative integer n. If T ∈ F [H] (or if T ∈ Fr [H]), then T n ∈ F [H] (or T n ∈ Fr [H]) and ind (T n ) = n ind (T ). Proof. The result holds trivially for n = 0 (the identity is Fredholm with index zero), and tautologically for n = 1. Thus suppose n ≥ 2. The result holds for

5.1 Fredholm and Weyl Operators — Fredholm Index

139

n = 2 by Theorem 5.4, and a trivial induction ensures the claimed result for each n ≥ 2 by using Theorem 5.4 again.  The null operator on an infinite-dimensional space is not Fredholm, and so a compact operator may not be Fredholm. However, the sum of a Fredholm operator and a compact operator is a Fredholm operator with the same index. In other words, the index of a Fredholm operator remains unchanged under compact perturbation, a property which is referred to as index stability. Theorem 5.6. Take T ∈ B[H] and K ∈ B∞[H]. (a) T ∈ F [H] ⇐⇒ T + K ∈ F [H]

and

T ∈ Fr [H] ⇐⇒ T + K ∈ Fr [H].

In particular, T ∈ F[H] ⇐⇒ T + K ∈ F[H]

and

T ∈ SF[H] ⇐⇒ T + K ∈ SF[H].

(b) Moreover, for every T ∈ SF[H], ind (T + K) = ind (T ). Proof. (a) If T ∈ F [H], then there exist A ∈ B[H] and K1 ∈ B∞[H] such that A T = I + K1 . Hence A ∈ Fr [H]. Take any K ∈ B∞[H]. Set K2 = K1 + AK, which lies in B∞[H] because B∞[H] is an ideal of B[H]. Since A(T + K) = I + K2 , then T + K ∈ F [H]. Therefore T ∈ F [H] implies T + K ∈ F [H]. The converse also holds because T = (T + K) − K. Hence T ∈ F [H]

⇐⇒

T + K ∈ F [H].

Dually, if T ∈ Fr [H] and K ∈ B∞[H], then T ∗ ∈ F [H] and K ∗ ∈ B∞[H], so that T ∗ + K ∗ ∈ F [H], and hence T + K = (T ∗ + K ∗)∗ ∈ Fr [H]. Therefore T ∈ Fr [H]

⇐⇒

T + K ∈ Fr [H].

(b) First suppose T ∈ F [H] as above. By the Fredholm Alternative of Corollary 1.20, the operators I + K1 and I + K2 are both Fredholm with index zero (see Remark 5.3(e)). Therefore A T = I + K1 ∈ F[H] with ind (A T ) = 0 and A(T + K) = I + K2 ∈ F[H] with ind (A(T +K)) = 0. Recall: since T ∈ F [H], then A ∈ Fr [H]. If T ∈ F[H], then T + K ∈ F[H] by item (a). In this case, applying Theorem 5.4 for operators in Fr [H], ind (T + K) + ind (A) = ind (A(T + K)) = 0 = ind (A T ) = ind (A) + ind (T ). Since ind (T ) is finite (whenever T ∈ F[H]), then ind (A) = −ind (T ) is finite as well, and so we may subtract ind (A) to get ind (T + K) = ind (T ). On the other hand, if T lies in F [H]\F[H] = F [H]\Fr [H], then T + K lies in F [H]\Fr [H] by (a). In this case (see Remark 5.3(f)),

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ind (T + K) = −∞ = ind (T ). If T ∈ Fr [H]\F[H] and K ∈ B∞[H], then T ∗ ∈ F [H]\F[H] and K ∗ ∈ B∞[H]. So ind (T + K) = −ind (T ∗ + K ∗ ) = −ind (T ∗ ) = ind (T ).



Remark 5.7. (a) Index of A. We have established the following assertion in the preceding proof. If T ∈ F[H] and A ∈ B[H] are such that A T = I + K for some K ∈ B∞[H], then A ∈ F[H] and ind (A) = −ind (T ) = ind (T ∗ ). But if T ∈ F[H], then T ∈ Fr [H], so that T A = I + K for some K ∈ B∞[H]. Thus A ∈ F [H]. And a similar argument leads to the symmetric statement. If T ∈ F[H] and A ∈ B[H] are such that T A = I + K for some K ∈ B∞[H], then A ∈ F[H] and ind (A) = −ind (T ) = ind (T ∗ ). (b) Weyl Operator. A Weyl operator is a Fredholm operator with null index (equivalently, a semi-Fredholm operator with null index). Let   W[H] = T ∈ F[H] : ind (T ) = 0 denote the class of all Weyl operators from B[H]. Since T ∈ F[H] if and only if T ∗ ∈ F[H] and ind (T ∗ ) = −ind (T ), we conclude T ∈ W[H]

T ∗ ∈ W[H].

⇐⇒

According to items (d), (e), and (g) in Remark 5.3 we get: (i) every operator on a finite-dimensional space is a Weyl operator, dim H < ∞

W[H] = B[H],

=⇒

(ii) the Fredholm Alternative can be rephrased as K ∈ B∞[H] and λ = 0

=⇒

λI − K ∈ W[H],

(iii) every nonzero multiple of a Weyl operator is again a Weyl operator, T ∈ W[H]

=⇒

γ T ∈ W[H] for every γ ∈ C \{0}.

In particular, every nonzero scalar operator is a Weyl operator. In fact, the product of two Weyl operators is again a Weyl operator (by Theorem 5.4), S, T ∈ W[H]

=⇒

S T ∈ W[H].

Thus integral powers of Weyl operators are Weyl operators (Corollary 5.5), T ∈ W[H]

=⇒

T n ∈ W[H] for every n ∈ N 0 .

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141

On an infinite-dimensional space, the identity is Weyl but not compact; and the null operator is compact but not semi-Fredholm. Actually, T ∈ F[H] ∩ B∞[H]

⇐⇒

dim H < ∞

( ⇐⇒

T ∈ W[H] ∩ B∞[H] ).

(Set T = −K in Theorem 5.6, and for the converse see Remark 5.3(d).) Also, if T is normal, then N (T ) = N (T ∗ T ) = N (T T ∗ ) = N (T ∗ ) by Lemma 1.4, and so every normal Fredholm operator is Weyl, T normal in F[H]

=⇒

T ∈ W[H].

Since T ∈ B[H] is invertible if and only N (T ) = {0} and R(T ) = H (Theorem 1.1), which happens if and only if T ∗ ∈ B[H] is invertible (Proposition 1.L), it follows by Corollary 5.2 that every invertible operator is Weyl, T ∈ G[H]

=⇒

T ∈ W[H].

Moreover, by Theorem 5.6, for every compact K ∈ B∞[H], T ∈ W[H]

=⇒

T + K ∈ W[H].

5.2 Essential (Fredholm) Spectrum and Spectral Picture An element a in a unital algebra A is left invertible if there is an element a in A (a left inverse of a) such that a a = 1 where 1 stands for the identity in A, and it is right invertible if there is an element ar in A (a right inverse of a) such that a ar = 1. An element a in A is invertible if there is an element a−1 in A (the inverse of a) such that a−1 a = a a−1 = 1. Thus a in A is invertible if and only if it has a left inverse a in A and a right inverse ar in A, which coincide with its inverse a−1 in A (since ar = a a ar = a ). Lemma 5.8. Take an operator S in the unital Banach algebra B[H]. (a) S is left invertible if and only if it is injective with a closed range (i.e., N (S) = {0} and R(S)− = R(S)). (b) S is left (right) invertible if and only if S ∗ is right (left) invertible. (c) S is right invertible if and only if it is surjective (i.e., R(S) = H). Proof. An operator S ∈ B[H] has a bounded inverse on its range R(S) if there exists S −1 ∈ B[R(S), H] such that S −1S = I ∈ B[H], and this implies SS −1 = I ∈ B[R(S)] (cf. Section 1.2). Moreover, S ∈ B[H] has a bounded inverse on its range if and only if N (S) = {0} and R(S) is closed (Theorem 1.2). Claim. S is left invertible if and only if it has a bounded inverse on its range. Proof. If S ∈ B[H] is a left inverse of S ∈ B[H], then S S = I ∈ B[H]. Hence the restriction S |R(S) ∈ B[R(S), H] is such that S |R(S) S = I ∈ B[H]. Thus

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S −1 = S |R(S) ∈ B[R(S), H] is a bounded inverse of S on R(S). Conversely, if S −1 ∈ B[R(S), H] is a bounded inverse of S on R(S), then R(S) is closed (cf. Theorem 1.2). Take the orthogonal projection E ∈ B[H] onto the subspace R(S). Consider the extension S −1 E ∈ B[H] of S −1 ∈ B[R(S), H] over the whole space H, as in Proposition 1.G(b). Thus S −1 E S = S −1 S = I ∈ B[H]. Then S = S −1 E ∈ B[H] is a left inverse of S ∈ B[H]. This proves the claim. (a) But S has a bounded inverse on its range if and only if it is injective (i.e., N (S) = {0}) and R(S) is closed, according to Theorem 1.2. (b) Since S S = I (S Sr = I) if and only if S ∗ S∗ = I (Sr∗ S ∗ = I), it follows that S is left (right) invertible if and only if S ∗ is right (left) invertible. (c) Thus, according to items (a) and (b), S is right invertible if and only if N (S ∗ ) = {0} (i.e., N (S ∗ )⊥ = H) and R(S ∗ ) is closed, or equivalently (cf.  Lemmas 1.4 and 1.5) R(S) = R(S)− = H, which means R(S) = H. Let T be an operator in B[H]. The left spectrum σ (T ) and the right spectrum σr (T ) of T ∈ B[H] are the sets   σ (T ) = λ ∈ C : λI − T is not left invertible ,   σr (T ) = λ ∈ C : λI − T is not right invertible , and so the spectrum σ(T ) of T ∈ B[H] is given by   σ(T ) = λ ∈ C : λI − T is not invertible = σ (T ) ∪ σr (T ). Corollary 5.9. Take an arbitrary operator T ∈ B[H].   σ (T ) = λ ∈ C : R(λI − T ) is not closed or N (λI − T ) = {0} = σ(T )\σR1 (T ) = σAP (T ) ⊆ σ(T ),   σr (T ) = λ ∈ C : R(λI − T ) is not closed or N (λI − T ∗ ) = {0}   = λ ∈ C : R(λI − T ) is not closed or R(λI − T )− = H   = λ ∈ C : R(λI − T ) = H = σ(T )\σP1 (T ) = σAP (T ∗ )∗ ⊆ σ(T ). Proof. Take T ∈ B[H] and λ ∈ C . Set S = λI − T in B[H]. By Lemma 5.8, S is not left invertible if and only if R(S) is not closed or S is not injective (i.e., or N (S) = 0). This proves the first expression for σ (T ). By Lemma 5.8, S is not right invertible if and only if R(S) = H, which means R(S) = R(S)− or R(S)− = H, which is equivalent to R(S) = R(S)− or N (S ∗ ) = {0} (Lemma 1.5), thus proving the first expressions for σr (T ). The remaining identities and inclusions follow from Theorem 2.6 (see also the diagram of Section 2.2).  Therefore

5.2 Essential (Fredholm) Spectrum and Spectral Picture

143

σ (T ) and σr (T ) are closed subsets of σ(T ). Indeed, the approximate point spectrum σAP (T ) is a closed set (cf. Theorem 2.5). Recall from the remarks following Theorems 2.5 and 2.6 that σP1 (T ) and σR1 (T ) are open sets, and so σAP (T ) = σ(T )\σR1 (T ) = σ(T ) ∩ (C \σR1 (T )) and σAP (T ∗ )∗ = σ(T )\σP1 (T ) = σ(T ) ∩ (C \σP1 (T )) are closed (and bounded) sets in C , since σ(T ) is closed (and bounded); and so is the intersection  , σ (T ) ∩ σr (T ) = σ(T ) σP1 (T ) ∪ σR1 (T ) = σAP (T ) ∩ σAP (T ∗ )∗ . Actually, as we saw before, σP1 (T ) and σR1 (T ) are open subsets of C , and so σ (T ) and σr (T ) are closed subsets of C (and this follows because T lies in the unital Banach algebra B[H], and therefore σ(T ) is a compact set, which in fact happens in any unital Banach algebra). Also, since (S S)∗ = S ∗S∗ and (S Sr )∗ = Sr∗ S ∗, then S is a left inverse of S if and only if S∗ is a right inverse of S ∗, and Sr is a right inverse of S if and only if Sr∗ is a left inverse of S ∗. Therefore (as we can infer from Corollary 5.9 as well) σ (T ) = σr (T ∗ )∗

and

σr (T ) = σ (T ∗ )∗ .

Consider the Calkin algebra B[H] /B∞[H] — the quotient algebra of B[H] modulo the ideal B∞[H] of all compact operators. If dim H < ∞, then all operators are compact, and so B[H] /B∞[H] is trivially null. Thus if the Calkin algebra is brought into play, then the space H is assumed infinite-dimensional (i.e., dim H = ∞). Since B∞[H] is a subspace of B[H], then B[H] /B∞[H] is a unital Banach algebra whenever H is infinite-dimensional . Moreover, consider the natural map (or the natural quotient map) π : B[H] → B[H] /B∞[H] which is defined by (see, e.g., [78, Example 2.H])   π(T ) = [T ] = S ∈ B[H] : S = T + K for some K ∈ B∞[H] = T + B∞[H] for every T in B[H]. The origin of the linear space B[H] /B∞[H] is [0] = B∞[H] ∈ B[H] /B∞[H], the kernel of the natural map π is   N (π) = T ∈ B[H] : π(T ) = [0] = B∞[H] ⊆ B[H], and π is a unital homomorphism. Indeed, since B∞[H] is an ideal of B[H], π(T + T  ) = (T + T  ) + B∞[H] = (T + B∞[H]) + (T  + B∞[H]) = π(T ) + π(T  ), π(T T  ) = (T T  ) + B∞[H] = (T + B∞[H]) (T  + B∞[H]) = π(T ) π(T  ),

for every T, T  ∈ B[H], and π(I) = [I] is the identity element of the algebra B[H] /B∞[H]. Furthermore, the norm on B[H] /B∞[H] is given by [T ] =

inf

K∈B∞[H]

T + K ≤ T ,

so that π is a contraction (see, e.g., [78, Section 4.3]). Theorem 5.10. Take any operator T ∈ B[H]. T ∈ F [H] (or T ∈ Fr [H]) if and only if π(T ) is left (or right) invertible in the Calkin algebra B[H]/B∞[H].

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5. Fredholm Theory in Hilbert Space

Proof. We show the equivalence of the following assertions. (a) T ∈ F [H]. (b) There is a pair of operators {A, K} with A ∈ B[H] and K ∈ B∞[H] such that A T = I + K. (c) There exists a quadruple of operators {A, K1 , K2 , K3 } with A ∈ B[H] and K1 , K2 , K3 ∈ B∞[H] such that (A + K1 )(T + K2 ) = I + K3 . (d) π(A)π(T ) = π(I), identity in B[H]/B∞[H], for some π(A) in B[H]/B∞[H]. (e) π(T ) is left invertible in B[H]/B∞[H]. By definition, (a) and (b) are equivalent, and (b) implies (c) trivially. If (c) holds, then there exist B ∈ [A] = π(A), S ∈ [T ] = π(T ), and J ∈ [I] = π(I) for which BS = J. Therefore π(A)π(T ) = [A] [T ] = [B] [S] = π(B)π(S) = π(BS) = π(J) = [J] = [I] = π(I), and so (c) implies (d). Now observe that X ∈ π(A)π(T ) = π(A T ) if and only if X = A T + K1 for some K1 ∈ B∞[H], and X ∈ π(I) if and only if X = I + K2 for some K2 ∈ B∞[H]. Therefore if π(A)π(T ) = π(I), then X = A T + K1 if and only if X = I + K2 , and hence A T = I + K with K = K2 − K1 ∈ B∞[H]. Thus (d) implies (b). Finally, (d) and (e) are equivalent by the definition of left invertibility. Outcome: T ∈ F [H] if and only if π(T ) is left invertible in B[H]/B∞[H]. Dually, T ∈ Fr [H] if and only if π(T ) is right invertible in B[H]/B∞[H].  The essential spectrum (or the Calkin spectrum) σe (T ) of T ∈ B[H] is the spectrum of π(T ) in the unital Banach algebra B[H] /B∞[H], σe (T ) = σ(π(T )), and so σe (T ) is a compact subset of C . Similarly, the left essential spectrum σe (T ) and the right essential spectrum σre (T ) of T ∈ B[H] are defined as the left and the right spectrum of π(T ) in the Calkin algebra B[H] /B∞[H]: σe (T ) = σ (π(T )) Hence

and

σre (T ) = σr (π(T )).

σe (T ) = σe (T ) ∪ σre (T ).

By using only the definitions of left and right semi-Fredholm operators, and of left and right essential spectra, we get the following characterization. Corollary 5.11. If T ∈ B[H], then   σe (T ) = λ ∈ C : λI − T ∈ B[H]\F [H] ,   σre (T ) = λ ∈ C : λI − T ∈ B[H]\Fr [H] .

5.2 Essential (Fredholm) Spectrum and Spectral Picture

145

Proof. Take T ∈ B[H]. According to Theorem 5.10, λI − T ∈ F [H] if and only if π(λI − T ) is not left invertible in B[H]/B∞[H], which means (definition of left spectrum) λ ∈ σ (π(T )). But σe (T ) = σ (π(T )) — definition of left essential spectrum. Thus λ ∈ σe (T ) if and only if λI − T ∈ F [H]. Dually,  λ ∈ σre (T ) if and only if λI − T ∈ Fr [H]. Corollary 5.12. (Atkinson Theorem).   σe (T ) = λ ∈ C : λI − T ∈ B[H]\F[H] . Proof. The expressions for σe (T ) and σre (T ) in Corollary 5.11 lead to the claimed identity since σe (T ) = σe (T ) ∪ σre (T ) and F[H] = F [H] ∩ Fr [H].  Thus the essential spectrum is the set of all scalars λ for which λI − T is not Fredholm. The following equivalent version is also frequently used [8]. Corollary 5.13. (Atkinson Theorem). An operator T ∈ B[H] is Fredholm if and only if its image π(T ) in the Calkin algebra B[H] /B∞[H] is invertible. Proof. Straightforward from Theorem 5.10: T is Fredholm if and only if it is both left and right semi-Fredholm, and π(T ) is invertible in B[H]/B∞[H] if  and only if it is both left and right invertible in B[H]/B∞[H]. This is usually referred to by saying that T is Fredholm if and only if T is essentially invertible. Thus the essential spectrum σe (T ) is the set of all scalars λ for which λI − T is not essentially invertible (i.e., λI − T is not Fredholm), and so the essential spectrum is also called the Fredholm spectrum. As we saw above (and as happens in any unital Banach algebra — in particular, in B[H]), the sets σ (T ) and σr (T ) are closed in C because σ(T ) is a compact subset of C . Similarly, in the unital Banach algebra B[H] /B∞[H], σe (T ) and σre (T ) are closed subsets of σe (T ) because σe (T ) = σ(π(T )) is a compact set in C . From Corollary 5.11 we get  C \σe (T ) = λ ∈ C : λI − T ∈ F [H]},  C \σre (T ) = λ ∈ C : λI − T ∈ Fr [H]}, which are open subsets of C . Thus, since SF[H] = F [H] ∪ Fr [H], ,        C σe (T )∩σre (T ) = C \σe (T ) ∪ C \σre (T ) = λ ∈ C : λI−T ∈ SF[H] , which is again an open subset of C . Therefore the intersection   σe (T ) ∩ σre (T ) = λ ∈ C : λI − T ∈ B[H]\SF[H] is a closed subset of C . Since F[H] = F [H] ∩ Fr [H], the complement of the union σe (T ) ∪ σre (T ) is given by (cf. Corollary 5.12)

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5. Fredholm Theory in Hilbert Space

,

 σe (T ) ∪ σre (T )       = C \σe (T ) ∩ C \σre (T ) = λ ∈ C : λI − T ∈ F[H] ,

C \σe (T ) = C

which is an open subset of C . Corollary 5.14. If T ∈ B[H], then   σe (T ) = λ ∈ C : R(λI −T ) is not closed or dim N (λI −T ) = ∞ ⊆ σ(T ),   σre (T ) = λ ∈ C : R(λI−T ) is not closed or dim N (λI−T ∗ ) = ∞ ⊆ σ(T ). Proof. Theorem 5.1 and Corollary 5.11. For the inclusions in σ(T ) see the diagram of Section 2.2. (See also Remark 5.3(b).)  By Corollaries 5.9 and 5.14, σe (T ) ⊆ σ (T ) Thus

σre (T ) ⊆ σr (T ).

and

σe (T ) ⊆ σ(T ).

Also, still by Corollary 5.14, σe (T ) = σre (T ∗ )∗ and hence

σre (T ) = σe (T ∗ )∗ ,

and

σe (T ) = σe (T ∗ )∗ .

Moreover, by the previous results and the diagram of Section 2.2, σC (T ) ⊆ σe (T ) ⊆ σ (T ) = σ(T )\σR1 (T ) = σAP (T ), σC (T ) = σC (T ∗ )∗ ⊆ σe (T ∗ )∗ = σre (T ) ⊆ σr (T ) = σ(T )\σP1(T ) = σAP (T ∗ )∗, and so

σe (T ) ∩ σre (T ) ⊆ σ(T )

,

 σP1 (T ) ∪ σR1 (T ) ,

which is an inclusion of closed sets. Therefore , ,   σP1 (T ) ∪ σR1 (T ) ⊆ σ(T ) σe (T ) ∩ σre (T ) ⊆ C σe (T ) ∩ σre (T ) , which is an inclusion of open sets. Remark 5.15. (a) Finite Dimension. For any T ∈ B[H], σe (T ) = ∅

⇐⇒

dim H = ∞.

Actually, since σe (T ) was defined as the spectrum of π(T ) in the Calkin algebra B[H]/B∞[H], which is a unital Banach algebra if and only if H is infinitedimensional (in a finite-dimensional space all operators are compact), and since spectra are nonempty in a unital Banach algebra, we get

5.2 Essential (Fredholm) Spectrum and Spectral Picture

dim H = ∞

=⇒

147

σe (T ) = ∅.

However, if we take the equivalent expression of the essential spectrum in Corollary 5.12, namely σe (T ) = {λ ∈ C : λI − T ∈ B[H]\F[H]}, then the converse holds by Remark 5.3(d): dim H < ∞

=⇒

σe (T ) = ∅.

(b) Atkinson Theorem. An argument similar to the one in the proof of Corollary 5.13, using each expression for σe (T ) and σre (T ) in Corollary 5.11 separately, leads to the following result. An operator T ∈ B[H] lies in F [H] (or in Fr [H]) if and only if its image π(T ) in the Calkin algebra B[H] /B∞[H] is left (right) invertible. (c) Compact Perturbation. For every K ∈ B∞[H], σe (T + K) = σe (T ). Indeed, take an arbitrary compact operator K ∈ B∞[H]. Since π(T + K) = π(T ), we get σe (T ) = σ(π(T )) = σ(π(T + K)) = σe (T + K) by the definition of σe (T ) in B[H]/B∞[H]. Similarly, by the definitions of σe (T ) and σre (T ) in B[H]/B∞[H] we get σe (T ) = σ (π(T )) = σ (π(T + K)) = σe (T + K) and σre (T ) = σr (π(T )) = σr (π(T + K)) = σre (T + K). Thus σe (T + K) = σe (T )

and

σre (T + K) = σre (T ).

Such invariance is a major feature of Fredholm theory. Now take any operator T in B[H]. For each k ∈ Z \{0} set   σk (T ) = λ ∈ C : λI − T ∈ SF[H] and ind (λI − T ) = k . As we have seen before (Section 2.7), a component of a set in a topological space is any maximal connected subset of it. A hole of a set in a topological space is any bounded component of its complement. If a set has an open complement (i.e., if it is closed), then a hole of it must be open. We show that each σk (T ) with finite index is a hole of σe (T ) which lies in σ(T ) (i.e., if k ∈ Z \{0}, then σk (T ) ⊆ σ(T )\σe (T ) is a bounded component of the open set C \σe (T )), and hence it is an open set. Moreover, we also show that σ+∞ (T ) and σ−∞ (T ) are holes of σre (T ) and of σe (T ) which lie in σe (T ) and σre (T ) (in fact, σ+∞ (T ) = σe (T )\σre (T ) is a bounded component of the open set C \σre (T ), and σ−∞ (T ) = σre (T )\σe (T ) is a bounded component of the open set C \σe (T )), and hence they are open sets. The sets σ+∞ (T ) and σ−∞ (T ) — which are holes of σre (T ) and of σe (T ) but are not holes of σe (T ) — are called pseudoholes of σe (T ). Let σPF (T ) denote the set of all eigenvalues of T of finite multiplicity,

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5. Fredholm Theory in Hilbert Space

  σPF (T ) = λ ∈ σP (T ) : dim N (λI − T ) < ∞   = λ ∈ C : 0 < dim N (λI − T ) < ∞ . According to the diagram of Section 2.2 and Corollary 5.14, σAP (T ) = σe (T ) ∪ σPF (T ). Theorem 5.16. Take an arbitrary operator T ∈ B[H]. For each k ∈ Z \{0}, σk (T ) = σ−k (T ∗ )∗ .

(a)

If k ∈ Z \{0}, then   (b) σk (T ) = λ ∈ C : λI − T ∈ F[H] and ind (λI − T ) = k + 0 < k, σPF (T ), ⊆ ∗ ∗ σPF (T ) , k < 0, so that σk (T ) ⊆ σP (T ) ∪ σR (T ) ⊆ σ(T ), and    σk (T ) = λ ∈ C : λI − T ∈ F[H] and ind (λI − T ) = 0 k∈Z \{0}   = λ ∈ C : λI − T ∈ F[H]\W[H]   ⊆ λ ∈ C : λI − T ∈ F[H] = C \σe (T ). If k = ±∞, then (c)

σ+∞ (T ) ⊆ σP (T )\σPF (T ) Thus

 k∈Z \{0}

and

σ−∞ (T ) ⊆ σP (T ∗ )∗ \σPF (T ∗ )∗ .

σk (T ) ⊆ σP (T ) ∪ σR (T ) ⊆ σ(T ).

Moreover ,   σ+∞ (T ) = λ ∈ C : λI − T ∈ SF[H] and ind (λI − T ) = +∞   = λ ∈ C : λI − T ∈ Fr [H]\F [H] = σe (T )\σre (T ) = σe (T )\σre (T ) ⊆ C \σre (T ),   σ−∞ (T ) = λ ∈ C : λI − T ∈ SF[H] and ind (λI − T ) = −∞   = λ ∈ C : λI − T ∈ F [H]\Fr [H] = σre (T )\σe (T ) = σe (T )\σe (T ) ⊆ C \σe (T ), which are all open subsets of C , and hence σ+∞ (T ) ⊆ σe (T ) Furthermore, with

and

σ−∞ (T ) ⊆ σe (T ).

5.2 Essential (Fredholm) Spectrum and Spectral Picture

149

+

(d) Z \{0} = N = N ∪ {+∞} (the set of all extended positive integers), and − Z \{0} = −N = −N ∪ {−∞} (the set of all extended negative integers), we get  σk (T ) σP1 (T ) ⊆ + k∈Z \{0}   = λ ∈ C : λI − T ∈ SF[H] and 0 < ind (λI − T ) ,  σR1 (T ) ⊆ σk (T ) − k∈Z \{0}   = λ ∈ C : λI − T ∈ SF[H] and ind (λI − T ) < 0 , and so  σk (T ) σP1 (T ) ∪ σR1 (T ) ⊆ k∈Z \{0}   = λ ∈ C : λI − T ∈ SF[H] and ind (λI − T ) = 0 ,    ⊆ λ ∈ C : λI − T ∈ SF[H] = C σe (T ) ∩ σre (T ) , which are all open subsets of C . (e) For any k ∈ Z \{0} the set σk (T ) is a hole of σe (T ) ∩ σre (T ). If k is finite (i.e., if k ∈ Z \{0}), then σk (T ) is a hole of σe (T ) = σe (T ) ∪ σre (T ). Otherwise, σ+∞ (T ) is a hole of σre (T ) and σ−∞ (T ) is a hole of σe (T ). Thus σk (T ) is an open set for every k ∈ Z \{0}. Proof. (a) λI − T ∈ SF[H] if and only if λI − T ∗ ∈ SF[H], and ind (λI − T ) = −ind (λI − T ∗ ) (cf. Section 5.1). Thus λ ∈ σk (T ) if and only if λ ∈ σ−k (T ∗ ). (b) The expression for σk (T ) — with SF[H] replaced with F[H] — holds since for finite k all semi-Fredholm operators are Fredholm (cf. Remark 5.3(f)). Take any k ∈ Z \{0}. If k > 0 and λ ∈ σk (T ), then 0 < ind (λI − T ) < ∞. Hence 0 ≤ dim N (λI − T ∗ ) < dim N (λI − T ) < ∞, and so 0 < dim N (λI − T ) < ∞. Thus λ is an eigenvalue of T of finite multiplicity. Dually, if k < 0 (i.e., −k > 0) and λ ∈ σk (T ) = σ−k (T ∗ )∗ , then λ is an eigenvalue of T ∗ of finite multiplicity. Also,   C \σe (T ) = λ ∈ C : λI − T ∈ F[H] by Corollary 5.12. This closes the proof of (b). (c) According to Corollary 5.2 and by the definitions of σ+∞ (T ) and σ−∞ (T ), λ ∈ C lies in σ+∞ (T ) if and only if R(λI − T ) is closed, dim N (λI − T ∗ ) is finite, and dim N (λI − T ) is infinite. Thus if λ ∈ σ+∞ (T ), then 0 ≤ dim N (λI − T ∗ ) < dim N (λI − T ) = ∞, and so λ is an eigenvalue of T of infinite multiplicity. Dually, if λ ∈ σ−∞ (T ) = σ+∞ (T ∗ )∗ , then λ is an eigenvalue of T ∗ of infinite multiplicity. Thus the

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5. Fredholm Theory in Hilbert Space

expressions for σ+∞ (T ) and σ−∞ (T ) follow from Remark 5.3(f) and Corollary 5.11, since σe (T ) = σe (T ) ∪ σre (T ). Also, σ+∞ (T ) and σ−∞ (T ) are open sets in C because σe (T ), σre (T ) and so σe (T ) are all closed sets in C . (d) Recall: M− = H if and only if M⊥ = {0} (Section 1.3). So R(λI − T )− = H if and only if N (λI − T ∗ ) = {0} (Lemma 1.4) or, equivalently, if and only if dim N (λI − T ∗ ) = 0. Thus, according to the diagram of Section 2.2,   σP1(T ) = λ ∈ C : R(λI − T ) is closed, 0 = dim N (λI − T ∗ ) = dim N (λI − T )   = λ ∈ C : R(λI − T ) is closed, 0 = dim N (λI − T ∗ ), ind (λI − T ) > 0 . Therefore, by Corollary 5.2,   σP1 (T ) = λ ∈ C : λI − T ∈ SF[H], 0 = dim N (λI − T ∗ ), ind (λI − T ) > 0    σk (T ). ⊆ λ ∈ C : λI − T ∈ SF[H], ind (λI − T ) > 0 = + k∈Z \{0}

Dually (cf. item (a) and Theorem 2.6), 

∗ σR1 (T ) = σP1 (T ∗ )∗ ⊆ σk (T ∗ ) + k∈Z \{0} 

∗  = σ−k (T )∗ = + − k∈Z \{0}

k∈Z \{0}

σk (T ).

Finally, as we had verified before, 

C \σe (T ) ∩ σre (T ) = λ ∈ C : λI − T ∈ SF[H]



according to Corollary 5.11. (e) By item (d) we get    σk (T ) ⊆ C \σe (T ) ∩ σre (T ) = λ ∈ C : λI − T ∈ SF[H] . k∈Z \{0}

Take an arbitrary λ ∈ k∈Z \{0} σk (T ) so that λI − T ∈ SF[H]. By Proposition 5.D, ind (ζI − T ) is constant for every ζ in a punctured disk Bε (λ)\{λ} centered at λ for some positive radius ε. Thus, for every pair of distinct extended integers j, k ∈ Z \{0}, the sets σj (T ) and σk (T ) are not only disjoint, but they are (bounded) components (i.e., maximal connected subsets) of the open set C \σe (T ) ∩ σre (T ), and hence they are holes of the closed set σe (T ) ∩ σre (T ). By item (b), if j, k are finite (i.e., if j, k ∈ Z \{0}), then σj (T ) and σk (T ) are (bounded) components of the open set C \σe (T ) = C \σe (T ) ∪ σre (T ) and, in this case, they are holes of the closed set σe (T ). By item (c), σ+∞ (T ) and σ−∞ (T ) are (bounded) components of the open sets C \σre (T ) and C \σe (T ), and so σ+∞ (T ) and σ−∞ (T ) are holes of the closed sets σre (T ) and σe (T ), respectively. Thus in any case σk (T ) is an open subset of C . Therefore

 k∈Z \{0} σk (T ) is open as claimed in (d).

5.2 Essential (Fredholm) Spectrum and Spectral Picture

151

The following figure illustrates a simple instance of holes and pseudoholes of the essential spectrum. Observe that the right and left essential spectra σe (T ) and σre (T ) differ only by pseudoholes σ±∞ (T ). σe (T ) = σe (T ) ∪ σre (T ) | | | |

| |  || σk (T )

σe (T )

| | | |

♣

 | | | |

| |  ||

♠

| |  ||

σre (T )

σe (T ) ∩ σre (T )

♣

| | | |

σ−∞ (T ) = σre (T )\σe (T )

♣

♠

| |  ||

♠

σ+∞ (T ) = σe (T )\σre (T )

Fig. § 5.2. Holes and pseudoholes of the essential spectrum Now for k = 0 we define the set σ0 (T ) as the following subset of σ(T ).   σ0 (T ) = λ ∈ σ(T ) : λI − T ∈ SF[H] and ind (λI − T ) = 0   = λ ∈ σ(T ) : λI − T ∈ F[H] and ind (λI − T ) = 0   = λ ∈ σ(T ) : λI − T ∈ W[H] . Corollary 5.2, the diagram of Section 2.2, and Theorem 2.6 give an equivalent description for the set σ0 (T ) ⊆ σ(T ):  σ0 (T ) = λ ∈ σ(T ) : R(λI − T ) is closed and  dim N (λI − T ) = dim N (λI − T ∗ ) < ∞  = λ ∈ σP (T ) : R(λI − T ) = R(λI − T )− = H and  dim N (λI − T ) = dim N (λI − T ∗ ) < ∞   = λ ∈ σP4 (T ) : dim N (λI − T ) = dim N (λI − T ∗ ) < ∞ ,   where σP4 (T ) = λ ∈ σP (T ) : R(λI − T )− = R(λI − T ) = H . In fact,

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5. Fredholm Theory in Hilbert Space

σ0 (T ) ⊆ σPF (T ) ∩ σPF (T ∗ )∗ ⊆ σP (T ) ∩ σP (T ∗ )∗ . Thus

σ0 (T ) = σ0 (T ∗ )∗ .

Observe that if λ ∈ σ0 (T ), then the definition of σ0 (T ) forces λ to be in σ(T ), while in the definition of σk (T ) for k = 0 this happens naturally. In fact, the definition of σ0 (T ) forces λ to be in σP (T ) (specifically, in σP4 (T )). Remark 5.17. (a) Fredholm Alternative. If K ∈ B∞[H] and λ = 0, then λI − K is Fredholm with ind (λI − K) = 0 (i.e., λI − K is Weyl; in symbols, λI − K ∈ W[H]). This is a restatement of the Fredholm Alternative as in Remark 5.3(e). Since σ0 (K) = {λ ∈ σ(K) : λI − K ∈ W[H]}, the Fredholm Alternative can be restated once again as follows (compare with Theorem 2.18): K ∈ B∞[H]

=⇒

σ(K)\{0} = σ0 (K)\{0}.

(b) Finite Dimension. If H is finite-dimensional, then every operator is Weyl (i.e., if dim H < ∞, then W[H] = B[H] — cf. Remark 5.3(d) or 5.7(c)). Thus, dim H < ∞

=⇒

σ(T ) = σ0 (T ).

(c) Compact Perturbation. The expressions for σk (T ), σ+∞ (T ), and σ−∞ (T ), viz., σk (T ) = {λ ∈ C : λI − T ∈ F[H] and ind (λI − T ) = k} for k ∈ Z \{0}, σ+∞ (T ) = σe (T )\σre (T ), and σ−∞ (T ) = σre (T )\σe (T ) as in Theorem 5.16, together with the results of Theorem 5.4, ensure that the sets σk (T ) for each k ∈ Z \{0}, σ+∞ (T ), and σ−∞ (T ), are invariant under compact perturbation as well (see Remark 5.15(c)). In other words, for every K ∈ B∞[H] σk (T + K) = σk (T ) for every k ∈ Z \{0}. Such invariance, however, does not apply to σ0 (T ). Example: if dim H < ∞ (where all operators are compact), and if T = I, then σ0 (T ) = σ(T ) = {1} and σ0 (T − T ) = σ(T − T ) = {0}. Thus there is a compact K = −T for which {0} = σ0 (T + K) = σ0 (T ) = {1}. This will be generalized in the proof Theorem 5.24, where it is proved that a nonempty σ0 (T ) is never invariant under compact perturbation. The partition of the spectrum σ(T ) obtained in the next corollary is called the Spectral Picture of T [97]. Corollary 5.18. (Spectral Picture). If T ∈ B[H], then  σk (T ) σ(T ) = σe (T ) ∪ k∈Z and   σe (T ) = σe (T ) ∩ σre (T ) ∪ σ+∞ (T ) ∪ σ−∞ (T ),

5.3 Riesz Points and Weyl Spectrum

where σe (T ) ∩

 k∈Z

153

σk (T ) = ∅,

σk (T ) ∩ σj (T ) = ∅ for every j, k ∈ Z such that j = k, and

    σe (T ) ∩ σre (T ) ∩ σ+∞ (T ) ∪ σ+∞ = ∅.

Proof. The collection {σk (T )}k∈Z is clearly pairwise disjoint. By Theorem 5.16(b) and the definition of σ0 (T ),    σk (T ) = λ ∈ σ(T ) : λI − T ∈ F[H] . k∈Z

Since σe (T ) ⊆ σ(T ) we get   / F[H] σe (T ) = λ ∈ σ(T ) : λI − T ∈ according to Corollary 5.12. This leads to the following partition of the spec trum: σ(T ) = σe (T ) ∪ k∈Z σk (T ) with σe (T ) ∩ k∈Z σk (T ) = ∅. Moreover, since σe (T ) = σe (T ) ∪ σre (T ) ⊆ σ(T ) we also get   / SF[H] ⊆ σe (T ) σe (T ) ∩ σre (T ) = λ ∈ σ(T ) : λI − T ∈ from Corollary 5.11. Furthermore, according to Theorem 5.16(c),   σ+∞ (T ) ∪ σ−∞ (T ) = λ ∈ σ(T ) : λI − T ∈ SF[H]\F[H] partition of the and σ+∞ (T ) ∩ σ−∞ (T ) = ∅. Thus we  obtain the following  (T ) = σ (T ) ∩ σ (T ) ∪ σ (T ) ∪ σ−∞ (T ) essential spectrum as well: σ e e re +∞      with σe (T ) ∩ σre (T ) ∩ σ+∞ (T ) ∪ σ+∞ = ∅. As we saw above, the collection {σk (T )}k∈Z \{0} consists of pairwise disjoint open sets (cf. Theorem 5.16), which are subsets of σ(T )\σe (T ). These are the holes of the essential spectrum σe (T ). Moreover, σ±∞ (T ) also are open sets (cf. Theorem 5.16), which are subsets of σe (T ). These are the pseudoholes of σe (T ) (they are holes of σre (T ) and σe (T )). Thus the spectral picture of T consists of the essential spectrum σe (T ), the holes σk (T ) and pseudoholes σ±∞ (T ) (to each is associated a nonzero index k in Z \{0}), and the set σ0 (T ). It is worth noticing that any spectral picture can be attained [28] by an operator in B[H] (see also Proposition 5.G).

5.3 Riesz Points and Weyl Spectrum The set σ0 (T ) will play a rather important role in the sequel. It consists of an open set τ0 (T ) and the set π0 (T ) of isolated points of σ(T ) the Riesz idempotents associated with which have finite rank. The next proposition says that

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5. Fredholm Theory in Hilbert Space

π0 (T ) is precisely the set of all isolated points of σ(T ) which lie in σ0 (T ). Let Eλ ∈ B[H] be the Riesz idempotent associated with an isolated point λ of the spectrum σ(T ) of an operator T ∈ B[H]. Theorem 5.19. If λ is an isolated point of σ(T ), then the following assertions are pairwise equivalent. (a) λ ∈ σ0 (T ). (b) λ ∈ σe (T ). (c) λI − T ∈ F[H]. (d) R(λI − T ) is closed and dim N (λI − T ) < ∞. (e) λ ∈ / σe (T ) ∩ σre (T ). (f) dim R(Eλ ) < ∞. Proof. Let T be an operator on an infinite-dimensional complex Hilbert space. (a) ⇒(b) ⇒(c). By definition, we have σ0 (T ) = {λ ∈ σ(T ) : λI − T ∈ W[H]} = {λ ∈ σ(T ) : λI − T ∈ F[H] and ind (λI − T ) = 0}. Thus (a) implies (c) tautologically, and (c) is equivalent to (b) by Corollary 5.12. (c) ⇒(d). By Corollary 5.2, λI − T ∈ F[H] if and only if R(λI − T ) is closed, dim N (λI − T ) < ∞, and dim N (λI − T ∗ ) < ∞. Thus (c) implies (d) trivially. (d) ⇒(e). By Corollary 5.14, σe (T ) ∩ σre (T ) = {λ ∈ C : R(λI − T ) is not closed or dim N (λI − T ) = dim N (λI − T ∗ ) = ∞}. Thus (d) implies (e). From now on suppose λ is an isolated point of the spectrum σ(T ) of T .

(e) ⇒(a). If λ ∈ σ(T )\σ0 (T ), then λ ∈ σe (T ) ∪ k∈Z \{0} σk (T ) by Corollary

5.18, where k∈Z \{0} σk (T ) is an open subset of C according to Theorem 5.16, and so it has no isolated point. Hence if λ is an isolated pont of σ(T )\σ0 (T ), then λ lies in σe (T ) = (σe (T ) ∩ σre (T )) ∪ (σ+∞ (T ) ∪ σ−∞ (T )), where the set σ+∞ (T ) ∪ σ−∞ (T ) is open in C , and hence it likewise has no isolated point (cf. Theorem 5.16 and Corollary 5.18 again). Therefore λ ∈ σe (T ) ∩ σre (T ). Outcome: if λ is an isolated point of σ(T ) and λ ∈ / σe (T ) ∩ σre (T ), then λ ∈ σ0 (T ). Thus (e) implies (a). So assertions (a), (b), (c), (d), and (e) are pairwise equivalent. As we will see next, (f) is also equivalent to them. (f) ⇒(d). If λ is an isolated point of σ(T ), then Δ = {λ} is a spectral set for the operator T . Thus (f) implies (d) by Corollary 4.22. (a) ⇒(f). Let π : B[H] → B[H] /B∞[H] be the natural map of B[H] into the Calkin algebra B[H] /B∞[H], which is a unital homomorphism of the unital Banach algebra B[H] to the unital Banach algebra B[H]/B∞[H], whenever the complex Hilbert space H is infinite-dimensional.

5.3 Riesz Points and Weyl Spectrum

155

Claim. If T ∈ B[H] and ζ ∈ ρ(T ), then ζ ∈ ρ(π(T )) and   −1  π (ζI − T )−1 = ζ π(I) − π(T ) . Proof. Since B∞[H] is an ideal of B[H],      π (ζI − T )−1 π(ζI − T ) = (ζI − T )−1 + B∞[H] (ζI − T ) + B∞[H] = (ζI − T )−1 (ζI − T ) + B∞[H] = I + B∞[H] = π(I) = [I], where π(I) = [I] is the identity in B[H] /B∞[H]. Therefore   −1   −1 −1  π (ζI − T )−1 = π(ζI − T ) = ζI − T + B∞[H] = ζ π(I) − π(T ) , completing the proof of the claimed result. Suppose an isolated point λ of σ(T ) lies in σ0 (T ). According to Corollary 5.18, λ ∈ σe (T ), which means (by definition) λ ∈ σ(π(T )), and hence λ ∈ ρ(π(T )). Thus λ π(I) − π(T ) is invertible in B[H] /B∞[H]. Since the resolvent function of an element in a unital complex Banach algebra is analytic on the resolvent set (cf. proof of Claim 1 in the proof of Lemma 4.13), it then follows that (ζ π(I) − π(T ))−1 : Λ → B[H] /B∞[H] is analytic on any nonempty open subset Λ of C for which ( ins Υλ )− ⊂ Λ, where Υλ is any simple closed rectifiable positively oriented curve (e.g., any positively oriented circle) enclosing λ but no other point of σ(T ). Hence, $by the Cauchy Theorem (cf. Claim 2 in the proof of Lemma 4.13), we get Υλ(ζ π(I) − π(T ))−1 dζ = [O], where [O] = π(O) = B$∞[H] is the origin in the Calkin algebra B[H] /B∞[H]. Then, with 1 Eλ = 2πi (ζI − T )−1 dζ standing for the Riesz idempotent associated with Υλ λ, and since π : B[H] → B[H] /B∞[H] is linear and bounded, % %   −1  −1 1 1 dζ = 2πi ζ π(I) − π(T ) π (ζI − T ) dζ = [O]. π(Eλ ) = 2πi Υλ

Υλ

Therefore Eλ ∈ B∞[H]. But the restriction of a compact operator to an invariant subspace is again compact (cf. Proposition 1.V), and so the identity I = Eλ |R(Eλ ) : R(Eλ ) → R(Eλ ) on R(Eλ ) is compact. Then dim R(Eλ ) < ∞ (cf. Proposition 1.Y). Thus (a) implies (f).  A Riesz point of an operator T is an isolated point λ of σ(T ) for which the Riesz idempotent Eλ has finite rank (i.e., for which dim R(Eλ ) < ∞). Let σiso (T ) denote the set of all isolated points of the spectrum σ(T ),   σiso (T ) = λ ∈ σ(T ) : λ is an isolated point of σ(T ) . So its complement in σ(T ), σacc (T ) = σ(T )\σiso (T ),

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5. Fredholm Theory in Hilbert Space

is the set of all accumulation points of σ(T ). Let π0 (T ) denote the set of all isolated points of σ(T ) lying in σ0 (T ): π0 (T ) = σiso (T ) ∩ σ0 (T ). Since σ0 (T ) ⊆ σP4 (T ) ⊆ σP (T ), it follows by Theorem 5.19 that   π0 (T ) = λ ∈ σiso (T ) : dim R(Eλ ) < ∞ , and so π0 (T ) is precisely the set of all Riesz points of T , and these are eigenvalues of T . Indeed, σiso (T )\σe (T ) = σiso (T ) ∩ σ0 (T ) ⊆ σP4 (T ) ⊆ σP (T ). Summing up (cf. Theorem 5.19 and the diagram of Section 2.2):   π0 (T ) = λ ∈ σP (T ) : λ ∈ σiso (T ) and λ ∈ σ0 (T ) = σiso (T ) ∩ σ0 (T )   = λ ∈ σP (T ) : λ ∈ σiso (T ) and λ ∈ σe (T ) = σiso (T )\σe (T )   = λ ∈ σP (T ) : λ ∈ σiso (T ) and dim R(Eλ ) < ∞    / σe (T ) ∩ σre (T ) = λ ∈ σP (T ) : λ ∈ σiso (T ) and λ ∈   = λ ∈ σP (T ) : λ ∈ σiso (T ), dim N (λI − T ) < ∞, and R(λI − T ) is closed   = λ ∈ σP4 (T ) : λ ∈ σiso (T ) and dim N (λI − T ) = dim N (λI − T ∗ ) < ∞ . Let τ0 (T ) denote the complement of π0 (T ) in σ0 (T ), τ0 (T ) = σ0 (T )\π0 (T ) ⊆ σP4 (T ), and so {π0 (T ),τ0 (T )} is a partition of σ0 (T ). That is, τ0 (T ) ∩ π(T )0 = ∅ and σ0 (T ) = τ0 (T ) ∪ π0 (T ). Corollary 5.20.

τ0 (T ) is an open subset of C .

Proof. First, τ0 (T ) = σ0 (T )\π0 (T ) ⊆ σ(T ) has no isolated point. Moreover, if λ ∈ τ0 (T ), then λI − T ∈ W[H] and λ ∈ C \σe (T ) ∩ σre (T ) (by Corollary 5.18). Thus (since λ is not an isolated point), there exists a nonempty open ball Bε (λ) centered at λ and included in τ0 (T ) (cf. Proposition 5.D). Therefore τ0 (T ) is  an open subset of C . As usual, let ∂σ(T ) denote the boundary of σ(T ). Corollary 5.21. If λ ∈ ∂σ(T ), then either λ is an isolated point of σ(T ) in π0 (T ) or λ ∈ σe (T ) ∩ σre (T ).   Proof. If λ ∈ ∂σ(T ), then λ ∈ k∈Z \{0} σk (T )∪σ+∞ (T )∪σ−∞ (T ) (since this set is open by Theorem 5.16 and is included in σ(T ), which is closed). Thus

5.3 Riesz Points and Weyl Spectrum

157

λ ∈ (σe (T ) ∩ σre (T )) ∪ σ0 (T ) by Corollary 5.18. But σ0 (T ) = τ0 (T ) ∪ π0 (T ), where π0 (T ) = σiso (T ) ∩ σ0 (T ), and so π0 (T ) consists of isolated points only. However λ ∈ τ0 (T ) because τ0 (T ) is again an open set (Corollary 5.20) in cluded in σ(T ), which is closed. Thus λ ∈ (σe (T ) ∩ σre (T )) ∪ π0 (T ). Let π00 (T ) denote the set of all isolated eigenvalues of finite multiplicity, π00 (T ) = σiso (T ) ∩ σPF (T ). Since σ0 (T ) ⊆ σPF (T ), then π0 (T ) ⊆ π00 (T ). Indeed, π0 (T ) = σiso (T ) ∩ σ0 (T ) ⊆ σiso (T ) ∩ σPF (T ) = π00 (T ). Corollary 5.22.

  π0 (T ) = λ ∈ π00 (T ) : R(λI − T ) is closed .

Proof. If λ ∈ π0 (T ), then λ ∈ π00 (T ) and R(λI −T ) is closed (since λ ∈ σ0 (T )). Conversely, if R(λI − T ) is closed and λ ∈ π00 (T ), then R(λI − T ) is closed and dim N (λI − T ) < ∞ (since λ ∈ σPF (T )), which means λ ∈ σ0 (T ) by  Theorem 5.19 (since λ ∈ σiso (T )). Thus λ ∈ σiso (T ) ∩ σ0 (T ) = π0 (T ). The set π0 (T ) of all Riesz points of T is sometimes referred to as the set of isolated eigenvalues of T of finite algebraic multiplicity (sometimes also called normal eigenvalues of T [64, p. 5], but we reserve this terminology for eigenvalues satisfying the inclusion in Lemma 1.13(a)) — see also the first paragraph of Section 2.7. The set π00 (T ) of all isolated eigenvalues of T of finite multiplicity is sometimes referred to as the set of isolated eigenvalues of T of finite geometric multiplicity. The Weyl spectrum of an operator T ∈ B[H] is the set ! σw (T ) = σ(T + K), K∈B∞[H]

which is the largest part of σ(T ) that remains unchanged under compact perturbations. In other words, σw (T ) is the largest part of σ(T ) such that σw (T + K) = σw (T )

for every

K ∈ B∞[H].

Another characterization of σw (T ) will be given in Theorem 5.24. Lemma 5.23. (a) If T ∈ SF[H] with ind (T ) ≤ 0, then T ∈ F [H] and there exists a compact operator K  ∈ B∞[H] for which T + K  is left invertible (i.e., such that there exists an operator A ∈ B[H] for which A (T + K  ) = I). (b) If T ∈ SF[H] with ind (T ) ≥ 0, then T ∈ Fr [H] and there exists a compact operator K  ∈ B∞[H] for which T + K  is right invertible (i.e., such that there exists an operator A ∈ B[H] for which (T + K  )A = I). (c) If T ∈ SF[H], then ind (T ) = 0 if and only if there exists a compact operator K ∈ B∞[H] for which T + K is invertible (i.e., such that there exists an operator A ∈ B[H] for which A(T + K) = (T + K)A = I).

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5. Fredholm Theory in Hilbert Space

Proof. Take an operator T in B[H]. Claim. If T ∈ SF[H] with ind (T ) ≤ 0, then T ∈ F [H] and there exists a compact (actually, a finite-rank) operator K ∈ B∞[H] for which N (T + K) = {0}. Proof. Take T ∈ SF[H]. If ind (T ) ≤ 0, then dim N (T ) − dim N (T ∗ ) ≤ 0. Thus dim N (T ) ≤ dim N (T ∗ ) and dim N (T ) < ∞ (and so T ∈ F [H] by Theorem 5.1). Let {ei }ni=1 be an orthonormal basis for N (T ), and let B be an orthonormal basis for N (T ∗ ) = R(T )⊥ (cf. Lemma 1.4) whose cardinality is not less than n (since dim N (T ) ≤ dim N (T ∗ )). Take any orthonormal set {fk }nk=1 ⊆ B and define a map K : H → H by n x ; ej fj for every x ∈ H. Kx = j=1

This is clearly linear and bounded (a contraction, actually). In fact, the map K is a finite-rank operator (thus compact — K lies in B0 [H] ⊆ B∞[H]) whose range is included in R(T )⊥. Indeed,   R(K) ⊆ {fj }nj=1 ⊆ B = N (T ∗ ) = R(T )⊥ . Take any x ∈ N (T ) and consider its Fourierseries expansion with respect to n the orthonormal basis {ei }ni=1 , namely x = i=1 x ; ei ei . Thus n x2 = |x ; ei |2 = Kx2 for every x ∈ N (T ). j=1

Now, if x ∈ N (T + K), then T x = −Kx, and hence T x ∈ R(T ) ∩ R(K) ⊆ R(T ) ∩ R(T )⊥ = {0}, so that T x = 0 (i.e., x ∈ N (T )). Thus x = Kx = T x = 0, and hence x = 0. Therefore we get the claimed result: N (T + K) = {0}. (a) If T ∈ SF[H] with ind (T ) ≤ 0, then T ∈ F [H] and there exists K ∈ B∞[H] such that N (T + K) = {0} by the preceding claim. Now T + K ∈ F [H] according to Theorem 5.6, and so R(T + K) is closed by Theorem 5.1. Thus T + K is left invertible by Lemma 5.8. (b) If T ∈ SF[H] has ind (T ) ≥ 0, then T ∗ ∈ SF[H] has ind (T ∗ ) = −ind (T ) ≤ 0. Thus according to item (a) T ∗ ∈ F [H] and so T ∈ Fr [H] and there exists K ∈ B∞[H] for which T ∗ + K is left invertible. Hence T + K ∗ = (T ∗ + K)∗ is right invertible with K ∗ ∈ B∞[H] (cf. Proposition 1.W in Section 1.9). (c) If T ∈ SF[H] with ind (T ) = 0, then T ∈ F[H] and there exists K ∈ B∞[H] such that N (T + K) = {0} by the preceding claim. Since ind (T + K) = ind (T ) = 0 (Theorem 5.6), we get N ((T + K)∗ ) = {0}. So R(T + K)⊥ = {0} (Lemma 1.4), and hence R(T + K) = H. Thus T + K is invertible (Theorem 1.1). Conversely, if there exists K ∈ B∞[H] such that T + K is invertible, then T + K is Weyl (since every invertible operator is Weyl), and so is T (Theorem 5.6); that is, T ∈ F[H] with ind (T ) = 0. 

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159

According to the claim in the preceding proof, the statement of Lemma 5.23 holds if “compact” is replaced with “finite-rank” (see Remark 5.3(c)). Theorem 5.24. (Schechter Theorem). If T ∈ B[H], then  σw (T ) = σe (T ) ∪ σk (T ) = σ(T )\σ0 (T ). k∈Z \{0}

Proof. Take an operator T in B[H]. Claim. If λ ∈ σ0 (T ), then there is a K ∈ B∞[H] for which λ ∈ σ(T + K). Proof. If λ ∈ σ0 (T ), then λI − T ∈ SF[H] with ind (λI − T ) = 0. Thus according to Lemma 5.23(c) λI − (T + K) is invertible for some K ∈ B∞[H], and therefore λ ∈ ρ(T + K). This completes the proof of the claimed result. From Corollary 5.18, σ(T ) = σe (T ) ∪

 k∈Z \{0}

σk (T ) ∪ σ0 (T ),

where the above sets are all pairwise disjoint, and so -

 σk (T ) . σ0 (T ) = σ(T ) σe (T ) ∪ k∈Z \{0}

Take an arbitrary λ ∈ σ(T ). If λ lies in σe (T ) ∪ k∈Z \{0} σk (T ), then λ lies in

σe (T + K) ∪ k∈Z \{0} σk (T + K) for every K ∈ B∞[H] since σe (T + K) = σe (T )

and

σk (T + K) = σk (T ) for every k ∈ Z \{0}

for every K ∈ B∞[H] (cf. Remarks 5.15(c) and 5.17(c)). On the other hand, if λ ∈ σ0 (T ), then the preceding claim ensures the existence of a K ∈ B∞[H] for which λ ∈ σ(T + K). As the largest part of σ(T ) that remains invariant under compact perturbation is σw (T ), we get  σw (T ) = σe (T ) ∪ σk (T ) = σ(T )\σ0 (T ).  k∈Z \{0}

Since σw (T ) ⊆ σ(T ) by the very definition of σw (T ), σe (T ) ⊆ σw (T ) ⊆ σ(T ). Chronologically, Theorem 5.24 precedes the spectral picture of Corollary 5.18 [105]. Since σk (T ) ⊆ σ(T )\σe (T ) for all k ∈ Z , we also get  σw (T )\σe (T ) = σk (T ) k∈Z \{0}

which is the collection of all holes of the essential spectrum. Thus  σk (T ) = ∅ σe (T ) = σw (T ) ⇐⇒ k∈Z \{0}

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5. Fredholm Theory in Hilbert Space

(i.e., if and only if the essential spectrum has no holes). Since σw (T ) and σ0 (T ) are both subsets of σ(T ), it follows by Theorem 5.24 that σ0 (T ) is the complement of σw (T ) in σ(T ), σ0 (T ) = σ(T )\σw (T ), and therefore {σw (T ), σ0 (T )} forms a partition of the spectrum σ(T ): σ(T ) = σw (T ) ∪ σ0 (T )

and

σw (T ) ∩ σ0 (T ) = ∅.

σw (T ) = σ(T )

⇐⇒

σ0 (T ) = ∅,

Thus and so, by Theorem 5.24 again, σe (T ) = σw (T ) = σ(T )

⇐⇒

 k∈Z

σk (T ) = ∅.

Moreover, σw (T ) is always compact (it is the intersection K∈B [H] σ(T + K) ∞ of compact sets in C ), and so σ0 (T ) is closed in C if and only if σ0 (T ) = π0 (T ) ⊆ σiso (T ) (since {σw (T ), σ0 (T )} is a partition of σ(T ) and {π0 (T ), τ0 (T )} is a partition of σ0 (T )). Here is a third characterization of the Weyl spectrum. Corollary 5.25. For every operator T ∈ B[H],   σw (T ) = λ ∈ C : λI − T ∈ B[H]\W[H] . Proof. If λ ∈ ρ(T ), then λI − T is invertible, and so λI − T ∈ W[H] (since invertible operators are Weyl). Therefore if λI − T ∈ W[H], then λ ∈ σ(T ), and the result follows from Theorem 5.24 and the definition of σ0 (T ):   σw (T ) = σ(T )\σ0 (T ) and σ0 (T ) = λ ∈ σ(T ) : λI − T ∈ W[H] .  Then the Weyl spectrum σw (T ) is the set of all scalars λ for which λI − T is not a Weyl operator (i.e., for which λI − T is not a Fredholm operator of index zero). Since an operator lies in W[H] together with its adjoint, it follows by Corollary 5.25 that λ ∈ σw (T ) if and only if λ ∈ σw (T ∗ ): σw (T ) = σw (T ∗ )∗ . Theorem 5.24 also gives us another characterization of the set W[H] of all Weyl operators from B[H] in terms of the set σ0 (T ) = σ(T )\σw (T ). Corollary 5.26.     W[H] = T ∈ B[H] : 0 ∈ ρ(T ) ∪ σ0 (T ) = T ∈ F[H] : 0 ∈ ρ(T ) ∪ σ0 (T ) . Proof. Take T ∈ B[H]. If T ∈ W[H], then T ∈ F[H] and T + K is invertible for some K ∈ B∞[H] by Lemma 5.23(c), which means 0 ∈ ρ(T + K) or, equivalently, 0 ∈ σ(T + K), and so 0 ∈ σw (T + K). Thus 0 ∈ σw (T ) (cf. definition

5.3 Riesz Points and Weyl Spectrum

161

of the Weyl spectrum preceding Lemma 5.23), and hence 0 ∈ ρ(T ) ∪ σ0 (T ) by Theorem 5.24. The converse is (almost) trivial: if 0 ∈ ρ(T ) ∪ σ0 (T ), then either 0 ∈ ρ(T ), so that T ∈ G[H] ⊂ W[H] by Remark 5.7(b); or 0 ∈ σ0 (T ), which means 0 ∈ σ(T ) and T ∈ W[H] by the very definition of σ0 (T ). This proves the first identity, which implies the second one since W[H] ⊆ F[H].  Remark 5.27. (a) Finite Dimension. Take any operator T ∈ B[H]. Since σe (T ) ⊆ σw (T ), then by Remark 5.15(a) σw (T ) = ∅

dim H < ∞

=⇒

and, by Remark 5.3(d) and Corollary 5.25, Thus

dim H < ∞

=⇒

σw (T ) = ∅.

σw (T ) = ∅

⇐⇒

dim H = ∞.

(b) More on Finite Dimension. Let T ∈ B[H] be an operator on a finitedimensional space. Then B[H] = W[H] by Remark 5.3(d) and so σ0 (T ) = σ(T ) by the definition of σ0 (T ). Moreover, according to Corollary 2.9, σ(T ) = σiso (T ) (since #σ(T ) < ∞), and σ(T ) = σPF (T ) (since dim H < ∞). Hence dim H < ∞

=⇒

σ(T ) = σPF (T ) = σiso (T ) = σ0 (T ) = π0 (T ) = π00 (T )

and so, either by Theorem 5.24 or by item (a) and Remark 5.15(a), dim H < ∞

=⇒

σe (T ) = σw (T ) = ∅.

(c) Fredholm Alternative. The Fredholm Alternative has been restated again and again in Corollary 1.20, Theorem 2.18, Remark 5.3(e), Remark 5.7(b), and Remark 5.17(a). Here is an ultimate restatement of it. If K is compact (i.e., if K ∈ B∞[H]), then σ(K)\{0} = σ0 (K)\{0} by Remark 5.17(a). But σ(K)\{0} ⊆ σiso (K) by Corollary 2.20, and π0 (K) = σiso (K) ∩ σ0 (K). Therefore (compare with Theorem 2.18 and Remark 5.17(a)), K ∈ B∞[H]

=⇒

σ(K)\{0} = π0 (K)\{0}.

As π0 (K) = σiso (K) ∩ σ0 (K) ⊆ σiso (K) ∩ σPF (K) = π00 (K) ⊆ σPF (T ) ⊆ σP (T ), σ(K)\{0} = σPF (K)\{0} = π00 (K)\{0} = π0 (K)\{0}. (d) More on Compact Operators. Take a compact operator K ∈ B∞[H]. By Remark 5.3(e) (Fredholm Alternative) if λ = 0, then λI − K ∈ W[H]. Thus, according to Corollary 5.25, σw (K)\{0} = ∅. If dim H = ∞, then σw (K) = {0} by item (a). Since σe (K) ⊆ σw (K), it follows by Remark 5.15(a) that if dim H = ∞, then ∅ = σe (K) ⊆ {0}. Hence K ∈ B∞[H] and dim H = ∞

=⇒

σe (K) = σw (K) = {0}.

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5. Fredholm Theory in Hilbert Space

Therefore, by the Fredholm Alternative of item (c), K ∈ B∞[H] and dim H = ∞

σ(K) = π0 (K) ∪ {0},

=⇒

σw (K) = {0}.

(e) Normal Operators. Recall: T is normal if and only if λI − T is normal. Since every normal Fredholm operator is Weyl (see Remark 5.7(b)), we may invoke Corollaries 5.12 and 5.25 to get T normal

=⇒

σe (T ) = σw (T ).

Take any T ∈ B[H] and consider the following expressions (cf. Theorem 5.24 or Corollary 5.25 and definitions of σ0 (T ), π0 (T ), and π00 (T )). σ0 (T ) = σ(T )\σw (T ),

π0 (T ) = σiso (T )∩σ0 (T ),

π00 (T ) = σiso (T )∩σPF (T ).

The equivalent assertions in the next corollary define an important class of operators. An operator for which any of those equivalent assertions holds is said to satisfy Weyl’s Theorem. This will be discussed later in Section 5.5. Corollary 5.28. The assertions below are equivalent. (a) σ0 (T ) = π00 (T ). (b) σ(T )\π00 (T ) = σw (T ). Moreover, if σ0 (T ) = σiso (T ) (i.e., if σw (T ) = σacc (T )), then the above equivalent assertions hold true. Conversely, if either of the above equivalent assertions holds true, then π0 (T ) = π00 (T ). Proof. By Theorem 5.24, σ(T )\σ0 (T ) = σw  implies (b). If (b) ,(T ), and so (a) holds, then σ0 (T ) = σ(T )\σw (T ) = σ(T ) σ(T )\π00 (T ) = π00 (T ) because π00 (T ) ⊆ σ(T ), and so (b) implies (a). Moreover, if σ0 (T ) = σiso (T ) (or, if their complements in σ(T ) coincide), then (a) holds because σ0 (T ) ⊆ σPF (T ). Conversely, if (a) holds, then π0 (T ) = σiso (T ) ∩ σ0 (T ) = σiso (T ) ∩ π00 (T ) =  π00 (T ) because π00 (T ) ⊆ σiso (T ).

5.4 Browder Operators and Browder Spectrum As usual, take T ∈ B[H] (or, more generally, take any linear transformation of a linear space into itself). In fact, part of this section is purely algebraic, and the entire section (as the whole chapter) has a natural counterpart in a Banach space. Let N 0 be the set of all nonnegative integers and consider the nonnegative integral powers T n of T. As is readily verified, N (T n ) ⊆ N (T n+1 )

and

R(T n+1 ) ⊆ R(T n )

for every n ∈ N 0 . Thus {N (T n )} and {R(T n )} are nondecreasing and nonincreasing (in the inclusion ordering) sequences of subsets of H, respectively.

5.4 Browder Operators and Browder Spectrum

163

Lemma 5.29. Let n0 be an arbitrary integer in N 0 . (a) If N (T n0 +1 ) = N (T n0 ), then N (T n+1 ) = N (T n ) for every n ≥ n0 . (b) If R(T n0 +1 ) = R(T n0 ), then R(T n+1 ) = R(T n ) for every n ≥ n0 . Proof. (a) Rewrite the statements in (a) as follows. N (T n0 +1 ) = N (T n0 )

=⇒

N (T n0 +k+1 ) = N (T n0 +k ) for every k ≥ 0.

The claimed result holds trivially for k = 0. Suppose it holds for some k ≥ 0. Take an arbitrary x ∈ N (T n0 +k+2 ) so that T n0 +k+1 (T x) = T n0 +k+2 x = 0. Thus T x ∈ N (T n0 +k+1 ) = N (T n0 +k ), and so T n0 +k+1 x = T n0 +k (T x) = 0, which implies x ∈ N (T n0 +k+1 ). Hence N (T n0 +k+2 ) ⊆ N (T n0 +k+1 ). However, N (T n0 +k+1 ) ⊆ N (T n0 +k+2 ) since {N (T n )} is nondecreasing, and therefore N (T n0 +k+2 ) = N (T n0 +k+1 ). Then the claimed result holds for k + 1 whenever it holds for k, which completes the proof of (a) by induction. (b) Rewrite the statements in (b) as follows. R(T n0 +1 ) = R(T n0 )

=⇒

R(T n0 +k+1 ) = R(T n0 +k ) for every k ≥ 1.

The claimed result holds trivially for k = 0. Suppose it holds for some integer k ≥ 0. Take an arbitrary y ∈ R(T n0 +k+1 ) so that y = T n0 +k+1 x = T (T n0 +k x) for some x ∈ H, and hence y = T u for some u ∈ R(T n0 +k ). If R(T n0 +k ) = R(T n0 +k+1 ), then u ∈ R(T n0 +k+1 ), and so y = T (T n0 +k+1 v) for some v ∈ H. Thus y ∈ R(T n0 +k+2 ). Therefore R(T n0 +k+1 ) ⊆ R(T n0 +k+2 ). Since the sequence {R(T n )} is nonincreasing we get R(T n0 +k+2 ) ⊆ R(T n0 +k+1 ). Hence R(T n0 +k+2 ) = R(T n0 +k+1 ). Thus the claimed result holds for k + 1 whenever it holds for k, which completes the proof of (b) by induction.  Let N 0 = N 0 ∪ {+∞} denote the set of all extended nonnegative integers with its natural (extended) ordering. The ascent and descent of an operator T ∈ B[H] are defined as follows. The ascent of T , asc (T ), is the least (extended) nonnegative integer for which N (T n+1 ) = N (T n ):   asc (T ) = min n ∈ N 0 : N (T n+1 ) = N (T n ) , and the descent of T , dsc (T ), is the least (extended) nonnegative integer for which R(T n+1 ) = R(T n ):   dsc (T ) = min n ∈ N 0 : R(T n+1 ) = R(T n ) . The ascent of T is null if and only if T is injective, and the descent of T is null if and only if T is surjective. Indeed, since N (I) = {0} and R(I) = H, asc (T ) = 0

⇐⇒

N (T ) = {0},

dsc(T ) = 0

⇐⇒

R(T ) = H.

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5. Fredholm Theory in Hilbert Space

Again, the notion of ascent and descent holds for every linear transformation of a linear space into itself, and so does the next result. Lemma 5.30. Take any operator T ∈ B[H]. (a) If asc(T ) < ∞ and dsc (T ) = 0, then asc (T ) = 0. (b) If asc (T ) < ∞ and dsc(T ) < ∞, then asc (T ) = dsc(T ). Proof. (a) Suppose dsc (T ) = 0 (i.e., suppose R(T ) = H). If asc (T ) = 0 (i.e., if N (T ) = {0}), then take 0 = x1 ∈ N (T ) ∩ R(T ) and x2 , x3 in R(T ) = H as follows: x1 = T x2 and x2 = T x3 , and so x1 = T 2 x3 . Continuing this way, we can construct a sequence {xn }n≥1 of vectors in H = R(T ) such that xn = T xn+1 and 0 = x1 = T n xn+1 lies in N (T ), and so T n+1 xn+1 = 0. Therefore xn+1 ∈ N (T n+1 )\N (T n ) for each n ≥ 1, and hence asc (T ) = ∞ by Lemma 5.29. Summing up: If dsc (T ) = 0, then asc(T ) = 0 implies asc (T ) = ∞. (b) Set m = dsc(T ), so that R(T m ) = R(T m+1 ), and set S = T |R(T m ) . By Proposition 1.E(a), R(T m ) is T -invariant, and so S ∈ B[R(T m )]. Also R(S) = S(R(T m )) = R(S T m ) = R(T m+1 ) = R(T m ). Thus S is surjective, which means dsc(S) = 0. Moreover, since asc (S) < ∞ (because asc(T ) < ∞), we get asc (S) = 0 by (a), which means N (S) = {0}. Take x ∈ N (T m+1 ) and set y = T m x in R(T m ). Thus Sy = T m+1 x = 0, and so y = 0. Therefore x ∈ N (T m ). Then N (T m+1 ) ⊆ N (T m ) which implies N (T m+1 ) = N (T m ) since {N (T n )} is nondecreasing. Consequently, asc (T ) ≤ m by Lemma 5.29. To prove the reverse inequality, suppose m = 0 (otherwise apply (a)) and take a vector z in R(T m−1 )\R(T m ) so that z = T m−1 u and T z = T (T m−1 u) = T m u lies in R(T m ) for some u in H. Since T m (R(T m )) = R(T 2m ) = R(T m ), we get T z = T m v for some v in R(T m ). Now observe that T m (u − v) = T z − T z = 0 and T m−1 (u − v) = z − T m−1 v = 0 (reason: T m−1 v ∈ R(T 2m−1 ) = R(T m ) since / R(T m )). Therefore (u − v) ∈ N (T m )\N (T m−1 ), and so v ∈ R(T m ), and z ∈ asc (T ) ≥ m. Outcome: If dsc (T ) = m and asc(T ) < ∞, then asc(T ) = m.  Lemma 5.31. If T ∈ F[H], then asc (T ) = dsc (T ∗ ) and dsc(T ) = asc (T ∗ ). Proof. Take any T ∈ B[H]. We will use freely the relations between range and kernel involving adjoints and orthogonal complements of Lemma 1.4. Claim 1. asc (T ) < ∞ if and only if dsc (T ∗ ) < ∞. dsc (T ) < ∞ if and only if asc (T ∗ ) < ∞. Proof. Take an arbitrary n ∈ N 0 . If asc(T ) = ∞, then N (T n ) ⊂ N (T n+1 ), and so N (T n+1 )⊥ ⊂ N (T n )⊥. Equivalently, R(T ∗(n+1) )− ⊂ R(T ∗n )−, which ensures the proper inclusion R(T ∗(n+1) ) ⊂ R(T ∗n ), and hence dsc(T ∗ ) = ∞. Thus asc (T ) = ∞ implies dsc(T ∗ ) = ∞. Dually (since T ∗∗ = T ), asc(T ∗ ) = ∞ implies dsc (T ) = ∞: asc (T ) = ∞ =⇒ dsc(T ∗ ) = ∞

and

asc(T ∗ ) = ∞ =⇒ dsc (T ) = ∞.

5.4 Browder Operators and Browder Spectrum

165

Conversely, if dsc(T ) = ∞, then R(T n+1 ) ⊂ R(T n ). Therefore R(T n )⊥ ⊂ R(T n+1 )⊥. Equivalently, N (T ∗n ) ⊂ N (T ∗(n+1) ), and so asc(T ∗ ) = ∞. Thus dsc(T ) = ∞ implies asc (T ∗ ) = ∞. Dually, dsc(T ∗ ) = ∞ implies asc (T ) = ∞: dsc(T ) = ∞ =⇒ asc(T ∗ ) = ∞

and

dsc(T ∗ ) = ∞ =⇒ asc (T ) = ∞.

Summing up: asc (T ) = ∞ if and only if dsc (T ∗ ) = ∞ and dsc(T ) = ∞ if and only if asc (T ∗ ) = ∞, which completes the proof of Claim 1. Claim 2. If dsc(T ) < ∞, then asc (T ∗ ) ≤ dsc(T ). If dsc(T ∗ ) < ∞, then asc (T ) ≤ dsc(T ∗ ). Proof. If dsc (T ) < ∞, then set n0 = dsc (T ) ∈ N 0 and take any integer n ≥ n0 . Thus R(T n ) = R(T n0 ), and hence R(T n )− = R(T n0 )− so that N (T ∗n )⊥ = N (T ∗n0 )⊥, which implies N (T ∗n ) = N (T ∗n0 ). Therefore asc (T ∗ ) ≤ n0 , and so asc(T ∗ ) ≤ dsc(T ). Dually, if dsc(T ∗ ) < ∞, then asc (T ) ≤ dsc (T ∗ ). This completes the proof of Claim 2. Claim 3. If asc(T ) < ∞ and R(T n ) is closed for every n ≥ asc (T ), then dsc (T ∗ ) ≤ asc (T ). If asc (T ∗ ) < ∞ and R(T n ) is closed for every n ≥ asc (T ∗ ), then dsc (T ) ≤ asc (T ∗ ). Proof. If asc(T ) < ∞, then set n0 = asc (T ) ∈ N 0 and take an arbitrary integer n ≥ n0 . Thus N (T n ) = N (T n0 ) or, equivalently, R(T ∗n )⊥ = R(T ∗n0 )⊥, so that R(T ∗n0 )− = R(T ∗n )−. Suppose R(T n ) is closed. Thus R(T ∗n ) = R(T n∗ ) is closed (Lemma 1.5). Then R(T ∗n ) = R(T ∗n0 ), and therefore dsc (T ∗ ) ≤ n0 , so that dsc(T ∗ ) ≤ asc (T ). Dually, if asc (T ∗ ) < ∞ and R(T n ) is closed (equivalently, R(T ∗n ) is closed — Lemma 1.5) for every integer n ≥ asc (T ∗ ), then dsc(T ) ≤ asc(T ∗ ), completing the proof of Claim 3. Outcome: If R(T n ) is closed for every n ∈ N 0 (so that we can apply Claim 3), then it follows by Claims 1, 2, and 3 that asc (T ) = dsc(T ∗ )

and

dsc(T ) = asc (T ∗ ).

In particular, if T ∈ F[H], then T n ∈ F[H] by Corollary 5.5, and therefore R(T n ) is closed for every integer n ∈ N 0 by Corollary 5.2. Thus if T is Fred holm, then asc (T ) = dsc (T ∗ ) and dsc(T ) = asc(T ∗ ). A Browder operator is a Fredholm operator with finite ascent and finite descent. Let Br [H] denote the class of all Browder operators from B[H]:   Br [H] = T ∈ F[H] : asc(T ) < ∞ and dsc (T ) < ∞ ; equivalently, according to Lemma 5.30(b),   Br [H] = T ∈ F[H] : asc (T ) = dsc (T ) < ∞

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5. Fredholm Theory in Hilbert Space

(i.e., Br [H] = {T ∈ F[H] : asc(T ) = dsc(T ) = m for some m ∈ N 0 }). Thus   F[H]\Br [H] = T ∈ F[H] : asc (T ) = ∞ or dsc(T ) = ∞ and, by Lemma 5.31, T ∈ Br [H]

if and only if

T ∗ ∈ Br [H].

Two linear manifolds of a linear space X , say R and N , are said to be algebraic complements of each other (or complementary linear manifolds) if they sum the whole space and are algebraically disjoint; that is, if R+N =X

and

R ∩ N = {0}.

A pair of subspaces (i.e., closed linear manifolds) of a normed space that are algebraic complements of each other are called complementary subspaces. Lemma 5.32. If T ∈ B[H] with asc (T ) = dsc (T ) = m for some m ∈ N 0 , then R(T m ) and N (T m ) are algebraic complements of each other . Proof. Take any T ∈ B[H]. (In fact, as will become clear during the proof, this is a purely algebraic result which holds for every linear transformation of a linear space into itself with coincident finite ascent and descent.) Claim 1. If asc(T ) = m, then R(T m ) ∩ N (T m ) = {0}. Proof. If y ∈ R(T m ) ∩ N (T m ), then y = T m x for some x ∈ H and T m y = 0, so that T 2m x = T m (T m x) = T m y = 0; that is, x ∈ N (T 2m ) = N (T m ) since asc (T ) = m. Hence y = T m x = 0, proving Claim 1. Claim 2. If dsc(T ) = m, then R(T m ) + N (T m ) = H. Proof. If dsc(T ) = m, then R(T m ) = R(T m+1 ). Set S = T |R(T m ) . According to Proposition 1.E(a), R(T m ) is T -invariant, and so S ∈ B[R(T m )] with R(S) = S(R(T m )) = R(S T m ) = R(T m+1 ) = R(T m ). Then S is surjective, and hence S m is surjective too (surjectiveness of S implies dsc (S) = 0) Therefore R(S m ) = R(T m ). Take an arbitrary vector x ∈ H. Thus T m x ∈ R(T m ) = R(S m ), and so there exists a vector u ∈ R(T m ) for which S m u = T m x. Since S m u = T m u (because S m = (T |R(T m ) )m = T m |R(T m ) ), we get T m u = T m x and hence v = x − u is in N (T m ). Then x = u + v lies in R(T m ) + N (T m ).  Consequently H ⊆ R(T m ) + N (T m ), proving Claim 2. Theorem 5.33. Take T ∈ B[H] and consider the following assertions. (a) T is Browder but not invertible

(i.e., T ∈ Br [H]/G[H]).

(b) T ∈ F[H] is such that R(T m ) and N (T m ) are complementary subspaces for some m ∈ N . (c) T ∈ W[H] Claim:

(i.e., T is Weyl, which means T is Fredholm of index zero).

(a) =⇒ (b) =⇒ (c).

5.4 Browder Operators and Browder Spectrum

167

Proof. (a) ⇒(b). An operator T ∈ B[H] is invertible (equivalently, R(T ) = H and N (T ) = {0}) if and only if T ∈ G[H] (i.e., T ∈ B[H] has a bounded inverse) by Theorem 1.1. Moreover, every invertible operator is Fredholm (by the definition of Fredholm operators), and R(T ) = H and N (T ) = {0} (i.e., T ∈ G[H]) if and only if asc (T ) = dsc(T ) = 0 (by the definition of ascent and descent). Thus invertible operators are Browder, which in turn are Fredholm, G[H] ⊂ Br [H] ⊆ F[H]. (The last inclusion is proper only in infinite-dimensional spaces.) Therefore if T ∈ Br [H] and T ∈ G[H], then asc (T ) = dsc(T ) = m for some integer m ≥ 1, and so N (T m ) and R(T m ) are complementary linear manifolds of H by Lemma 5.32. If T ∈ Br [H], then T ∈ F[H], so that T m ∈ F[H] by Corollary 5.5. Hence R(T m ) is closed by Corollary 5.2. But N (T m ) is closed since T m is bounded. Thus R(T m ) and N (T m ) are subspaces of H. (b) ⇒(c). If (b) holds, then T ∈ F[H]. Thus T n ∈ F[H] by Corollary 5.5, and so N (T n ) and N (T n∗ ) are finite-dimensional and R(T n ) is closed by Corollary 5.2, for every n ∈ N . In addition, suppose there exists m ∈ N such that R(T m ) + N (T m ) = H

and

R(T m ) ∩ N (T m ) = {0}.

Since R(T m ) is closed, R(T m ) + R(T m )⊥ = H (Projection Theorem), where R(T m )⊥ = N (T m∗ ) (Lemma 1.4). Then R(T m ) + N (T m∗ ) = H

and

R(T m ) ∩ N (T m∗ ) = {0}.

Thus N (T m ) and N (T m∗ ) are both algebraic complements of R(T m ), and so they have the same (finite) dimension (see, e.g., [78, Theorem 2.18] — it is in fact the codimension of R(T m )). Then ind (T m ) = 0. Since T ∈ F[H], we get m ind (T ) = ind (T m ) = 0 (see Corollary 5.5). Therefore (recalling that m = 0) ind (T ) = 0, and so T ∈ W[H].  Since G[H] ⊂ W[H] (cf. Remark 5.7(b)), G[H] ⊂ Br [H] ⊆ W[H] ⊆ F[H] ⊆ B[H]. (The last three inclusions are proper only in infinite-dimensional spaces — Remarks 5.3(d), 5.7(b) and 5.36(a)). The inclusion Br [H] ⊆ W[H] can be otherwise verified by Proposition 5.J. Also, the assertion “T is Browder and not invertible” (i.e., T ∈ Br [H]\G[H]) means “T ∈ Br [H] and 0 ∈ σ(T )”. According to Theorem 5.34 below, in this case 0 must be an isolated point of σ(T ). Precisely, T ∈ Br [H] and 0 ∈ σ(T ) if and only if T ∈ F[H] and 0 ∈ σiso (T ). Theorem 5.34. Take an operator T ∈ B[H]. (a) If T ∈ Br [H] and 0 ∈ σ(T ), then 0 ∈ π0 (T ). (b) If T ∈ F[H] and 0 ∈ σiso (T ), then T ∈ Br [H].

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5. Fredholm Theory in Hilbert Space

Proof. (a) As we have seen before, σ0 (T ) = {λ ∈ σ(T ) : λI − T ∈ W[H]} and Br [H] ⊂ W[H]. So if T ∈ Br [H] and 0 ∈ σ(T ), then 0 ∈ σ0 (T ). Moreover, Theorem 5.33 ensures the existence of an integer m ≥ 1 for which R(T m ) + N (T m ) = H

and

R(T m ) ∩ N (T m ) = {0}.

and

R(T m ) ∩ N (T m ) = {0}

Equivalently, R(T m ) ⊕ N (T m ) = H

(where the direct sum is not necessarily orthogonal and equality in fact means equivalence — i.e., equality means equality up to a topological isomorphism). Since both R(T m ) and N (T m ) are T m -invariant, T m = T m |R(T m ) ⊕ T m |N (T m ) . Since N (T m ) = {0} (otherwise T m would be invertible) and T m |N (T m ) = O acts on the nonzero space N (T m ), we get T m = T m |R(T m ) ⊕ O. Hence (see Proposition 2.F even though the direct sum is not orthogonal) σ(T )m = σ(T m ) = σ(T m |R(T m ) ) ∪ {0} by Theorem 2.7 (Spectral Mapping). Since T m |R(T m ) : R(T m ) → R(T m ) is surjective and injective, and since R(T m ) is closed because T m ∈ F[H], we also get T m |R(T m ) ∈ G[R(T m )]. Thus 0 ∈ ρ(T m |R(T m ) ) and so 0 ∈ σ(T m |R(T m ) ). Then σ(T m ) is disconnected, and therefore 0 is an isolated point of σ(T )m = σ(T m ). Then 0 is an isolated point of σ(T ). That is, 0 ∈ σiso (T ). Outcome: 0 ∈ π0 (T ) = σiso (T ) ∩ σ0 (T ). (b) Take an arbitrary m ∈ N . If 0 ∈ σiso (T ), then 0 ∈ σiso (T m ). Indeed, if 0 is an isolated pont of σ(T ), then 0 ∈ σ(T )m = σ(T m ) and hence 0 is an isolated pont of σ(T )m = σ(T m ). Therefore σ(T m ) = {0} ∪ Δ, where Δ is a compact subset of C which does not contain 0. Then {0} and Δ are complementary spectral sets for the operator T m (i.e., they form a disjoint partition for σ(T m )). Let E0 = E{0} ∈ B[H] be the Riesz idempotent associated with 0. By Corollary 4.22 N (T m ) ⊆ R(E0 ). Claim. If T ∈ F[H] and 0 ∈ σiso (T ), then dim R(E0 ) < ∞. Proof. Theorem 5.19.

5.4 Browder Operators and Browder Spectrum

169

Thus if T ∈ F[H] and 0 ∈ σiso (T ), then dim N (T m ) ≤ dim R(E0 ) < ∞. Since {N (T n )} is a nondecreasing sequence of subspaces, {dim N (T n )} is a nondecreasing sequence of positive integers, which is bounded by the (finite) integer dim R(E0 ). This ensures T ∈ F[H] and 0 ∈ σiso (T )

=⇒

asc (T ) < ∞.

Since σ(T )∗ = σ(T ∗ ) we get 0 ∈ σiso (T ) if and only if 0 ∈ σiso (T ∗ ), and so T ∈ F[H] and 0 ∈ σiso (T )

=⇒

asc (T ∗ ) < ∞,

because T ∈ F[H] if and only if T ∗ ∈ F[H]. Therefore, according to Lemma 5.31, asc (T ) < ∞ and dsc(T ) < ∞ which means (by definition) T ∈ Br [H].  Theorem 5.34 also gives us another characterization of the set Br [H] of all Browder operators from B[H] (compare with Corollary 5.26). Corollary 5.35.     Br [H] = T ∈ F[H] : 0 ∈ ρ(T ) ∪ σiso (T ) = T ∈ F[H] : 0 ∈ ρ(T ) ∪ π0 (T ) . Proof. Consider the set π0 (T ) = σiso (T ) ∩ σ0 (T ) of Riesz points of T . Claim. If T ∈ F[H], then 0 ∈ σiso (T ) if and only if 0 ∈ π0 (T ). Proof. Theorem 5.19. Thus the expressions for Br [H] follow from (a) and (b) in Theorem 5.34.



Remark 5.36. (a) Finite Dimension. Since {N (T n )} is nondecreasing and {R(T n )} is nonincreasing, we get dim N (T n ) ≤ dim N (T n+1 ) ≤ dim H and dim R(T n+1 ) ≤ dim R(T n ) ≤ dim H for every n ≥ 0. Thus if H is finite-dimensional, then asc (T ) < ∞ and dsc(T ) < ∞. Therefore (see Remark 5.3(d)) dim H < ∞

=⇒

Br [H] = B[H].

(b) Complements. By definition σ(T )\σiso (T ) = σacc (T ) and   σ(T )\π0 (T ) = σ(T )\ σiso (T ) ∩ σ0 (T )     = σ(T )\σiso (T ) ∪ σ(T )\σ0 (T ) = σacc (T ) ∪ σw (T ). Thus, since G[H] ⊂ Br [H] ⊆ W[H] ⊆ F[H], we get by Corollary 5.35      F[H]\Br [H] = T ∈ F[H] : 0 ∈ σacc (T ) = T ∈ F[H] : 0 ∈ σacc (T ) ∪ σw (T ) .  Since W[H]\B [H] = W[H] ∩ (F[H]\B [H]), then W[H]\B [H] = T ∈ W[H] : r r r  0 ∈ σacc (T ) . Moreover, since σ0 (T ) = τ0 (T ) ∪ π0 (T ) where τ0 (T ) is an open subset of C (according to Corollary 5.20), and since π0 (T ) = σiso (T  ) ∩ σ0 (T ), we get σ0(T )\σiso (T ) = τ0 (T ) ⊆ σacc (T ). Hence W[H]\Br [H] = T ∈ W[H] : 0 ∈ τ0 (T ) by Corollaries 5.26 and 5.35. Summing up:

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5. Fredholm Theory in Hilbert Space

    W[H]\Br [H] = T ∈ W[H] : 0 ∈ σacc (T ) = T ∈ W[H] : 0 ∈ τ0 (T ) . Furthermore, since σ(T )\σ0 (T ) = σw (T ) we get by Corollary 5.26   F[H]\W[H] = T ∈ F[H] : 0 ∈ σw (T ) . (c) Another Characterization. An operator T ∈ B[H] is Browder if and only if it is Fredholm and λI − T is invertible for sufficiently small λ = 0. Indeed, by the identity Br [H] = {T ∈ F[H] : 0 ∈ ρ(T ) ∪ σiso (T )} in Corollary 5.35, T ∈ Br [H] if and only if T ∈ F[H] and either 0 ∈ ρ(T ) or 0 is an isolated point of σ(T ) = C \ρ(T ). Since the set ρ(T ) is open in C , this means T ∈ Br [H] if and only if T ∈ F[H] and there exists an ε > 0 for which Bε (0)\{0} ⊂ ρ(T ), where Bε (0) is the open ball centered at 0 with radius ε. This can be restated as follows: T ∈ Br [H] if and only if T ∈ F[H] and λ ∈ ρ(T ) whenever 0 = |λ| < ε for some ε > 0. But this is precisely the claimed assertion. The Browder spectrum of an operator T ∈ B[H] is the set σb (T ) of all complex numbers λ for which λI − T is not Browder,   σb (T ) = λ ∈ C : λI − T ∈ B[H]\Br [H] (compare with Corollary 5.25). Since an operator lies in Br [H] together with its adjoint, it follows at once that λ ∈ σb (T ) if and only if λ ∈ σb (T ∗ ): σb (T ) = σb (T ∗ )∗ . Moreover, by the preceding definition if λ ∈ σb (T ), then λI − T ∈ / Br [H], and so either λI − T ∈ / F[H] or λI − T ∈ F[H] and asc (λI − T ) = dsc (λI − T ) = ∞ (cf. definition of Browder operators). In the former case λ ∈ σe (T ) ⊆ σ(T ) according to Corollary 5.12. In the latter case λI − T is not invertible (because N (λI − T ) = {0} if and only if asc(λI − T ) = 0 — also R(λI − T ) = H if and only if dsc(λI − T ) = 0), and so λ ∈ / ρ(T ); that is, λ ∈ σ(T ). Therefore σb (T ) ⊆ σ(T ). Since Br [H] ⊂ W[H], we also get σw (T ) ⊆ σb (T ) by Corollary 5.25. Hence σe (T ) ⊆ σw (T ) ⊆ σb (T ) ⊆ σ(T ). Equivalently, G[H] ⊆ Br [H] ⊆ W[H] ⊆ F[H]. Corollary 5.37. For every operator T ∈ B[H], σb (T ) = σe (T ) ∪ σacc (T ) = σw (T ) ∪ σacc (T ). Proof. Since σ(T ) = {λ} − σ(λI − T ) by Theorem 2.7 (Spectral Mapping), 0 ∈ σ(λI − T )\σiso (λI − T ) if and only if λ ∈ σ(T )\σiso (T ). Thus by the definition of σb (T ) and Corollaries 5.12 and 5.35,

5.4 Browder Operators and Browder Spectrum

171

  σb (T ) = λ ∈ C : λI − T ∈ F[H] or 0 ∈ ρ(λI − T ) ∪ σiso (λI − T )   = λ ∈ C : λ ∈ σe (T ) or 0 ∈ σ(λI − T )\σiso (λI − T )   = λ ∈ C : λ ∈ σe (T ) or λ ∈ σ(T )\σiso (T )   = σe (T ) ∪ σ(T )\σiso (T ) = σe (T ) ∪ σacc (T ). However, by Theorem 5.24 σw (T ) = σe (T ) ∪ κ(T ) where  σk (T ) ⊆ σ(T ) κ(T ) = k∈Z \{0}

is the collection of all holes of σe (T ), thus a union of open sets in C by Theorem 5.16, and therefore is itself an open set in C . Hence κ(T )\σacc (T ) = ∅ and so   σw (T )\σacc (T ) = σe (T ) ∪ κ(T ) \σacc (T )     = σe (T )\σacc (T ) ∪ κ(T )\σacc (T ) = σe (T )\σacc (T ).  The next result is particularly important: it gives another partition of σ(T ). Corollary 5.38. For every operator T ∈ B[H], σb (T ) = σ(T )\π0 (T ). Proof. By the Spectral Picture (Corollary 5.18) and Corollary 5.37,  ,    σ(T )\σb (T ) = σ(T ) σe (T ) ∪ σacc (T ) = σ(T )\σe (T ) ∩ (σ(T )\σacc (T ) 

= σk (T ) ∩ σiso (T ) = σ0 (T ) ∩ σiso (T ) = π0 (T ) k∈Z

according to the definition of π0 (T ), since σk (T ) is an open set in C for every  0 = k ∈ Z (Theorem 5.16) and so it has no isolated point. Since σb (T ) ∪ π0 (T ) ⊆ σ(T ), we get from Corollary 5.38 σb (T ) ∪ π0 (T ) = σ(T )

and

σb (T ) ∩ π0 (T ) = ∅,

and so {σb (T ), π0 (T )} forms a partition of the spectrum σ(T ). Thus   π0 (T ) = σ(T )\σb (T ) = λ ∈ σ(T ) : λI − T ∈ Br [H] , and σb (T ) = σ(T ) ⇐⇒ π0 (T ) = ∅. Corollary 5.39. For every operator T ∈ B[H], π0 (T ) = σiso (T )\σb (T ) = σiso (T )\σw (T ) = σiso (T )\σe (T ) = π00 (T )\σb (T ) = π00 (T )\σw (T ) = π00 (T )\σe (T ). Proof. The identities involving σb (T ) are readily verified. In fact, recall that π0 (T ) = σiso (T ) ∩ σ0 (T ) ⊆ σiso (T ) ∩ σPF (T ) = π00 (T ) ⊆ σiso (T ) ⊆ σ(T ). Since σb (T ) ∩ π0 (T ) = ∅ and σ(T )\π0 (T ) = σb (T ) (Corollary 5.38), we get

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5. Fredholm Theory in Hilbert Space

π0 (T ) = π0 (T )\σb (T ) ⊆ π00 (T )\σb (T ) ⊆ σiso (T )\σb (T ) ⊆ σ(T )\σb (T ) = π0 (T ). To prove the identities involving σw (T ) and σe (T ), proceed as follows. Since σ0 (T ) = σ(T )\σw (T ) (Theorem 5.24), we get   π0 (T ) = σiso (T ) ∩ σ0 (T ) = σiso (T ) ∩ σ(T )\σw (T ) = σiso (T )\σw (T ). Moreover, as we saw in the proof of Corollary 5.37, σw (T ) = σe (T ) ∪ κ(T ) where κ(T ) ⊆ σ(T ) is open in C (union of open sets). Hence σiso (T )\σw (T ) = σiso (T )\(σe (T ) ∪ κ(T )) = (σiso (T )\σe (T )) ∩ (σiso (T )\κ(T )). But σiso (T ) ∩ κ(T ) = ∅ because κ(T ) is a subset of σ(T ) which is open in C . Then σiso (T )\κ(T ) = σiso (T ) and so σiso (T )\σw (T ) = σiso (T )\σe (T ). Since π0 (T ) ⊆ π00 (T ) ⊆ σiso (T ), π0 (T ) ⊆ σ0 (T ) = σ(T )\σw (T ), σe (T ) ⊆ σw (T ) (cf. Theorem 5.24), and as we have just verified σiso (T )\σw (T ) = π0 (T ) and σiso (T )\σe (T ) = σiso (T )\σw (T ), then π00 (T )\σw (T ) ⊆ π00 (T )\σe (T ) ⊆ σiso (T )\σe (T ) = σiso (T )\σw (T ) = π0 (T ) = π0 (T )\σw (T ) ⊆ π00 (T )\σw (T ), and therefore π0 (T ) = π00 (T )\σw (T ) = π00 (T )\σe (T ).



Remark 5.40. (a) Finite Dimension. Take any T ∈ B[H]. Since every operator on a finite-dimensional space is Browder (cf. Remark 5.36(a)), and according to the definition of the Browder spectrum σb (T ), dim H < ∞

=⇒

σb (T ) = ∅.

The converse holds by Remark 5.27(a) since σw (T ) ⊆ σb (T ): dim H = ∞

=⇒

σb (T ) = ∅.

Thus σb (T ) = ∅

⇐⇒

dim H = ∞,

and σb (T ) = σ(T )\π0 (T ) (cf. Corollary 5.38) is a compact set in C because σ(T ) is compact and π0 (T ) ⊆ σiso (T ). Moreover, by Remark 5.27(b), dim H < ∞

=⇒

σe (T ) = σw (T ) = σb (T ) = ∅.

(b) Compact Operators. Since σacc (K) ⊆ {0} if K is compact (cf. Corollary 2.20), it follows by Remark 5.27(d), Corollary 5.37, and item (a) that K ∈ B∞[H] and dim H = ∞

=⇒

σe (K) = σw (K) = σb (K) = {0}.

5.4 Browder Operators and Browder Spectrum

173

(c) Finite Algebraic and Geometric Multiplicities. We saw in Corollary 5.28 that the set of Riesz points π0 (T ) = σiso (T ) ∩ σ0 (T ) and the set of isolated eigenvalues of finite multiplicity π00 (T ) = σiso (T ) ∩ σPF (T ) coincide if any of the equivalent assertions of Corollary 5.28 holds true. The previous corollary supplies a collection of conditions equivalent to π0 (T ) = π00 (T ). In fact, by Corollary 5.39 the following four assertions are pairwise equivalent. (i) π0 (T ) = π00 (T ). (ii) σe (T ) ∩ π00 (T ) = ∅

(i.e., σe (T ) ∩ σiso (T ) ∩ σPF (T ) = ∅ ).

(iii) σw (T ) ∩ π00 (T ) = ∅

(i.e., σw (T ) ∩ σiso (T ) ∩ σPF (T ) = ∅ ).

(iv) σb (T ) ∩ π00 (T ) = ∅

(i.e., σb (T ) ∩ σiso (T ) ∩ σPF (T ) = ∅ ).

Take any T ∈ B[H] and consider again the following relations: σ0 (T ) = σ(T )\σw (T ), π0 (T ) = σiso (T ) ∩ σ0 (T ), π00 (T ) = σiso (T ) ∩ σPF (T ). We have defined a class of operators for which the equivalent assertions of Corollary 5.28 hold true (namely, operators that satisfy Weyl’s Theorem). The following equivalent assertions define another important class of operators. An operator for which any of these equivalent assertions holds is said to satisfy Browder’s Theorem. This will be discussed later in Section 5.5. Corollary 5.41. The assertions below are pairwise equivalent. (a) σ0 (T ) = π0 (T ). (b) σ(T )\π0 (T ) = σw (T ). (c) σ(T ) = σw (T ) ∪ π00 (T ). (d) σ(T ) = σw (T ) ∪ σiso (T ). (e) σ0 (T ) ⊆ σiso (T ). (f) σ0 (T ) ⊆ π00 (T ). (g) σacc (T ) ⊆ σw (T ). (h) σw (T ) = σb (T ). Proof. Consider the partition {σw (T ), σ0 (T )} of σ(T ) and the inclusions π0 (T ) ⊆ π00 (T ) ⊆ σ(T ). Thus (a) and (b) are equivalent and, if (a) holds, σ(T ) = σw (T ) ∪ σ0 (T ) = σw (T ) ∪ π0 (T ) ⊆ σw (T ) ∪ π00 (T ) ⊆ σ(T ), and so (a) implies (c); and (c) implies (d) because π00 (T ) ⊆ σiso (T ): σ(T ) = σw (T ) ∪ π00 (T ) ⊆ σw (T ) ∪ σiso (T ) ⊆ σ(T ). If (d) holds, then (since σ0 (T ) = σ(T )\σw (T ))   σ0 (T ) = σ(T )\σw (T ) = σw (T ) ∪ σiso (T ) \σw (T ) ⊆ σiso (T ).

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5. Fredholm Theory in Hilbert Space

Thus (d) implies (e). Since π0 (T ) = σiso (T ) ∩ σ0 (T ), the inclusion in (e) is equivalent to the identity in (a). Since π00 (T ) = σiso (T ) ∩ σPF (T ) and since σ0 (T ) ⊆ σPF (T ), the inclusions in (e) and (f) are equivalent. Since σacc (T ) and σiso (T ) are complements of each other in σ(T ), and σw (T ) and σ0 (T ) are complements of each other in σ(T ), the inclusions in (e) and (g) are equivalent. Similarly, since σw (T ) and σ0 (T ) are complements of each other in σ(T ), and σb (T ) and π0 (T ) are complements of each other in σ(T ) (cf. Theorem 5.24 and Corollary 5.38), the identities in (h) and (a) are equivalent as well.  The Browder spectrum is not invariant under compact perturbation. In fact, if T ∈ B[H] is such that σb (T ) = σw (T ) (i.e., if T is such that any of the equivalent assertions in Corollary 5.41 fails — and so all of them fail), then there exists a compact K ∈ B∞[H] for which σb (T + K) = σb (T ). Indeed, if σb (T ) = σw (T ), then σw (T ) ⊂ σb (T ), and so σb (T ) is not invariant under compact perturbation because σw (T ) is the largest part of σ(T ) that remains unchanged under compact perturbation. However, as we will see in the forthcoming Theorem 5.44 (whose proof is based on Lemmas 5.42 and 5.43 below), σb (T ) is invariant under perturbation by compact operators that commute with T . We follow the framework set forth in [79]. Set A[H] = B[H] where H is an infinite-dimensional complex Hilbert space. Let A [H] be a unital closed subalgebra of the unital complex Banach algebra A[H], thus a unital complex Banach algebra itself. Take an arbitrary operator  [H] = B∞[H] ∩ A [H] denote the collection of all compact T in A [H]. Let B∞  operators from A [H], and let F  [H] denote the class of all Fredholm operators in the unital complex Banach algebra A [H]. That is (Proposition 5.C),    F  [H] = T ∈ A [H] : A T − I and T A − I are in B∞ [H] for some A ∈ A [H] . Therefore F  [H] ⊆ F[H] ∩ A [H]. The inclusion is proper in general (see, e.g., [11, p. 86]) but it becomes an identity if A [H] is a ∗-subalgebra of the C*algebra A[H]; that is, if the unital closed subalgebra A [H] is a ∗-algebra (see, e.g., [11, Theorem A.1.3]). However, if the inclusion becomes an identity, then by Corollary 5.12 the essential spectra in A[H] and in A [H] coincide: F  [H] = F[H] ∩ A [H] where

=⇒

σe (T ) = σe (T ),

  σe (T ) = λ ∈ σ  (T ) : λI − T ∈ A [H]\F  [H]

is the essential spectrum of T ∈ A [H] with respect to A [H], with σ  (T ) denoting the spectrum of T ∈ A [H] with respect to the unital complex Banach algebra A [H]. Similarly, let W  [H] denote the class of Weyl operators in A [H],   W  [H] = T ∈ F  [H] : ind (T ) = 0 ,

5.4 Browder Operators and Browder Spectrum

and let

175

   (T ) = λ ∈ σ  (T ) : λI − T ∈ A [H]\W  [H] σw

be the Weyl spectrum of T ∈ A [H] with respect to A [H]. If T ∈ A [H], set    σ0 (T ) = σ  (T )\σw (T ) = λ ∈ σ  (T ) : λI − T ∈ W  [H] . Moreover, let B r [H] stand for the class of Browder operators in A [H],    B r [H] = T ∈ F  [H] : 0 ∈ ρ (T ) ∪ σiso (T )  (T ) denotes the set of all isolated points according to Corollary 5.35, where σiso    of σ (T ), and ρ (T ) = C \σ (T ) is the resolvent set of T ∈ A [H] with respect to the unital complex Banach algebra A [H]. Let   σb (T ) = λ ∈ σ  (T ) : λI − T ∈ A [H]\B r [H]

be the Browder spectrum of T ∈ A [H] with respect to A [H]. Lemma 5.42. Consider the preceding setup. If T ∈ A [H], then σ  (T ) = σ(T )

=⇒

σb (T ) = σb (T ).

Proof. Take an arbitrary operator T in A [H]. Suppose σ  (T ) = σ(T ),   so that ρ (T ) = ρ(T ) and σiso (T ) = σiso (T ). Therefore if λ ∈ σiso (T ), then the Riesz idempotent associated with λ, % 1 Eλ = 2πi (ζI − T )−1 dζ, Υλ



 σiso (T )

= σiso (T ), and also (ζI − T )−1 ∈ A [H] whenlies in A [H] (because −1 ever (ζI − T ) ∈ A[H] = B[H] since ρ (T ) = ρ(T )). Thus by Theorem 5.19, λ ∈ σ0 (T )

⇐⇒

dim R(Eλ ) < ∞

⇐⇒

λ ∈ σ0 (T )

  whenever λ ∈ σiso (T ), since σiso (T ) = σiso (T ). Hence  π0 (T ) = σ0 (T ) ∩ σiso (T ) = σ0 (T ) ∩ σiso (T ) = π0 (T ).

Then by Corollary 5.38, σb (T ) = σ(T )\π0 (T ) = σ  (T )\π0 (T ) = σb (T ).



Let {T } denote the commutant of T ∈ A[H] (i.e., the collection of all operators in B[H] that commute with T ∈ B[H]). This is a unital subalgebra of the unital complex Banach algebra A[H] = B[H], which is weakly closed in B[H] (see, e.g., [75, Problem 3.7]), and hence uniformly closed (i.e., closed in the operator norm topology of B[H]). Therefore A [H] = {T } is a unital closed subalgebra of the unital complex Banach algebra A[H] = B[H].

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5. Fredholm Theory in Hilbert Space

Lemma 5.43. Consider the preceding setup with A [H] = {T }. If 0 ∈ σ  (T )  (T ). Equivalently, and T ∈ W  [H], then 0 ∈ σiso A [H] = {T } and 0 ∈ σ0 (T )

=⇒

 0 ∈ σiso (T ).

Proof. Take T ∈ B[H]. Let A [H] be a unital closed subalgebra of A[H] = B[H] that includes T . Suppose 0 ∈ σ0 (T ), which means 0 ∈ σ  (T ) and T ∈ W  [H].  [H] = B∞[H] ∩ A [H], actually a Since T ∈ W  [H], there is a compact K ∈ B∞  finite-rank operator, and an operator A ∈ A [H] such that (cf. Lemma 5.23(c)) A(T + K) = (T + K)A = I. So A ∈ A [H] is invertible with inverse A−1 = T + K ∈ A [H]. If A [H] = {T }, then A K = KA and so A−1 K = KA−1. Let A [H] be the unital closed commutative subalgebra of A [H] generated by A, A−1, and K. Since T = A−1 − K,   (T ) stand for the sets of isolated points (T ) and σiso A [H] includes T . Let σiso   of the spectra σ (T ) and σ (T ) of T with respect to the Banach algebras A [H] and A [H], respectively. Since A [H] ⊆ A [H], it follows by Proposition 2.Q(a) that σ  (T ) ⊆ σ  (T ). Hence 0 ∈ σ  (T ) because 0 ∈ σ0 (T ) ⊆ σ  (T ). Claim.

 0 ∈ σiso (T ).

Proof. Let A [H] denote the collection of all algebra homomorphisms of A [H] into C . From Proposition 2.Q(b) we get   σ  (A−1 ) = Φ(A−1 ) ∈ C : Φ ∈ A [H] , which is bounded away from zero (since 0 ∈ ρ (A−1 )), and     σ  (K) = Φ(K) ∈ C : Φ ∈ A [H] = {0} ∪ Φ(K) ∈ C : Φ ∈ AF [H] , where AF [H] ⊆ A [H] is a set of nonzero homomorphisms, which is finite since K is a finite-rank operator, and so it has a finite spectrum (Corollary 2.19). Also, 0 ∈ σ  (T ) = σ  (A−1−K) if and only if 0 = Φ(A−1−K) = Φ(A−1)−Φ(K) for some Φ ∈ A [H]. If Φ ∈ A [H]\AF [H], then Φ(K) = 0. So Φ(A−1 ) = 0, which is a contradiction (since 0 ∈ σ  (A−1 )). Thus Φ ∈ AF [H], and therefore Φ(A−1 − K) = Φ(A−1 ) − Φ(K) = 0. Then Φ(A−1 ) = Φ(K) for at most a finite number of homomorphisms Φ in AF [H]. Hence, since the set {Φ(A−1 ) ∈ C : Φ ∈ A [H]} = σ  (A−1 ) is bounded away from zero, and since the set     Φ ∈ AF [H] : Φ(A−1 ) = Φ(K) = Φ ∈ AF [H] : 0 ∈ σ  (A−1 − K) is finite, it follows that 0 is an isolated point of σ  (A−1 − K) = σ  (T ), which concludes the proof of the claimed result.  Since 0 ∈ σ  (T ) ⊆ σ  (T ), then 0 ∈ σiso (T ).



The next characterization of the Browder spectrum is the counterpart of the very definition of the Weyl spectrum. It says that σb (T ) is the largest

5.4 Browder Operators and Browder Spectrum

177

part of σ(T ) that remains unchanged under compact perturbations in the commutant of T . That is, σb (T ) is the largest part of σ(T ) such that [88, 70] σb (T + K) = σb (T ) for every K ∈ B∞[H] ∩ {T } . Theorem 5.44. For every T ∈ B[H], ! σb (T ) =

K∈B∞[H]∩{T }

σ(T + K).

Proof. Take T ∈ B[H]. Let A [H] be a unital closed subalgebra of the unital complex Banach algebra A[H] = B[H] (thus a unital complex Banach algebra  (T ) be the spectrum, the itself). If T ∈ A [H], then let σ  (T ), σb (T ), and σw Browder spectrum, and the Weyl spectrum of T with respect to A [H]. Claim 1. If A [H] = {T }, then σ  (T ) = σ(T ). Proof. Suppose A [H] = {T }. Trivially, T ∈ A [H]. Let P[H] = P(T ) be the collection of all polynomials p(T ) in T with complex coefficients, which is a unital commutative subalgebra of A[H] = B[H]. Consider the collection T of all unital commutative subalgebras of A[H] containing T. Every element of T is included in A [H], and T is partially ordered (in the inclusion ordering) and nonempty (e.g., P[H] ∈ T — and so P[H] ⊆ A [H]). Moreover, every chain in T has an upper bound in T (the union of all subalgebras in a given chain of subalgebras in T is again a subalgebra in T ). Thus according to Zorn’s Lemma, T has a maximal element, say A [H] = A (T ) ∈ T. Hence there is a maximal commutative subalgebra A [H] of B[H] containing T (which is unital and closed — see the paragraph that precedes Proposition 2.Q). Therefore A [H] ⊆ A [H] ⊆ A[H]. Let σ  (T ) denote the spectrum of T with respect to A [H]. If the preceding inclusions hold true, and since A [H] is a maximal commutative subalgebra of A[H], then Proposition 2.Q(a,b) ensures the following relations: σ(T ) ⊆ σ  (T ) ⊆ σ  (T ) = σ(T ). Claim 2. If A [H] = {T }, then σb (T ) = σb (T ). Proof. Claim 1 and Lemma 5.42.  Claim 3. If A [H] = {T }, then σb (T ) = σw (T ).

Proof. By definition λ ∈ σ0 (T ) if and only if λ ∈ σ  (T ) and λI − T ∈ W  [H]. But λ ∈ σ  (T ) if and only if 0 ∈ σ  (λI − T ) by the Spectral Mapping Theorem (Theorem 2.7). Hence λ ∈ σ0 (T )

⇐⇒

0 ∈ σ0 (λI − T ).

Since A [H] = {T } = {λI − T } we get by Lemma 5.43

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5. Fredholm Theory in Hilbert Space

0 ∈ σ0 (λI − T )

=⇒

 0 ∈ σiso (λI − T ).

However, applying the Spectral Mapping Theorem again,  (λI − T ) 0 ∈ σiso

Therefore

 (T ). λ ∈ σiso

⇐⇒

 σ0 (T ) ⊆ σiso (T ),

which means (cf. Corollary 5.41)  σw (T ) = σb (T ).

Claim 4.

 (T ) = σw

K∈B∞[H]∩A [H]

σ(T + K).

Proof. This is the definition of the Weyl spectrum of T with respect to A [H]:    (T ) = K∈B∞ σw  [H] σ(T + K), where B∞ [H] = B∞[H] ∩ A [H]. By Claims 2, 3, and 4 we have ! σb (T ) =

K∈B∞[H]∩{T }

σ(T + K).



5.5 Remarks on Browder and Weyl Theorems This final section consist of a brief survey on Weyl’s and Browder’s Theorems. As such, and unlike the previous sections, some of the assertions discussed here, instead of being fully proved, will be accompanied by a reference to indicate where the proof can be found. Take any operator T ∈ B[H] and consider the partitions {σw (T ), σ0 (T )} and {σb (T ), π0 (T )} of the spectrum of T in terms of Weyl and Browder spectra σw (T ) and σb (T ) and their complements σ0 (T ) and π0 (T ) in σ(T ) as in Theorem 5.24 and Corollary 5.38, so that σw (T ) = σ(T )\σ0 (T )

and

σb (T ) = σ(T )\π0 (T ),

where   σ0 (T ) = λ ∈ σ(T ) : λI − T ∈ W[H]

and

π0 (T ) = σiso (T ) ∩ σ0 (T ).

Although σ0 (T ) ⊆ σPF (T ) and σacc (T ) ⊆ σb (T ) (i.e., π0 (T ) ⊆ σiso (T ) ) for every T , in general σ0 (T ) may not be included in σiso (T ); equivalently, σacc (T ) may not be included in σw (T ). Recall the definition of π00 (T ): π00 (T ) = σiso (T ) ∩ σPF (T ). Definition 5.45. An operator T is said to satisfy Weyl’s Theorem (or Weyl’s Theorem holds for T ) if any of the equivalent assertions of Corollary 5.28 holds true. Therefore an operator T satisfies Weyl’s Theorem if

5.5 Remarks on Browder and Weyl Theorems

179

σ0 (T ) = σiso (T ) ∩ σPF (T ). In other words, if σ0 (T ) = π00 (T ) or, equivalently, if σ(T )\π00 (T ) = σw (T ). Further necessary and sufficient conditions for an operator to satisfy Weyl’s Theorem can be found in [51]. See also [69, Chapter 11]. Definition 5.46. An operator T is said to satisfy Browder’s Theorem (or Browder’s Theorem holds for T ) if any of the equivalent assertions of Corollary 5.41 holds true. Therefore an operator T satisfies Browder’s Theorem if σ0 (T ) ⊆ σiso (T ) ∩ σPF (T ). That is, σ0 (T ) ⊆ π00 (T ) or, equivalently, σ0 (T ) = π0 (T ),

σ0 (T ) ⊆ σiso (T ),

or

σacc (T ) ⊆ σw (T ),

or

or

σw (T ) = σb (T ).

A word on terminology. The expressions “T satisfies Weyl’s Theorem” and “T satisfies Browder’s Theorem” have become standard and are often used in current literature and we will stick to them, although saying T “satisfies Weyl’s (or Browder’s) property” rather than “satisfies Weyl’s (or Browder’s) Theorem” would perhaps sound more appropriate. Remark 5.47. (a) Sufficiency for Weyl’s. σw (T ) = σacc (T )

=⇒

T satisfies Weyl’s Theorem,

according to Corollary 5.28. (b) Browder’s not Weyl’s. Consider Definitions 5.45 and 5.46. If T satisfies Weyl’s Theorem, then it obviously satisfies Browder’s Theorem. An operator T satisfies Browder’s but not Weyl’s Theorem if and only if the proper inclusion σ0 (T ) ⊂ σiso (T ) ∩ σPF (T ) holds true, and so there exists an isolated eigenvalue of finite multiplicity not in σ0 (T ) (i.e., there exists an isolated eigenvalue of finite multiplicity in σw (T ) = σ(T )\σ0 (T )). Outcome: If T satisfies Browder’s Theorem but not Weyl’s Theorem, then σw (T ) ∩ σiso (T ) ∩ σPF (T ) = ∅. (c) Equivalent Condition. The preceding result can be extended as follows (cf. Remark 5.40(c)). Consider the equivalent assertions of Corollary 5.28 and of Corollary 5.41. If Browder’s Theorem holds and π0 (T ) = π00 (T ), then Weyl’s Theorem holds (i.e., if σ0 (T ) = π0 (T ) and π0 (T ) = π00 (T ), then σ0 (T ) = π00 (T ) tautologically). Conversely, if Weyl’s Theorem holds, then π0 (T ) = π00 (T ) (see Corollary 5.28), and Browder’s Theorem holds trivially. Summing up: Weyl’s Theorem holds ⇐⇒ Browder’s Theorem holds and π0 (T ) = π00 (T ).

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5. Fredholm Theory in Hilbert Space

That is, Weyl’s Theorem holds if and only if Browder’s Theorem and any of the equivalent assertions of Remark 5.40(c) hold true. (d) Trivial Cases. By Remark 5.27(a), if dim H < ∞, then σ0 (T ) = σ(T ). Thus, according to Corollary 2.19, dim H < ∞ =⇒ σ0 (T ) = π00 (T ) = σ(T ): Every operator on a finite-dimensional space satisfies Weyl’s Theorem. This extends to finite-rank but not to compact operators (see examples in [44]). On the other hand, if σP (T ) = ∅, then σ0 (T ) = π00 (T ) = ∅: Every operator without eigenvalues satisfies Weyl’s Theorem. Since σ0 (T ) ⊆ σPF (T ), this extends to operators with σPF (T ) = ∅. Investigating quadratic forms with compact difference, Hermann Weyl proved in 1909 that Weyl’s Theorem holds for self-adjoint operators (i.e., every self-adjoint operator satisfies Weyl’s Theorem) [114]. We present next a contemporary proof of Weyl’s original result. First recall from Sections 1.5 and 1.6 the following elementary (proper) inclusion of classes of operators: Self-Adjoint

⊂

Normal

⊂

Hyponormal.

Theorem 5.48. (Weyl’s Theorem). If T ∈ B[H] is self-adjoint, then σ0 (T ) = σiso (T ) ∩ σPF (T ). Proof. Take an arbitrary operator T ∈ B[H]. Claim 1. If T is self-adjoint, then σ0 (T ) = π0 (T ). Proof. If T is self-adjoint, then σ(T ) ⊂ R (Proposition 2.A). Thus no subset of σ(T ) is open in C , and hence σ0 (T ) = τ0 (T ) ∪ π0 (T ) = π0 (T ) because τ0 (T ) is open in C (Corollary 5.20). Claim 2. If T is hyponormal and λ ∈ π00 (T ), then R(λI − T ) is closed. Proof. λ ∈ π00 (T ) = σiso (T ) ∩ σPF (T ) if and only if λ is an isolated point of σ(T ) and 0 < dim N (λI − T ) < ∞. Thus, if λ ∈ π00 (T ) and T is hyponormal, then dim R(Eλ ) < ∞ by Proposition 4.L, where Eλ is the Riesz idempotent associated with λ. Then R(λI − T ) is closed by Corollary 4.22. Claim 3. If T is hyponormal, then π0 (T ) = π00 (T ). Proof. By Claim 2 if T is hyponormal, then (cf. Corollary 5.22)   π0 (T ) = λ ∈ π00 (T ) : R(λI − T ) is closed = π00 (T ). In particular, if T is self-adjoint, then π0 (T ) = π00 (T ) and so, by Claim 1 σ0 (T ) = π0 (T ) = π00 (T ) = σiso (T ) ∩ σPF (T ).



5.5 Remarks on Browder and Weyl Theorems

181

What the preceding proof says is this: if T is self-adjoint, then it satisfies Browder’s Theorem, and if T is hyponormal, then π0 (T ) = π00 (T ). Thus if T is self-adjoint, then T satisfies Weyl’s Theorem by Remark 5.47(c). Theorem 5.48 was extended to normal operators in [107]. This can be verified by Proposition 5.E, according to which if T is normal, then σ(T )\σe (T ) = π00

(T ). But Corollary 5.18 says σ(T )\σe (T ) = k∈Z \{0} σk (T ) ∪ σ0 (T ), where k∈Z \{0} σk (T ) is open in C (cf. Theorem 5.16) and π00 (T ) = σiso (T ) ∩ σPF (T ) is closed in C . Thus σ0 (T ) = π00 (T ). Therefore every normal operator satisfies Weyl’s Theorem. Moreover, Theorem 5.48 was further extended to hyponormal operators in [27] and to seminormal operators in [16]. In other words: If T or T ∗ is hyponormal, then T satisfies Weyl’s Theorem. Some additional definitions and terminology will be needed. An operator is isoloid if every isolated point of its spectrum is an eigenvalue; that is, T is isoloid if σiso (T ) ⊆ σP (T ). A Hilbert-space operator T is said to be dominant if R(λI − T ) ⊆ R(λI − T ∗ ) for every λ ∈ C or, equivalently, if for each λ ∈ C there is an Mλ ≥ 0 for which (λI − T )(λI − T )∗ ≤ Mλ (λI − T )∗ (λI − T ) [37]. Therefore every hyponormal operator is dominant (with Mλ = 1) and isoloid (Proposition 4.L). Recall that a subspace M of a Hilbert space H is invariant for an operator T ∈ B[H] (or T -invariant) if T (M) ⊆ M, and reducing if it is invariant for both T and T ∗. A part of an operator is a restriction of it to an invariant subspace, and a direct summand is a restriction of it to a reducing subspace. The main result in [16] reads as follows (see also [17]). Theorem 5.49. If each finite-dimensional eigenspace of an operator on a Hilbert space is reducing, and if every direct summand of it is isoloid, then it satisfies Weyl’s Theorem. This is a fundamental result that includes many of the previous results along this line, and has also been frequently applied to yield further results, mainly through the following corollary. Corollary 5.50. If an operator on a Hilbert space is dominant and every direct summand of it is isoloid, then it satisfies Weyl’s Theorem. Proof. Take any T ∈ B[H]. The announced result can be restated as follows. If R(λI − T ) ⊆ R(λI − T ∗ ) for every λ ∈ C , and the restriction T |M of T to each reducing subspace M is such that σiso (T |M ) ⊆ σP (T |M ), then T satisfies Weyl’s Theorem. To prove it we need the following elementary result, which extends Lemma 1.13(a) from hyponormal to dominant operators. Take an arbitrary scalar λ ∈ C . Claim. If R(λI − T ) ⊆ R(λI − T ∗ ), then N (λI − T ) ⊆ N (λI − T ∗ ). Proof. Take any S ∈ B[H]. If R(S) ⊆ R(S ∗ ), then R(S ∗ )⊥ ⊆ R(S)⊥, which is equivalent to N (S) ⊆ N (S ∗ ) by Lemma 1.4, thus proving the claimed result.

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5. Fredholm Theory in Hilbert Space

If R(λI − T ) ⊆ R(λI − T ∗ ), then N (λI − T ) reduces T by the preceding Claim and Lemma 1.14(b). Therefore if R(λI − T ) ⊆ R(λI − T ∗ ) for every λ ∈ C , then every eigenspace of T is reducing and so is, in particular, every finite-dimensional eigenspace of T . Thus the stated result is a straightforward consequence of Theorem 5.49.  Since every hyponormal operator is dominant, every direct summand of a hyponormal operator is again hyponormal (in fact, every part of a hyponormal operator is again hyponormal — Proposition 1.P), and every hyponormal operator is isoloid (Proposition 4.L), then Corollary 5.50 offers another proof that every hyponormal operator satisfies Weyl’s Theorem. We need a few more definitions and terminologies. An operator T ∈ B[H] is paranormal if T x2 ≤ T 2 x x for every x ∈ H, and totally hereditarily normaloid (THN) if all parts of it are normaloid, as well as the inverse of all invertible parts. (Tautologically, totally hereditarily normaloid operators are normaloid.) Hyponormal operators are paranormal and dominant, but paranormal operators are not necessarily dominant. These classes are related by proper inclusion (see [43, Remark 1] and [44, Proposition 2]):   Hyponormal ⊂ Paranormal ⊂ THN ⊂ Normaloid ∩ Isoloid . Weyl’s Theorem has been extended to classes of nondominant operators that properly include the hyponormal operators. For instance, it was extended to paranormal operators in [112] and beyond to totally hereditarily normaloid operators in [42]. In fact [42, Lemma 2.5]: If T is totally hereditarily normaloid, then both T and T ∗ satisfy Weyl’s Theorem (variations along this line in [40, Theorem 3.9], [41, Corollary 2.16]). So Weyl’s Theorem holds for paranormal operators and their adjoints and, in particular, Weyl’s Theorem holds for hyponormal operators and their adjoints. Let T and S be arbitrary operators acting on Hilbert spaces. First we consider their (orthogonal) direct sum. The spectrum of a direct sum coincides with the union of the spectra of the summands by Proposition 2.F(b), σ(T ⊕ S) = σ(T ) ∪ σ(S). For the Weyl spectra, only inclusion is ensured. In fact, the Weyl spectrum of a direct sum is included in the union of the Weyl spectra of the summands, σw (T ⊕ S) ⊆ σw (T ) ∪ σw (S), but equality does not hold in general [61]. However, the equality holds if σw (T ) ∩ σw (S) has empty interior [90],

5.5 Remarks on Browder and Weyl Theorems

 ◦ σw (T ) ∩ σw (S) = ∅

=⇒

183

σw (T ⊕ S) = σw (T ) ∪ σw (S).

In general, Weyl’s Theorem does not transfer from T and S to their direct sum T ⊕ S. The above identity involving the Weyl spectrum of a direct sum, viz., σw (T ⊕ S) = σw (T ) ∪ σw (S), when it holds, plays an important role in establishing sufficient conditions for a direct sum to satisfy Weyl’s Theorem. This was recently investigated in [89] and [44]. As for the problem of transferring Browder’s Theorem from T and S to their direct sum T ⊕ S, the following necessary and sufficient condition was proved in [61, Theorem 4]. Theorem 5.51. If both operators T and S satisfy Browder’s Theorem, then the direct sum T ⊕ S satisfies Browder’s Theorem if and only if σw (T ⊕ S) = σw (T ) ∪ σw (S). Now consider the tensor product T ⊗ S of a pair of Hilbert-space operators T and S (for an expository paper on tensor products which will suffice for our needs see [76]). As is known from [22], the spectrum of a tensor product coincides with the product of the spectra of the factors, σ(T ⊗ S) = σ(T ) · σ(S). For the Weyl spectrum the following inclusion was proved in [68]: σw (T ⊗ S) ⊆ σw (T ) · σ(S) ∪ σ(T ) · σw (S). However, it remained an open question whether such an inclusion might be an identity; that is, it was not known if there existed a pair of operators T and S for which the above inclusion was proper. This question was solved recently, as we will see later. Sufficient conditions ensuring that the equality holds were investigated in [83]. For instance, if σe (T )\{0} = σw (T )\{0}

and

σe (S)\{0} = σw (S)\{0}

(which holds, in particular, for compact operators T and S), or if σw (T ⊗ S) = σb (T ⊗ S) (the tensor product satisfies Browder’s Theorem), then [83, Proposition 6] σw (T ⊗ S) = σw (T ) · σ(S) ∪ σ(T ) · σw (S). Again, Weyl’s Theorem does not necessarily transfer from T and S to their tensor product T ⊗ S. The preceding identity involving the Weyl spectrum of a tensor product, namely, σw (T ⊗ S) = σw (T ) · σ(S) ∪ σ(T ) · σw (S), when it holds, plays a crucial role in establishing sufficient conditions for a tensor product to satisfy Weyl’s Theorem, as was recently investigated in [108], [83], and [84]. As for the problem of transferring Browder’s Theorem from T and S

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5. Fredholm Theory in Hilbert Space

to their tensor product T ⊗ S, the following necessary and sufficient condition was proved in [83, Corollary 6]. Theorem 5.52. If both operators T and S satisfy Browder’s Theorem, then the tensor product T ⊗ S satisfies Browder’s Theorem if and only if σw (T ⊗ S) = σw (T ) · σ(S) ∪ σ(T ) · σw (S). According to Theorem 5.52, if there exist operators T and S satisfying Browder’s Theorem for which T ⊗ S does not satisfy Browder’s Theorem, then the Weyl spectrum identity, viz., σw (T ⊗ S) = σw (T ) · σ(S) ∪ σ(T ) · σw (S), does not hold for such a pair of operators. An example of a pair of operators that satisfy Weyl’s Theorem (and thus satisfy Browder’s Theorem) but whose tensor product does not satisfy Browder’s Theorem was recently supplied in [73]. Therefore, [73] and [83] together ensure that there exists a pair of operators T and S for which the inclusion σw (T ⊗ S) ⊂ σw (T ) · σ(S) ∪ σ(T ) · σw (S) is proper; that is, for which the Weyl spectrum identity fails.

5.6 Supplementary Propositions Proposition 5.A. Both classes of left and right semi-Fredholm operators F [H] and Fr [H] are open sets in B[H], and so are SF[H] and F[H]. Proposition 5.B. The mapping ind ( · ) : SF[H] → Z is continuous, where the topology on SF[H] is the topology inherited from B[H], and the topology on Z is the discrete topology. Proposition 5.C. Take T ∈ B[H]. The following assertions are equivalent. (a) T ∈ F[H]. (b) There exists A ∈ B[H] such that I − A T and I − T A are compact. (c) There exists A ∈ B[H] such that I − A T and I − T A are finite-rank . Proposition 5.D. Take an operator T in B[H]. If λ ∈ C \σe (T ) ∩ σre (T ), then there exists an ε > 0 such that dim N (ζI − T ) and dim N (ζI − T ∗ ) are constant in the punctured disk Bε (λ)\{λ} = {ζ ∈ C : 0 < |ζ − λ| < ε}. Proposition 5.E. If T ∈ B[H] is normal, then σe (T ) = σre (T ) = σe (T ) and σ(T )\σe (T ) = σiso (T ) ∩ σPF (T )

(i.e., σ(T )\σe (T ) = π00 ).

(Thus σe (T ) = σw (T ) = σb (T ) and σ0 (T ) = π00 (T) = π0 (T ) — no holes or pseudoholes — cf. Theorems 5.18, 5.24, 5.38, 5.48 and Remark 5.27(e).)

5.6 Supplementary Propositions

185

Proposition 5.F. Let D denote the open unit disk centered at the origin of 2 ] is a unilatthe complex plane, and let T = ∂ D be the unit circle. If S+ ∈ B[+ 2 2 eral shift on + = + (C ), then σe (S+ ) = σre (S+ ) = σe (S+ ) = σC (S+ ) = ∂σ(S+ ) = T , and ind (λI − S+ ) = −1 if |λ| < 1 (i.e., if λ ∈ D = σ(S+ )\∂σ(S+ )). Proposition 5.G. If ∅ = Ω ⊂ C is a compact set, {Λk }k∈Z is a collection of open sets included in Ω whose nonempty sets are pairwise disjoint, and Δ ⊆ Ω is a discrete set (i.e., containing only isolated points), then there exists T ∈ B[H] such that σ(T ) = Ω, σk (T ) = Λk for each k = 0, σ0 (T ) = Λ0 ∪ Δ, and for each λ ∈ Δ there is a positive integer nλ ∈ N such that dim R(Eλ ) = nλ (where Eλ is the Riesz idempotent associated with the isolated point λ ∈ Δ). The above proposition shows that every spectral picture is attainable [28]. If K ∈ B∞[H] is compact, then by Remark 5.27(b,d) either σw (K) = ∅ if dim H < ∞, or σw (K) = {0} if dim H = ∞. According to the next result, on an infinite-dimensional separable Hilbert space, every compact operator is a commutator . (An operator T is a commutator if there are operators A and B such that T = A B − BA.) Proposition 5.H. If T ∈ B[H] is an operator on an infinite-dimensional separable Hilbert space H, then 0 ∈ σw (T )

=⇒

T is a commutator.

Proposition 5.I. Take any operator T in B[H]. (a) If dim N (T ) < ∞ or dim N (T ∗ ) < ∞, then (a1 ) asc (T ) < ∞ =⇒ dim N (T ) ≤ dim N (T ∗ ), (a2 ) dsc (T ) < ∞ =⇒ dim N (T ∗ ) ≤ dim N (T ). (b) If dim N (T ) = dim N (T ∗ ) < ∞, then asc (T ) < ∞ ⇐⇒ dsc(T ) < ∞. Proposition 5.J. Suppose T ∈ B[H] is a Fredholm operator (i.e., T ∈ F[H] ). (a) If asc(T ) < ∞ and dsc (T ) < ∞, then ind (T ) = 0. (b) If ind (T ) = 0, then asc (T ) < ∞ if and only if dsc(T ) < ∞. Therefore

  Br [H] ⊆ T ∈ W[H] : asc(T ) < ∞ ⇐⇒ dsc (T ) < ∞ ,   W[H]\Br [H] = T ∈ W[H] : asc (T ) = dsc (T ) = ∞ .

The next result is an extension of the Fredholm Alternative of Remark 5.7(b), and also of the ultimate form of it in Remark 5.27(c).

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5. Fredholm Theory in Hilbert Space

Proposition 5.K. Take any compact operator K in B∞[H]. (a) If λ = 0, then λI − K ∈ Br [H]. (b) σ(K)\{0} = π0 (K)\{0} = σiso (K)\{0}. Proposition 5.L. Take T ∈ B[H]. If σw (T ) is simply connected (so it has no holes), then T + K satisfies Browder’s Theorem for every K ∈ B∞[H]. Notes: Propositions 5.A to 5.C are standard results on Fredholm and semiFredholm operators. For instance, see [97, Proposition 1.25] or [30, Proposition XI.2.6] for Proposition 5.A, and [97, Proposition 1.17] or [30, Proposition XI.3.13] for Proposition 5.B. For Proposition 5.C see [7, Remark 3.3.3] or [58, Problem 181]. The locally constant dimension of kernels is considered in Proposition 5.D (see, e.g., [30, Theorem XI.6.7]), and a finer analysis of the spectra of normal operators and of unilateral shifts is discussed in Propositions 5.E and 5.F (see, e.g., [30, Proposition XI.4.6] and [30, Example XI.4.10]). Every spectral picture is attainable [28], and this is described in Proposition 5.G — see [30, Proposition XI.6.13]. For Proposition 5.H see [17, §7]. The results of Proposition 5.I are from [26, p. 57] (see also [36]), and Proposition 5.J is an immediate consequence of Proposition 5.I. Regarding the Fredholm Alternative version of Proposition 5.K, for item (a) see [96, Theorem 1.4.6], and item (b) follows from Corollary 5.35, which goes back to Corollary 2.20. The compact perturbation result of Proposition 5.L is from [10, Theorem 11].

Suggested Readings Aiena [3] Arveson [7] Barnes, Murphy, Smyth, and West [11] Caradus, Pfaffenberger, and Yood [26] Conway [30, 32] Douglas [38] Halmos [58] Harte [60]

Istrˇ a¸tescu [69] Kato [72] M¨ uller [95] Murphy [96] Pearcy [97] Schechter [106] Sunder [109] Taylor and Lay [111]

Appendices

Appendix A Aspects of Fredholm Theory in Banach Space

Appendix B A Glimpse at Multiplicity Theory

A Aspects of Fredholm Theory in Banach Space

Chapter 5, whose framework was borrowed from [77], dealt with Fredholm theory in Hilbert space. The Hilbert-space geometry yields a tremendously rich structure leading to remarkable simplifications. In a Banach space, not all subspaces are complemented. This drives the need for a finer analysis of semi-Fredholm operators on Banach spaces where, in addition to the notions of left and right semi-Fredholmness of Section 5.1, the concepts of upper and lower semi-Fredholmness are also required. Semi-Fredholm operators can be defined either as the union of the class of all left semi-Fredholm and the class of all right semi-Fredholm operators (as in Section 5.1) on the one hand or, on the other hand, as the union of the class of all upper semi-Fredholm and the class of all lower semi-Fredholm operators (which will be defined in Section A.4). These two ways of handling semi-Fredholm operators are referred to as left-right and upper-lower approaches. In a Hilbert space these approaches coincide, and the reason they do coincide is precisely the fact that every subspace of a Hilbert space is complemented (as a consequence of the Projection Theorem of Section 1.3). Such complementation may, however, fail in a Banach space (as was briefly commented in Remark 5.3(a)). And this is the source of the difference between these two approaches, which arises when investigating semi-Fredholm operators on a Banach space. This appendix draws a parallel between the left-right and upper-lower ways of handling semi-Fredholm operators, and provides an analysis of some consequences of the lack of complementation in Banach spaces. The framework here follows [85].

A.1 Quotient Space Let X be a linear space over a field F and let M be a linear manifold of X . An algebraic complement of M is any linear manifold N of X such that M + N = X and M ∩ N = {0}. Every linear manifold has an algebraic complement, and two linear manifolds M and N satisfying the above identities are referred to as a pair of complementary linear manifolds. See Section 1.1. Take a linear manifold M of a linear space X . The quotient space X /M of X modulo M is the collection of all translations of the linear manifold M © Springer Nature Switzerland AG 2020 C. S. Kubrusly, Spectral Theory of Bounded Linear Operators, https://doi.org/10.1007/978-3-030-33149-8

189

190

A. Aspects of Fredholm Theory in Banach Space

(also called affine spaces or linear varieties). For each vector x ∈ X the set [x] = x + M ∈ X /M (i.e., the translate of M by x) is referred to as the coset of x modulo M (which is an equivalence class [x] ⊂ X of vectors in X , where the equivalence relation ∼ is given by x ∼ x if x − x ∈ M). The quotient space X /M is the collection of all cosets [x] modulo a given M for all x ∈ X , which becomes a linear space under the natural definitions of vector addition and scalar product in X /M (viz., [x] + [y] = [x + y] and α[x] = [αx] for x, y ∈ X and α ∈ F — see, e.g., [78, Example 2.H] ). The natural map (or the natural quotient map) π : X → X /M of X onto X /M is defined by π(x) = [x] = x + M for every x ∈ X . The origin [0] of the linear space X /M is [0] = M ∈ X /M

(i.e., [y] = [0] ⇐⇒ y ∈ M).

The kernel N (π) and range R(π) of the natural map π are given by   N (π) = π −1 ([0]) = x ∈ X : π(x) = [0]     = x ∈ X : [x] = [0] = x ∈ X : x ∈ M = M ⊆ X ,   R(π) = π(X ) = [y] ∈ X /M : [y] = π(x) for some x ∈ X   = [y] ∈ X /M : y + M = x + M for some x ∈ X = X /M, so that π is in fact surjective. The map π is a linear transformation between the linear spaces X and X /M. If S is an arbitrary linear manifold of X , then   π(S) = [y] ∈ X /M : [y] = π(s) for some s ∈ S   = [y] ∈ X /M : [y] = [s] for some s ∈ S   = [y] ∈ X /M : [y] − [s] = [0] for some s ∈ S   = [y] ∈ X /M : [y − s] = [0] for some s ∈ S   = [y] ∈ X /M : y − s ∈ M for some s ∈ S   = [y] ∈ X /M : y ∈ M + s for some s ∈ S   = [y] ∈ X /M : y ∈ M + S . (In particular, π(X ) = {[y] ∈ X /M : y ∈ X } = X /M since M + X = X ; that is, again, π is surjective). Hence the inverse image of π(S) under π is     π −1 (π(S)) = x ∈ X : π(x) ∈ π(S) = x ∈ X : [x] ∈ π(S)   = x ∈ X : x ∈ M + S = M + S. If N is a linear manifold of X for which X = M + N , then     π(N ) = [y] ∈ X /M : y ∈ M + N = [y] ∈ X /M : y ∈ X = X /M, and in this case the restriction π|N : N → X /M of π to N is surjective as well. Moreover, if M ∩ N = ∅, then

A.1 Quotient Space

191

  −1 N (π|N ) = π|N ([0]) = v ∈ N : π(v) = [0]    = v ∈ N : [v] = [0] = v ∈ N : v ∈ M} = {0}, and in this case the restriction π|N : N → X /M of π to N is injective. Since π is linear, its restriction to a linear manifold is again linear. Remark. A word on terminology. By an isomorphism (or an algebraic isomorphism, or a linear-space isomorphism) we mean a linear invertible transformation between linear spaces. A topological isomorphism is a continuous invertible linear transformation with a continuous inverse between topological vector spaces (i.e., an isomorphism and a homeomorphism). When actin between normed spaces this is also called a normed-space isomorphism. Dimension is preserved by (plain) isomorphisms, closedness is preserved by topological isomorphisms. An isometric isomorphism is an isomorphism and an isometry between normed spaces (a particular case of topological isomorphism). A unitary transformation is an isometric isomorphism between inner product spaces. Thus if N is an algebraic complement of M, then π|N : N → X /M is an algebraic isomorphism (sometimes referred to as the natural quotient isomorphism) between the linear spaces N and X /M. Hence every algebraic ∼ complement of M is isomorphic to the quotient space X /M. Thus, with = standing for “algebraically isomorphic to”, N ∼ = X /M for every algebraic complement N of M. Then every algebraic complement of M has the same (constant) dimension: dim N = dim X /M for every algebraic complement N of M, and so two algebraic complements of M are algebraically isomorphic. Such an invariant (dimension of any algebraic complement) is the codimension of M (notation: codim M — see, e.g., [78, Lemma 2.17, Theorem 2.18] ). Thus codim M = dim X /M. Remark A.1. Let M, N , and R be linear manifolds of a linear space X . (a) Codimension. The codimension of M is the dimension of any algebraic complement of M. In other words, the codimension of M is the dimension of any linear manifold N for which M + N = X and M ∩ N = {0}. However, M + R = X and dim R < ∞

=⇒

codim M < ∞

even if R is not algebraically disjoint from M (i.e., even if M ∩ R = {0}). In fact, if M + R = X , then the linear manifold N = M ∩ R ⊆ R is an algebraic complement of M. Thus dim R < ∞ implies dim N < ∞ so that codim M < ∞. (b) Another Proof. Another way to show dim N = dim X /M

192

A. Aspects of Fredholm Theory in Banach Space

if N is an algebraic complement of M without using the natural map π goes as follows. Let M be a linear manifold of X and let N be any algebraic complement of M. Let Δ be an index set and consider the sets {eδ }δ∈Δ ⊆ N and {[eδ ]}δ∈Δ ⊆ X \M, where [eδ ] = eδ + M for each δ ∈ Δ. Thus ⇐⇒

{eδ }δ∈Δ is linearly independent

{[eδ ]}δ∈Δ is linearly independent.

Indeed, take an arbitrary eδ0 in {eδ }δ∈Δ , consider the respective arbitrary [eδ0 ] in {[eδ ]}δ∈Δ , and let {αδ }δ∈Δ be an arbitrary set of scalars. If  eδ0 = α δ eδ , (∗) δ=δ0

then [eδ0 ] =



 δ=δ0

α δ eδ =



δ=δ0 [αδ

[eδ0 ] =



eδ ] =

 δ=δ0

αδ [eδ ]. Conversely, if

αδ [eδ ],

(∗∗)

δ=δ0

    then [eδ0 ] =  δ=δ0 [αδ eδ ] = δ=δ0 αδ eδ ∈ M δ=δ0 αδ eδ and hence eδ0 − so that eδ0 = δ=δ0αδ eδ because M ∩ N = {0}. Thus (∗) holds true if and only if (∗∗) holds true. However, {eδ }δ∈Δ is linearly independent if and only if the only set of scalars for which (∗) holds is αδ = 0 for all δ ∈ Δ, and {[eδ ]}δ∈Δ is linearly independent if and only if the only set of scalars for which (∗∗) holds is αδ = 0 for all δ ∈ Δ. This proves the above equivalence. Moreover, {eδ }δ∈Δ spans N

⇐⇒

{[eδ ]}δ∈Δ spans X \M.

In fact, take an arbitrary [x] ∈ X \M, where x = u + v ∈ M + N is an arbitrary vector in X = M + N , with u being an arbitrary vector in M and v being an arbitrary vector in N . If {eδ }δ∈Δspans N, then there exists a sequence of scalars {αδ (v)}δ∈Δ such that v = δ∈Δ αδ (v) eδ . Hence [x] = x + M = u + v + M = v + M = [v]    αδ (v)(eδ + M) = αδ (v)eδ + M = = δ∈Δ

δ∈Δ

δ∈Δ

αδ (v)[eδ ],

and so {[eδ ]}δ∈Δ spans X \M. Conversely, if {[eδ ]}δ∈Δ spans X \M, then there exists a sequence of scalars {αδ ([x])}δ∈Δ such that v + M = v + u + M = x + M = [x]    αδ ([x]) [eδ ] = αδ ([x]) (eδ + M) = = δ∈Δ

δ∈Δ

αδ ([x]) eδ + M.

δ∈Δ

 Thus v = δ∈Δ αδ ([x]) eδ (because M ∩ N = {0}), and so {eδ }δ∈Δ spans N . Hence the above equivalence also holds true. By both equivalences {eδ }δ∈Δ is a Hamel basis for N ⇐⇒ {[eδ ]}δ∈Δ is a Hamel basis for X \M, and so dim N = dim X /M as claimed.

A.1 Quotient Space

193

(c) Complementary Dimension. Let M and N be algebraic complements in a linear space X (i.e., M and N are complementary linear manifolds). Then dim X = dim M + dim N (see, e,g., [111, Theorem 6.2] ). In other words, for any linear manifold, dimension plus codimension coincides with the dimension of the linear space: dim X = dim M + dim X /M = dim M + codim M. This is another consequence of the rank and nullity identity of linear algebra, dim X = dim R(L) + dim N (L) for every linear transformation L : X → Y between linear spaces X and Y. (In fact, if M and N are complementary linear manifolds of a linear space X , then there is a projection E : X → X with R(E) = M and N (E) = R(I − E) = N — as will be revisited below in the first paragraph of Section A.2 — and R(E) ∼ = X /N (E) — as will be seen in Remark A.2(a) below.) Recall: a subspace is a closed linear manifold of a normed space.  Let M be a linear manifold of a normed space X . Consider the map  ·  : X /M → R defined for each coset [x] in the quotient space X /M by     [x] = x + M = inf x + u = d(x, M) ≤ x. u∈M

This gauges the distance of x to M (with is invariant for all representatives in the equivalence class [x] ). It is a seminorm in X /M which becomes a norm (the quotient norm) in X /M if M is closed. So if M is a subspace, then equip the quotient space X /M with its quotient norm. When we refer to the normed space X /M, it is understood that M is a subspace and the quotient space X /M is equipped with the quotient norm. Thus if M is a subspace of a normed space X, then the natural quotient map π : X → X /M is a (linear) contraction (i.e., π(x) ≤ x for every x ∈ X ), and so π is continuous. Let X be a normed space. Take the normed algebra B[X ]. Let X ∗ = B[X , F ] be the dual space of X (so that X ∗ is a Banach space), and let T ∗ ∈ B[X ∗ ] stand for the normed-space adjoint of an operator T ∈ B[X ], defined as the unique bounded linear operator on X ∗ for which f ◦ T = T ∗f ∈ X ∗ for every f ∈ X ∗ (i.e., f (T x) = (T ∗f )(x) for every f ∈ X ∗ and every x ∈ X ; see, e.g., [93, Section 3.1] or [106, Section 3.2] ). The usual basic properties of Hilbert-space adjoints are transferred to normed-space adjoints. For instance, I ∗ is the identity in B[X ∗ ] (where I is the identity in B[X ]), T ∗  = T , (T + S)∗ = T ∗+ S ∗, (T S)∗ = S ∗ T ∗ and, slightly different, (αT )∗ = αT ∗ for α ∈ F and T, S ∈ B[X ] (see, e.g., [93, Proposition 3.1.2, 3.1.4, 3.1.6, 3.1.10] ). Remark A.2. Let X and Y be linear spaces over the same field.

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(a) The First Isomorphism Theorem. If L : X → Y is a linear transformation, R(L) ∼ = X /N (L). ∼ means an algebraic isomorphism (see, e.g., [101, Theorem 3.5] ). Here = This holds tautologically for bounded linear transformations between normed spaces. In particular, if T ∈ B[X ] where X is a Banach space and R(T ) is closed, then R(T ) ∼ = X /N (T ). ∼ means a topological isomorphism (see, e.g., [93, Theorem 1.7.14] ). Now = (b) Basic Dual Results. Let X be a normed space, and let T ∗ ∈ B[X ∗ ] be the normed-space adjoint of T ∈ B[X ]. If X is a Banach space, then the range of T ∗ is closed in X ∗ if and only if the range of T is closed in X (see, e.g., [93, Lemma 3.1.21] — compare with Lemma 1.5): R(T ∗ )− = R(T ∗ )

⇐⇒

R(T )− = R(T ).

The dual of the result in (a) holds for normed-space operators,  ∗ N (T ∗ ) ∼ = X /R(T ) (see, e.g., [93, Theorems 1.10.17, Lemma 3.1.16]). Here ∼ = means an isometric isomorphism. Moreover, if X is a normed space, then (see, e.g., [93, 1.10.8] ), dim X < ∞

⇐⇒

dim X ∗ < ∞.

In this finite-dimensional case we have dim X ∗ = dim X , since if dim X < ∞, then X ∗ = B[X , F ] = L[X , F ] with L[X , F ] denoting the linear space of all linear functionals on X (see, e.g,. [101, Corollary 3.13] ). (c) Range and Kernel Duality. For a normed space X we get from (b) dim N (T ∗ ) < ∞

⇐⇒

codim R(T ) < ∞.

Similarly, if X is a Banach space, then dim N (T ) < ∞

⇐⇒

codim R(T ∗ ) < ∞.

This needs an explanation since T ∗∗ may not be identified with T for normedspace adjoints (on nonreflexive spaces). The second equivalence, however, comes from the first one by using the identity (which holds in a Banach space) dim N (T ∗∗ ) = dim N (T ). To verify the above identity proceed as follows. Let X be a normed space and let Θ be the natural map (the natural second dual map). That is, for

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195

each x ∈ X take ϕx ∈ X ∗∗ such that ϕx (f ) = f (x) for every f ∈ X ∗ and define Θ : X → R(Θ) ⊆ X ∗∗ by Θ(x) = ϕx .This is an isometric isomorphism of X onto R(Θ), and Θ has a closed range if X is Banach. If T ∗∗ ∈ B[X ∗∗] is the normedspace adjoint of T ∗ ∈ B[X ∗ ], then T ∗∗ (Θ(X )) ⊆ Θ(X ) (i.e., R(Θ) ⊆ X ∗∗ is T ∗∗-invariant) [93, Proposition 3.1.13]. As Θ ∈ B[X , X ∗∗] is also injective, let Θ−1 ∈ B[R(Θ), X ] be its inverse on its closed range. So [93, Proposition 3.1.13] Θ−1 T ∗∗ Θ = T. Therefore x ∈ N (T ) if and only if Θ(x) ∈ N (T ∗∗ ). (In fact, by the above identity, x ∈ N (T ) ⇒ Θ−1 (T ∗∗ Θx) = 0 ⇒ (T ∗∗ Θx) = 0 (since Θ−1 is injective) ⇒ Θx ∈ N (T ∗∗ ) ⇒ T x = 0 ⇒ x ∈ N (T ).) This implies Θ(N (T )) = N (T ∗∗ ). Indeed, since x ∈ N (T ) ⇒ Θx ∈ N (T ∗∗ ), then Θ(N (T )) ⊆ N (T ∗∗ ) ∩ R(Θ); equivalently, N (T ) ⊆ Θ−1 (N (T ∗∗ )). On the other hand, since Θx ∈ N (T ∗∗ ) ⇒ x ∈ N (T ) ∩ R(Θ), then Θ−1 (N (T ∗∗ )) ⊆ N (T ). So N (T ) = Θ−1 (N (T ∗∗ )), proving the identity. Now consider the restriction of Θ to N (T ), Θ|N (T ) : N (T ) → R(Θ|N (T ) ) ⊆ R(Θ) ⊆ X ∗∗ , which is linear and injective since Θ is. Actually, N (Θ|N (T ) ) ⊆ N (Θ) = {0} and so dim N (Θ|N (T ) ) = 0. Moreover, by the above identity, R(Θ|N (T ) ) = N (T ∗∗ ) and so dim R(Θ|N (T ) ) = dim N (T ∗∗ ). Thus by linearity dim N (T ∗∗ ) = dim R(Θ|N (T ) ) + dim N (Θ|N (T ) ) = dim N (T ). (d) Fredholm Alternative. Let X be a Banach space. If λ = 0 and K ∈ B∞[X ] (i.e., if K is compact), then R(λI − K) is closed and dim N (λI − K) = dim N (λI − K ∗ ) < ∞. This is the Fredholm Alternative in Corollary 1.20 (which as commented in Section 1.8 still holds in a Banach space — see, e.g., [30, Theorem VII.7.9] or [93, Lemma 3.4.21] ). Thus with the help of (a) and (b) we get codim R(λI − K ∗ ) = dim N (λI − K) = dim N (λI − K ∗ ) = codim R(λI − K). Moreover, this leads to the Fredholm Alternative version in Theorem 2.18 according to the diagram of Section 2.2: σ(K)\{0} = σP (K)\{0} = σP4(K)\{0}. (e) Three-Space Property. If M is a subspace of a Banach space X (and so M is a Banach space), then the quotient space X /M is a Banach space (see, e.g., [78, Proposition 4.10], [30, Theorem III.4.2(b)], [93, Theorem 1.7.7] ). Conversely, if M is a Banach space (i.e., if M is complete in a normed space X , and so M is closed) and the quotient space X /M is a Banach space, then X is a Banach space (see, e.g., [78, Problem 4.13], [30, Exercise III.4.5] ). If X /M is a normed space equipped with the quotient norm, then M must be a subspace of X ; if X is Banach, then so is M. Thus if any two of M, X and X /M are Banach spaces, then so is the other (see also [93, Theorem 1.7.9] ).

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A.2 Complementation If E : X → X is a projection (i.e., an idempotent linear transformation on a linear space X ), then R(E) and N (E) are complementary linear manifolds, and conversely if M and N are complementary linear manifolds, then there is a unique projection E : X → X with R(E) = M = N (I − E) and N (E) = N = R(I − E), where I − E : X → X is the complementary projection of E. Proposition A.3. On a Banach space, a projection with a closed range and a closed kernel is continuous. Proof. Let E : X → X be a projection on a normed space X such that M = R(E) and N = N (E) are subspaces (i.e., complementary closed linear manifolds) of X . Take the natural mapping Π : M ⊕ N → M + N of the direct sum M ⊕ N (equipped with any of its usual p-norms  · p ) onto the ordinary sum X = M + N , which is given by Π(u, v) = u + v for every (u, v) in M ⊕ N . Consider the mapping P : M ⊕ N → M ⊆ X defined by P (u, v) = u for every (u, v) in M ⊕ N . Since P is a linear contraction (as P (u, v)p = up ≤ up + vp = (u, v)pp for every (u, v) ∈ M ⊕ N and any p ≥ 1), since Π is a topological isomorphism whenever X is a Banach space (by Theorem 1.1 — see, e.g., [78, Problem 4.34]), and since E = P ◦ Π −1 : M + N = X → X is the composition of two continuous functions, then E is continuous.  A subspace M of a normed space X is complemented if it has a subspace as an algebraic complement; that is, a closed linear manifold M of a normed space X is complemented if there is a closed linear manifold N of X such that M and N are algebraic complements. In this case, M and N are complementary subspaces. Remark A.4. Complementary Subspace and Continuous Projection. If X is a normed space and E : X → X is a continuous projection, then R(E) and N (E) are complementary subspaces of X . This is straightforward since R(E) and N (E) are complementary linear manifolds (whenever E is an idempotent linear transformation) and R(E) = N (I − E). Conversely, if M and N are complementary subspaces of a Banach space X , then the (unique) projection E : X → X with R(E) = M and N (E) = N is continuous by Proposition A.3. Therefore if X is a Banach space, then the assertions below are equivalent. (a1 ) A subspace M of X is complemented. (a2 ) There exists a projection E ∈ B[X ] with R(E) = M. (a3 ) There exists a projection I − E ∈ B[X ] with N (I − E) = M. Since in a finite-dimensional normed space every linear manifold is closed, then in a finite-dimensional normed space every subspace is complemented . (b) If M and N are complementary subspaces of a Banach space X , then M∼ = X /N (topologically isomorphic).

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197

(Let E ∈ B[X ] be the continuous projection with R(E) = M and N (E) = N . By the First Isomorphism Theorem of Remark A.2(a), R(E) ∼ = X /N (E)). Linear manifolds of a Banach space with finite dimension are complemented subspaces. On the other hand, not all linear manifolds of a Banach space with finite codimension are subspaces. If a linear manifold of a Banach space is, however, the range of an operator with finite codimension, then it is a subspace. These facts are discussed next. Remark A.5. Finite-Dimensional Complemented Subspaces. (a) Finite-dimensional subspaces of a Banach space are complemented. In fact, if M is a finite-dimensional subspace of a Banach space X , then we show next the existence of a continuous projection E : X → X with R(E) = M. Thus M is complemented by Remark A.4 (since X is Banach and M is closed). To verify the existence of such a continuous projection proceed as follows. Suppose M is nonzero to avoid trivialities. Let {ej }m j=1 be any Hamel basis of  = for where m = dim M.Thus every u ∈ M is uniquely 1) M unit vectors (e j m written as u = j=1 βj (u)ej . For each j consider the functional βj : M → F of the vector expansion in M. By uniqueness these functionals are linear, thus continuous since M is finite-dimensional. By the Hahn–Banach Theorem (see, e.g., [78, Theorem 4.62]) let βj : X → F be a norm-preserving (βj  = βj ) βj (u) = βj (u) for bounded linear extension of each βj over the whole space X ( m u ∈ M). Take the transformation E : X → X defined by Ex = j=1 βj (x)ej for m every x ∈ X . Then R(E) ⊆ M and Eu = j=1 βj (u)ej = u for every u ∈ M, and so M ⊆ R(E). Therefore R(E) = M. This E is linear since the functionals βj : X → F are linear. It is bounded since these functionals are bounded, m   m

  Ex =  βj  x, βj (x)ej  ≤ j=1

j=1

and idempotent since E is linear,

m m m E2x = E βj (x)ej = βj (x)Eej = βj (x)ej = Ex, j=1

j=1

j=1

for every x ∈ X . Thus E is a bounded (i.e., continuous) projection. By the way, since Ex = 0 if and only if βj (x) = 0 for all j, the resulting complementary m subspace of M = R(E) is N = N (E) = j=1 N (βj ). Here is another way to state the same result. Suppose X is a Banach space. (a ) Every finite-dimensional subspace of X has a closed algebraic complement, which does not mean that every algebraic complement of a finite-dimensional subspace of a Banach space is closed. Equivalently, these do not say that if a linear manifold has a finite codimension, then it is a complemented subspace. (a ) There are nonclosed linear manifolds of X with finite codimension.

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Example 1 . Let f : X → F be a nonzero linear functional on a nonzero normed space X . Then dim R(f ) = 1 (i.e., R(f ) = F ). Also R(f ) ∼ = X /N (f ) by the algebraic version of the First Isomorphism Theorem of Remark A.2(a) (where ∼ = means algebraic isomorphism) and so codim N (f ) = dim X /N (f ) = dim R(f ) = 1. Moreover (see, e.g., [93, Proposition 1.7.16, Theorem 1.7.15]), N (f )− = N (f ) ⇐⇒ f is unbounded ⇐⇒ N (f )− = X . Therefore, if f is unbounded, then N (f ) is a nonclosed (and dense) linear manifold of the normed space X (in particular, of a Banach space) with a finite codimension (i.e., with a one-dimensional complementary subspace). Note. On the other hand, as we will see in Corollary A.9, for the special case where the linear manifold is the range of an operator (bounded and linear) on a Banach space, if it has a finite codimension, then it must be closed: If T is a Banach-space operator, and if codim R(T ) < ∞, then R(T ) is closed . The above example does not show that assertion (a) fails if X is not Banach. (a ) If a normed space is incomplete, then there may exist an uncomplemented (nonzero) finite-dimensional subspace of it. Example 2 . Take the normed space C[0, 1] of all (equivalence classes of) scalarvalued continuous functions on the interval [0, 1] equipped with the norm  · 1. Let v : [0, 1] → R be a discontinuous function defined by v(t) = 0 for t ∈ [0, 21 ) and v(t) = 1 for t ∈ [ 12 , 1]. Take the normed space X = C[0, 1] + v equipped with the norm  · 1 (with v = span {v}). This X is not Banach (since C[0, 1] is dense in L1 [0, 1] and so is X  L1 [0, 1]). Also, C[0, 1] is a linear manifold of X , it is an algebraic complement of v in X , and it is not closed in X : for any scalar α there is a sequence {un } with un ∈ C[0, 1] such that un → αv (i.e., for each ε > 0 there is a kε ∈ N such that if k ≥ kε then uk − αv1 < ε). Claim. Every algebraic complement M of v in X is dense in X . Proof. Since C[0, 1] ⊆ M + v = X , then uk = mk + αk v with mk ∈ M. Thus d(mk ,v) = inf w∈v uk − αk v − w1 = inf w∈v uk − w1 ≤ uk − αv1 < ε for every k ≥ kε . Hence d(M,v) = 0 and so M− = X = M + v, proving the claimed assertion. Then every algebraic complement of v is not closed in X . (b) Subspaces of a normed space with finite codimension are complemented. This goes along the same line of the previous statements, it does not require completeness, and it is trivially verified once closedness is assumed a priori . Indeed, if M is a linear manifold of a normed space X which is already closed, and if it has a finite-dimensional algebraic complement N , then N is closed, and so M and N are complementary subspaces. But again this does not mean that a linear manifold (even if X were a Banach space) of finite codimension is a subspace (i.e., is closed), as we saw in Example 1 above.

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199

An ordinary sum of closed linear manifolds may not be closed (even if they are algebraically disjoint; even in a Hilbert space if they are not orthogonal). But if one of them is finite-dimensional, then the sum is closed. Proposition A.6. Suppose M and N are subspaces of a normed space. If dim N < ∞, then M + N is closed . Proof. Let M and N be subspaces of a normed space X . Suppose M and N are nontrivial to avoid trivialities. Since M is a subspace, then X /M is a normed space equipped with the quotient norm. Take the natural map π : X → X /M and consider its restriction π|N : N → π(N ) ⊆ X /M to N, which is linear. Thus dim N (π|N ) + dim R(π|N ) = dim N. If N is finite-dimensional, then dim π(N ) = dim R(π|N ) ≤ dim N < ∞. Being finite-dimensional, the linear manifold π(N ) is in fact a subspace of X (i.e., it is closed in X /M). So the inverse image π −1 (π(N )) of π(N ) under π is closed as well because π is  continuous. But π −1 (π(N )) = N + M as we saw in Section A.1. Proposition A.7. Suppose M and N are subspaces of a Banach space. If dim N < ∞ and M is complemented, then M + N is a complemented subspace. Proof. Let M and N be subspaces of a Banach space X . Again, suppose M and N are nontrivial to avoid trivialities. If M is complemented, then there exists a subspace M of X (a complement of M in X ) such that X = M + M

with

M ∩ M = {0}.

Then M + N ⊆ M + M. Set S = N ∩ M, again a subspace of X, and so M+N =M+S

with

M ∩ S = {0}.

If dim N < ∞, then dim S ≤ dim N < ∞ (as S ⊆ N ). Thus S is complemented (Remark A.5(a)). But S is also included in M. So there is a subspace S  of X including the complement M of M, which is a complement of S in X , X = S + S

with

S ∩ S  = {0},

S ⊆ M

and

M ⊆ S .

Since M is complemented in X with complement M, there exists a projection E ∈ B[X ] for which R(E) = M and N (E) = M (Remark A.4). Similarly, since S is complemented in X with complement S , there exists a projection P ∈ B[X ] for which R(P ) = S and N (P ) = S . Since R(E) = M ⊆ S  = N (P ) and R(P ) = S ⊆ M = N (E), then P + E ∈ B[X ] is a projection. Moreover, R(P + E) = R(E) + R(P ) = M + S (since R(E) ∩ R(P ) = M ∩ S = ∅), where M + S is a subspace of X (Proposition A.6). Therefore the subspace M + N = M + S is complemented (Remark A.4).  Consider the converse to Proposition A.6. Is M closed whenever M + N is closed and dim N < ∞ ? If M is the range of a bounded linear transformation between Banach spaces X and Y, then the answer is yes.

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Proposition A.8. Suppose X and Y are Banach spaces, take any T ∈ B[X , Y ], and let N be a finite-dimensional subspace of Y. If R(T ) + N is closed, then R(T ) is closed as well . Proof. Take T ∈ B[X , Y ] where X and Y are Banach spaces. Let N be a finitedimensional subspace of Y. Let S be an algebraic complement of R(T ) ∩ N in N (i.e., (R(T ) ∩ N ) + S = N and (R(T ) ∩ N ) ∩ S = {0}). Then R(T ) + S = R(T ) + N . (In fact, R(T ) + N = R(T ) + (R(T ) ∩ N ) + S ⊆ R(T ) + S ⊆ R(T ) + N ). Since N (T ) is a subspace, X /N (T ) is a Banach space. Thus, since S is finite-dimensional, X /N (T ) ⊕ S is a Banach space. Consider the transformation θ : X /N (T ) ⊕ S → Y given by θ(x + N (T ), s) = T (x) + s for every x ∈ X and s ∈ S. This θ is linear, bounded, and injective (as R(T ) ∩ S = {0}). Moreover, R(θ) = R(T ) + S = R(T ) + N . Hence if R(T ) + N is closed, then R(θ) ⊆ Y is closed. Therefore the linear, bounded, and injective θ with a closed range acting between Banach spaces must have a bounded inverse on its range (according to Theorem 1.2). Since X /N (T ) ⊕ {0} is closed in X /N (T ) ⊕ S, then (θ−1 )−1 (X /N (T ) ⊕ {0}) = θ(X /N (T ) ⊕ {0}) = R(T ) is closed in Y.  Corollary A.9. Let X and Y be Banach spaces. Take any T ∈ B[X , Y ]. If codim R(T ) < ∞, then R(T ) is closed and so it is a complemented subspace. Proof. If codim R(T ) < ∞, then dim N < ∞ for any algebraic complement N of R(T ). Since R(T )+N =Y is closed, then so is R(T ) by Proposition A.8.  A finite-rank linear transformation between Banach spaces does not need to be continuous (but if it is, then it must be compact). Finite-rank projections, however, are continuous (thus compact). Corollary A.10. A finite-rank projection on a Banach space is continuous. Proof. Let E : X → X be a finite-rank projection on a normed space X . Set M = R(E) so that dim M < ∞ and hence M is closed (i.e., a subspace of X ). Take the linear manifold N (E) of X , which is an algebraic complement of M. Suppose X is a Banach space. By Remark A.5(a) the linear manifold  N (E) is closed in X and so E is continuous by Proposition A.3. Remark. However, R(T ) is not necessarily closed if codim R(T )− < ∞. There are T ∈ B[X , Y ] between Banach spaces X and Y for which codim R(T )− = 0 and R(T ) is not closed. Example: if T = diag{ k1 } on X = 2 , then 0 ∈ σC (T ) (0 is in the continuous spectrum of T ), and so R(T ) is not closed but is dense in 2 . Thus the unique algebraic complement of R(T )− = 2 is {0}, and hence codim R(T )− = 0. Therefore codim R(T )− = 0 and R(T ) is not closed.

A.3 Range-Kernel Complementation A normed space is complemented if every subspace has a complementary subspace (i.e., if every subspace of it is complemented — recall: by a subspace we

A.3 Range-Kernel Complementation

201

mean a closed linear manifold). According to the Projection Theorem of Section 1.3, every Hilbert space is complemented. However, as we saw in Remark 5.3(a), if a Banach space is complemented, then it is isomorphic (i.e., topologically isomorphic) to a Hilbert space [91] (see also [71]). Thus complemented Banach spaces are identified with Hilbert spaces — only Hilbert spaces (up to an isomorphism) are complemented. Definition A.11. Let X and Y be normed spaces and define the following classes of transformations from B[X , Y ].   ΓR [X , Y] = T ∈ B[X , Y ] : R(T )− is a complemented subspace of Y ,   ΓN [X , Y] = T ∈ B[X , Y ] : N (T ) is a complemented subspace of X , Γ [X , Y] = ΓR [X , Y] ∩ ΓN [X , Y]   = T ∈ B[X , Y ] : R(T )− and N (T ) are complemented . If X = Y, then write ΓR [X ] = ΓR [X , X ],

ΓN [X ] = ΓN [X , X ],

Γ [X ] = ΓR [X ] ∩ ΓN [X ],

˜ ) in [25]. which are subsets of B[X ]. The class ΓR [X ] has been denoted by ζ(X A Banach space X is range complemented if ΓR [X ] = B[X ], kernel complemented if ΓN [X ] = B[X ], and range-kernel complemented if Γ [X ] = B[X ] (i.e., if ΓR [X ] = ΓN [X ] = B[X ]). Hilbert spaces are complemented, and consequently they are range-kernel complemented. If a Banach space X is complemented (i.e., if X is essentially a Hilbert space), then it is trivially range-kernel complemented. Is the converse true? At the very least, for an arbitrary Banach space the set Γ [X ] = ΓR [X ] ∩ ΓN [X ] is algebraically and topologically large in the sense that it includes nonempty open groups. For instance, the group G[X ] of all invertible operators from B[X ] (i.e., of all operators from B[X ] with a bounded inverse) is open in B[X ] and trivially included in Γ [X ]. Proposition A.12. Let X and Y be a Banach spaces. (a) If T ∈ B[X , Y ] is has a finite-dimensional range, then it lies in ΓR [X , Y]. If T ∈ B[X , Y ] has a finite-dimensional kernel, then it lies in ΓN [X , Y]. (b) If range of T ∈ B[X , Y ] has a finite codimension, then T lies in ΓR [X , Y]. If kernel of T ∈ B[X , Y ] has a finite codimension, then T lies in ΓN [X , Y]. (c) Every invertible operator from B[X , Y ] has complemented closed range and kernel, and so lies in Γ [X , Y] = ΓR [X , Y] ∩ ΓN [X , Y]. In particular G[X ] ⊆ Γ [X ] = ΓR [X ] ∩ ΓN [X ], and G[X ] is open in B[X ]. (d) T ∈ B[X , Y ] is left invertible (with a bounded left inverse) if and only if it is injective with a closed and complemented range.

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In other words, T ∈ ΓR [X , Y], N (T ) = {0}, and R(T )−= R(T ) (or T has a complemented range and satisfies any of the equivalent conditions in Theorem 1.2) if and only if T has a left inverse in B[Y, X ]. (Compare with the Hilbert-space counterpart in Lemma 5.8(a).) (e) T ∈ B[X , Y ] has a right inverse in B[Y, X ] if and only if it is surjective and lies in ΓN [X , Y]. Proof. Assertion (a) is a particular case of Remark A.5(a), and assertion (b) comes from Corollary A.9 and Remark A.5(b) since N (T ) is a subspace. (c) If T ∈ G[X , Y ], then R(T ) = Y and N (T ) = {0}. Moreover, G[X ] is open in the uniform topology of B[X ] (see, e.g., [78, Problem 4.48]). (d) Suppose T ∈ B[X , Y ] has a left inverse in B[Y, X ]. In other words, suppose there exists S ∈ B[Y, X ] such that S T = I ∈ B[X ], which trivially implies N (T ) = {0} (T x = 0 ⇒ x = Ix = S T x = 0); that is, T is injective. Since (T S)2 = T S T S = T S in B[Y] and R(T ) = R(T S T ) ⊆ R(T S) ⊆ R(T ), then T S ∈ B[Y] is a (continuous) projection with R(T S) = R(T ) ⊆ Y, and so R(T ) is closed and complemented in Y by Remark A.4. Conversely, suppose T ∈ B[X , Y ] is injective with a closed range. Thus T has a bounded inverse on its range according to Theorem 1.2, which means there exists S  ∈ B[R(T ), X ] for which S  T = I ∈ B[X ]. Moreover, if the closed range is complemented in Y, then there exists a (continuous) projection E ∈ B[Y] with R(E) = R(T ) by Remark A.4. Thus take the operator S = S  E ∈ B[Y, X ] so that S T = S  E T = S  T = I ∈ B[X ]; that is, T ∈ B[X , Y ] has a left inverse S in B[Y, X ]. (e) Suppose T ∈ B[X , Y ] has a right inverse in B[Y, X ]. In other words, suppose there exists S ∈ B[Y, X ] such that T S = I ∈ B[Y], which trivially implies R(T ) = Y (y ∈ Y ⇒ y = Iy = T Sy ∈ R(T )); that is, T is surjective. Since (S T )2 = S T S T = S T in B[X ], then S T ∈ B[X ] is a (continuous) projection and N (S T ) = N (T ) (T x = 0 ⇒ S T x = 0 ⇒ T x = T S T x = 0). Hence N (T ) = N (S T ) = R(I − S T ) is complemented by Remark A.4. Conversely, suppose T lies in ΓN [X , Y] so that N (T ) is a complemented subspace of X , which means there exists a subspace M of X for which X = N (T ) + M and N (T ) ∩ M = {0}. Thus the restriction T |M ∈ B[M, Y] is injective. If T is surjective, then so is T |M because R(T |M ) = R(T ) = Y. Hence T |M ∈ B[M, Y] has an inverse (T |M )−1 ∈ B[Y, M]. So T (T |M )−1 = T |M (T |M )−1 = I ∈ B[Y]. Since N (T ) and M are complementary, let J ∈ B[M, X ] be the natural embedding of M into X (i.e., J(u) = 0 + u ∈ N (T ) + M = X ). Therefore we get T J (T |M )−1 y = T (0 + (T |M )−1 y) = T (T |M )−1 y = y for every y ∈ Y. Then T J(T |M )−1 = I ∈ B[Y] and so J(T |M )−1 ∈ B[Y, X ] is a right inverse of T .  Proposition A.13. Take T ∈ B[X ] on a Banach space X . (a1 ) If T ∈ ΓR [X ], then T ∗ ∈ ΓN [X ∗ ]. (a2 ) If X is reflexive and T ∗ ∈ ΓN [X ∗ ], then T ∈ ΓR [X ].

A.3 Range-Kernel Complementation

203

(b1 ) If X is reflexive and T ∗ ∈ ΓR [X ∗ ], then T ∈ ΓN [X ]. (b2 ) If R(T ) is closed and T ∈ ΓN [X ], then T ∗ ∈ ΓR [X ∗ ]. 

Proof. See [82, Theorem 3.1].

A normed space is reflexive if the natural second dual map is an isometric isomorphism. Every reflexive normed space is a Banach space. Theorem A.14. If T ∈ B[X ] is a compact operator on a reflexive Banach space X with a Schauder basis, then T ∈ ΓR [X ] ∩ ΓN [X ]

and

T ∗ ∈ ΓR [X ∗ ] ∩ ΓN [X ∗ ].

Proof. (We borrow the proof of [82, Corollary 5.1].) Suppose T ∈ B[X ] is compact and X has a Schauder basis. Thus there is a sequence {Tn } of finite-rank u T (i.e., {Tn } converges uniformly to T ) operators Tn ∈ B[X ] such that Tn −→ − and R(Tn ) ⊆ R(Tn+1 ) ⊆ R(T ) (see, e.g., [78, Problem 4.58]). Since each Tn is finite-rank (i.e., dim R(Tn ) < ∞), we get R(Tn ) = R(Tn )−. Moreover, finite-dimensional subspaces of a Banach space are complemented (Remark A.5(a), also Proposition A.12(a)). Then Tn ∈ ΓR [X ]; equivalently, there exist continuous projections En ∈ B[X ] and I − En ∈ B[X ] (which as such have closed ranges) with R(En ) = R(Tn ) — Remark A.4(a,b), where {R(En )} is an increasing sequence of subspaces. Since {R(En )} is a monotone sequence of subspaces, limn R(En ) exists in the following sense: 

− !  ! lim R(En ) = R(Ek ) = R(Ek ) , n k≥n n≥1 n≥1 k≥n

 where k≥n R(Ek ) is the closure of the span of { k≥n R(Ek )} [25, Definition 1]. Thus regarding the complementary projections I − En , limn R(I − En ) also exists. Moreover, limn R(En ) is a subspace of X included in R(T )− (limn R(En ) ⊆ R(T )− because R(Tn ) ⊆ R(T )− ). Since X has a Schauder basis and T is compact, the sequence of operators {En } converges strongly (see, e.g., [78, Hint to Problem 4.58]) and so {En } is a bounded sequence. Thus s T (i.e., {Tn } converges strongly because it since (i) X is reflexive, (ii) Tn −→ converges uniformly), (iii) Tn ∈ ΓR [X ], (iv) R(En ) = R(Tn ), (v) {En } is bounded, (vi) limn R(Tn ) ⊆ R(T )−, and (vii) limn R(I − En ) exists, then T ∈ ΓR [X ] −

[25, Theorem 2]. Hence R(T ) is complemented if T is compact and X is reflexive with a Schauder basis. Since reflexivity for X is equivalent to reflexivity for X ∗ (see, e.g., [30, Theorem V.4.2]), since X ∗ has a Schauder basis whenever X has (see, e.g., [93, Theorem 4.4.1]), and since T is compact if and only if T ∗ is compact (see, e.g., [93, Theorem 3.4.15]), then R(T ∗ )− is also complemented: T ∗ ∈ ΓR [X ∗ ]. Therefore, since X is reflexive, Proposition A.13 ensures T ∈ ΓR [X ] =⇒ T ∗ ∈ ΓN [X ∗ ]

and

T ∗ ∈ ΓR [X ∗ ] =⇒ T ∈ ΓN [X ].



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A. Aspects of Fredholm Theory in Banach Space

A.4 Upper-Lower and Left-Right Approaches Consider the following classes Φ+ [X ], Φ− [X ], Φ[X ] of operators from B[X ]. Definition A.15. Let X be a Banach space. The set   Φ+ [X ] = T ∈ B[X ] : R(T ) is closed and dim N (T ) < ∞ is the class of upper semi-Fredholm operators from B[X ], while   Φ− [X ] = T ∈ B[X ] : R(T ) is closed and dim X /R(T ) < ∞ is the class of lower semi-Fredholm operators from B[X ], and their intersection Φ[X ] = Φ+ [X ] ∩ Φ− [X ] is the class of Fredholm operators from B[X ]. Since codim R(T ) = dim X /R(T ) and since R(T ) is closed whenever codim R(T ) < ∞ by Corollary A.9, then the class of lower semi-Fredholm operators can be equivalently written as   Φ− [X ] = T ∈ B[X ] : codim R(T ) < ∞ (cf. Remark 5.3(b)), and so the class of Fredholm operators can be written as   Φ[X ] = T ∈ B[X ] : dim N (T ) < ∞ and codim R(T ) < ∞   = T ∈ B[X ] : R(T ) is closed, dim N (T ) < ∞, dim X /R(T ) < ∞ . The sets Φ+ [X ] and Φ− [X ] are open (and Φ+ [X ]\Φ− [X ] and Φ− [X ]\Φ+ [X ] are closed) in B[X ] (see, e.g., [95, Proposition 16.11, Proof of Corollary 18.2]). Proposition A.16. Let X be a Banach space. (a) Φ+ [X ] ⊆ ΓN [X ]. (b) Φ− [X ] ⊆ ΓR [X ]. (c) Φ[X ] ⊆ Γ [X ]. Proof. If T ∈ Φ+ [X ], then dim N (T ) < ∞ and so N (T ) is complemented by Proposition A.12(a). Thus Φ+ [X ] ⊆ ΓN [X ]. On the other hand, if T ∈ Φ− [X ], then codim R(T ) < ∞, and hence R(T ) is complemented by Proposition  A.12(b). Thus Φ− [X ] ⊆ ΓR [X ]. Consequently, Φ[X ] ⊆ Γ [X ]. Proposition A.17. Let T ∈ B[X ] be an operator on a Banach space X and let T ∗ ∈ B[X ∗ ] be its normed-space adjoint on the Banach space X ∗. Then (a) T ∈ Φ+ [X ] if and only if T ∗ ∈ Φ− [X ∗ ], (b) T ∈ Φ− [X ] if and only if T ∗ ∈ Φ+ [X ∗ ], (c) T ∈ Φ[X ] if and only if T ∗ ∈ Φ[X ∗ ].

A.4 Upper-Lower and Left-Right Approaches

205

Proof. This follows from Definition A.15 since, according to Remark A.2(b,c), R(T ∗ )− = R(T ∗ )

⇐⇒

R(T )− = R(T ),

dim N (T ∗ ) < ∞ ⇐⇒ codim R(T ) < ∞, dim N (T ) = dim N (T ∗∗ ).



Proposition A.18. Let A, T ∈ B[X ] be operators on a Banach space X . (a) If A, T ∈ Φ− [X ], then A T ∈ Φ− [X ]. (b) If A, T ∈ Φ+ [X ], then A T ∈ Φ+ [X ]. (c) If A, T ∈ Φ[X ], then A T ∈ Φ[X ]. Proof. (a) If A, T ∈ Φ− [X ], then codim R(A) < ∞ and codim R(T ) < ∞ (Definition A.15). Thus there exist subspaces M and N of X such that dim M < ∞ and dim N < ∞ for which X = R(A) + M and X = R(T ) + N . Therefore X = A(X ) + M = A(R(T ) + N ) + M = A(R(T )) + A(N ) + M = R(A T ) + A(N ) + M = R(A T ) + R with R = A(N ) + M. Since dim N < ∞, then dim A(N ) < ∞ because A is linear. Thus dim R < ∞ because dim M < ∞. Hence codim R(A T ) < ∞ by Remark A.1(a), and so A T ∈ Φ− [X ] by Definition A.15. (b) Dually, if A, T ∈ Φ+ [X ], then A∗, T ∗ ∈ Φ− [X ∗ ] by Proposition A.17 so that T ∗A∗ ∈ Φ− [X ∗ ] by item (a) or, equivalently, (A T )∗ ∈ Φ− [X ∗ ] and so A T ∈ Φ+ [X ] applying Proposition A.17 again. (c) Assertions (a) and (b) imply assertion (c) according to Definition A.15.  Proposition A.19. Let A, T ∈ B[X ] be operators on a Banach space X . (a) If A T ∈ Φ− [X ], then A ∈ Φ− [X ]. (b) If A T ∈ Φ+ [X ], then T ∈ Φ+ [X ]. (c) If A T ∈ Φ[X ], then A ∈ Φ− [X ] and T ∈ Φ+ [X ]. Proof. (a) If A T ∈ Φ− [X ], then codim R(A T ) < ∞ by Definition A.15. Since R(A T ) ⊆ R(A) we get dim R(A T ) ≤ dim R(A) and therefore codim R(A) ≤ codim R(A T ). Hence codim R(A) < ∞ and so A ∈ Φ− [X ] by Definition A.15. (b) Dually, if A, T ∈ Φ+ [X ], then A T ∈ Φ+ [X ] by Proposition A.18. Hence T ∗A∗ = (A T )∗ ∈ Φ− [X ∗ ] by Proposition A.17. Thus T ∗ ∈ Φ− [X ∗ ] by item (a), and so T ∈ Φ+ [X ] by Proposition A.17. (c) Apply assertions (a) and (b) to get (c) by the definition of Φ[X ].



Next consider the classes F [X ], Fr [X ], F[X ] of operators from B[X ] as defined in Section 5.1, now for Banach-space operators.

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Definition A.20. Let X be a Banach space. The set   F [X ] = T ∈ B[X ] : T is left essentially invertible   = T ∈ B[X ] : A T = I + K for some A ∈ B[X ] and some K ∈ B∞[X ] is the class of left semi-Fredholm operators from B[X ], while   Fr [X ] = T ∈ B[X ] : T is right essentially invertible   = T ∈ B[X ] : T A = I + K for some A ∈ B[X ] and some K ∈ B∞[H] is the class of right semi-Fredholm operators from B[X ], and their intersection   F[X ] = F [X ] ∩ Fr [X ] = T ∈ B[X ] : T is essentially invertible is the class of Fredholm operators from B[X ]. Definitions of F [X ] and Fr [X ] can be equivalently stated if “compact” is replaced with “finite-rank” above (see, e.g., [7, Remark 3.3.3] or [95, Theorems 16.14 and 16.15] — cf. Remark 5.3(c) and Proposition 5.C). The sets F [X ] and Fr [X ] are open in B[X ] since they are inverse images under the quotient (continuous) map π : B[X ] → B[X ]/B∞ [X ] of the left and right invertible elements in the Calkin algebra B[X ]/B∞ [X ] of B[X ] modulo the ideal B∞ [X ] (see, e.g., [30, Proposition XI.2.6]) — cf. Proposition 5.A. Proposition A.21. Let T ∈ B[X ] be an operator on a Banach space X and let T ∗ ∈ B[X ∗ ] be its normed-space adjoint on the Banach space X ∗. (a) If T ∈ Fr [X ], then T ∗ ∈ F [X ∗ ]. (b) If T ∈ F [X ], then T ∗ ∈ Fr [X ∗ ]. (c) If T ∈ F[X ], then T ∗ ∈ F[X ∗ ]. The converses hold if X is reflexive. Proof. The implications follow from Definition A.20 since (AT )∗ = T ∗A∗, (T A)∗ = A∗ T ∗, (I + K)∗ = I ∗ + K ∗, and K ∗ is compact whenever K is (this is Schauder’s Theorem — see, e.g., [93, Theorem 3.4.15]). The converses hold if X is reflexive, when T ∗∗ = ΘT Θ−1 where Θ is the natural embedding of X into its second dual X ∗∗ (see, e.g., [93, Proposition 3.1.13]), which is surjective and thus an isometric isomorphism of X onto X ∗∗ whenever X is reflexive,  and B∞[X ] is an ideal of B[X ]. Left and right and upper and lower semi-Fredholm operators are linked by range and kernel complementation as follows (see, e.g., [95, Theorems 16.14 and 16.15]): T ∈ F [X ] if and only if T ∈ Φ+ [X ] and R(T ) is complemented ; T ∈ Fr [X ] if and only if T ∈ Φ− [X ] and N (T ) is complemented . Theorem A.22. Let X be a Banach space. Then

A.4 Upper-Lower and Left-Right Approaches

207

F [X ] = Φ+ [X ] ∩ ΓR [X ]   = T ∈ Φ+ [X ] : R(T ) is a complemented subspace of X , Fr [X ] = Φ− [X ] ∩ ΓN [X ]   = T ∈ Φ− [X ] : N (T ) is a complemented subspace of X . Proof. (a) Suppose T ∈ Φ+ [X ]. Then R(T ) is closed and dim N (T ) < ∞. Suppose T ∈ ΓR [X ] and take the projection E ∈ B[X ] with R(E) = R(T ) (Remark A.4). But dim N (T ) < ∞ implies T ∈ ΓN [X ] (Remark A.5(a), also Proposition A.12(a)). Then X = N (T ) + M for some subspace M of X for which N (T ) ∩ M = {0}, and the restriction T |M ∈ B[M, R(T )] is injective and surjective. Thus take its inverse (T |M )−1 ∈ B[R(T ), M] and set A = (T |M )−1 E in B[X ] (with R(A) = M). Hence A T = (T |M )−1 E T = (T |M )−1 E|R(T ) T = (T |M )−1 T : X → M so that A T |M = I ∈ B[M]. Then (I − A T )|M = O and so K = (I − A T ) ∈ B[X ] is finite-rank (since dim N (T ) < ∞), thus compact. Therefore A T = I + (−K), which means T ∈ F [X ] by Definition A.20. Hence Φ+ [X ] ∩ ΓR [X ] ⊆ F [X ]. (b) Suppose T ∈ F . Thus there exists A ∈ B[X ] such that A T = I − K for some K ∈ B∞[X ] (Definition A.20). By the Fredholm Alternative in Remark A.2(d) we get dim N (I − K) = codim R(I − K) < ∞ and so A T ∈ Φ[X ] (Definition A.15). Hence (Proposition A.19(c)) T ∈ Φ+ [X ]. Actually, the operator I − K is Weyl as in Remark 5.7(b,ii) — see, e.g., [111, Theorem V.7.11]. However, as in Proposition 5.K (an extension of the Fredholm Alternative), I − K is in fact Browder (recall: Browder operators are Weyl operators), meaning asc(I − K) = dsc(I − K) < ∞, which clearly survives from Hilbert to Banach spaces — as happens with the entire Section 5.4 (see, e.g., [111, Theorem V.7.9]). Therefore there exists an integer n ≥ 0 such that R((A T )n ) = R((A T )n+1 ). Thus set M = R((AT )n )

so that

AT (M) = M.

Since A T ∈ Φ[X ] we get (A T ) ∈ Φ[X ] (Proposition A.18(c)), and so n

M is a complemented subspace of X with codim M < ∞ (Corollary A.9). Hence X = M + N with M ∩ N = {0} for some subspace N of X with dim N < ∞. Let E ∈ B[X ] be the (continuous) projection for which R(E) = M and N (E) = N (Remark A.4) and consider the operator EA T |M = E(I − K)|M = (E|M − EK|M ) = I − K  ∈ B[M] since R(E) = M, where the first identity operator acts on X and the second on M, with K  = EK|M (which is compact since B∞[M] is an ideal of B[M] and restriction of compact is compact). Moreover, since A T (M) = M = R(E), R(I − K  ) = M.

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By the Fredholm Alternative (Remark A.2(d)) we get 1 ∈ ρ(K) and so I − K  is invertible (indeed, if 1 ∈ σP4(K), then R(I − K  ) = M). Therefore I = (I − K  )−1 (I − K  ) = (I − K  )−1 EA T |M . Thus T |M ∈ B[X ] is left invertible, and so T (M) = R(T |M ) is a complemented subspace of X (Proposition A.12(d)). Moreover, since X = M + N with M ∩ N = {0}, every x ∈ X has a unique representation as x = u + v with u ∈ M and v ∈ N , and so R(T ) = T (X ) = T (M) + T (N ). Furthermore, since dim N < ∞, then dim T (N ) < ∞. Summing up: R(T ) = T (M) + T (N ) is the sum of a complemented subspace and a finite-dimensional subspace. Since T (M) is a subspace and T (N ) is finite-dimensional, then R(T ) is a subspace (Proposition A.6). Moreover, since T (M) is complemented, and T (N ) is finite-dimensional, then R(T ) is complemented (Proposition A.7). Therefore F [X ] ⊆ Φ+ [X ] ∩ ΓR [X ]. (c) By the inclusions in (a) and (b) we get the first identity. The proof of the second identity is similar, following a symmetric argument. (However, for reflexive Banach spaces the second identity follows at once from the first one through Proposition A.13 and Propositions A.17 and A.21.)  In view of Theorem A.22, operators in F [X ] and Fr [X ] are also referred to as Atkinson operators [50, Theorem 2.3] (left and right respectively). Corollary A.23. Let X be a Banach space. Then F[X ] = Φ[X ]. Proof. F[X ] = Φ[X ] ∩ Γ [X ] = Φ[X ] (Theorem A.22 and Proposition A.16).  Remark A.24. Further Properties. Corollary A.23 justifies the same terminology, Fredholm operators, for the classes F[X ] and Φ[X ]. Further immediate consequences of Theorem A.22 and Proposition A.16 read as follows. (a) F [X ] ∪ Fr [X ] = (Φ+ [X ] ∪ Φ− [X ]) ∩ Γ [X ], F [X ]\Fr [X ] = (Φ+ [X ]\Φ− [X ]) ∩ ΓR [X ] = (Φ+ [X ]\Φ− [X ]) ∩ Γ [X ], Fr [X ]\F [X ] = (Φ− [X ]\Φ+ [X ]) ∩ ΓN [X ] = (Φ− [X ]\Φ+ [X ]) ∩ Γ [X ]. (b) If Φ+ [X ] = Φ− [X ], then F [X ] = Fr [X ] = Φ+ [X ] = Φ− [X ]. (c) If Φ+ [X ] ⊆ ΓR [X ] and Φ− [X ] ⊆ ΓN [X ], then Φ+ [X ] = F [X ] and Φ− [X ] = Fr [X ]. In particular, (d) if ΓR [X ] = ΓN [X ], then Φ+ [X ] = F [X ] and Φ− [X ] = Fr [X ]. In this case the classes ΓR [X ] and ΓN [X ] play no role in Theorem A.22.

A.5 Essential Spectrum and Spectral Picture Again

209

Corollary A.25. Let X be a Banach space and take T ∈ B[X ]. (a) If T ∈ ΓR [X ] is bounded below (or T satisfies any of the equivalent conditions in Theorem 1.2) and K ∈ B[X ] is compact, then R(T + K) is closed

and

T + K ∈ Γ [X ] = ΓR [X ] ∩ ΓN [X ].

(b) In particular, if G ∈ G[X ] and K ∈ B[X ] is compact, then R(G + K) is closed

and

G + K ∈ Γ [X ] = ΓR [X ] ∩ ΓN [X ].

(c) More particularly, for any compact K ∈ B[X ] and any nonzero scalar λ, R(λI − K) is closed

and

λI − K ∈ Γ [X ] = ΓR [X ] ∩ ΓN [X ].

Proof. (a) Suppose T ∈ B[X ] is bounded below. Equivalently, suppose T is injective with a closed range (Theorem 1.2). Moreover, suppose T ∈ ΓR [X ]. Thus T has a left inverse in B[X ] (Proposition A.12(d)), and therefore T ∈ F [X ] (Definition A.20). But F [X ] is invariant under compact perturbation (as in Theorem 5.6). Thus T + K ∈ F [X ] for every compact K ∈ B[X ], and so T + K ∈ ΓR [X ] ∩ Φ+ [X ] (Theorem A.22). Since Φ+ [X ] ⊆ ΓN [X ] (Proposition A.16), T + K ∈ ΓR [X ] ∩ ΓN [X ]. Since T + K ∈ Φ+ [X ], R(T + K) is closed (Definition A.15). (For a proof without using Theorem A.22, see [67, Theorem 2].) (b) Invertible operators are trivially bounded below with complemented range. (c) Set G = λI = O in (b), or use the Fredholm Alternative of Remark A.2(d) together with Proposition A.12(a,b). (For λ = 0 the result in (c) still holds if the Banach space X is reflexive with a Schauder basis, by Theorem A.14.) 

A.5 Essential Spectrum and Spectral Picture Again Let T be an operator on a Banach space X . Consider the characterizations in Corollaries 5.11 and 5.12, which equally hold in a Banach-space setting. Definition A.26. For each T ∈ B[X ] let   σe (T ) = λ ∈ C : λI − T ∈ B[X ]\F [X ]   = λ ∈ C : λI − T is not left essentially invertible be the left essential spectrum (or left semi-Fredholm spectrum) of T, and   σre (T ) = λ ∈ C : λI − T ∈ B[X ]\Fr [X ]   = λ ∈ C : λI − T is not right essentially invertible be the right essential spectrum (or right semi-Fredholm spectrum) of T. Let σe (T ) = σe (T ) ∪ σre (T )   = λ ∈ C : λI − T ∈ B[X ]\(F [X ] ∩ Fr [X ])   = λ ∈ C : λI − T ∈ B[X ]\F[X ] be the essential spectrum (or Fredholm spectrum) of T.

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As in Section 5.2, σe (T ) and σre (T ) are the left and right spectra of the natural image π(T ) of T in the Calkin algebra B[X ]/B∞ [X ], and so σe (T ), σre (T ), and σe (T ) are compact subsets of the spectrum σ(T ). Similarly, corresponding to the classes Φ+ [X ], Φ− [X ], and Φ[X ], define analogous sets. Definition A.27. For each T ∈ B[X ] let   σe+ (T ) = λ ∈ C : λI − T ∈ B[X ]\Φ+ [X ]   = λ ∈ C : R(λI − T ) is not closed or dim N (λI − T ) = ∞ be the upper semi-Fredholm spectrum of T , and   σe− (T ) = λ ∈ C : λI − T ∈ B[X ]\Φ− [X ]   = λ ∈ C : R(λI − T ) is not closed or dim X /R(λI − T ) = ∞   = λ ∈ C : codim R(λI − T ) = ∞ be the lower semi-Fredholm spectrum of T . Let σe (T ) = σe+ (T ) ∪ σe− (T )   = λ ∈ C : λI − T ∈ B[X ]\(Φ+ [X ] ∩ Φ− [X ])   = λ ∈ C : λI − T ∈ B[X ]\Φ[X ] be the essential spectrum (or Fredholm spectrum) of T . Since F[X ] = Φ[X ] (Corollary A.23), there is no ambiguity in the definition of the essential spectrum. Indeed, σe (T ) = σe (T ) ∪ σre (T ) = σe+ (T ) ∪ σe− (T ). Also σe+ (T ) and σe− (T ) are compact subsets of σ(T ) (see, e.g., [95, Proposition 6.2 and Theorems 16.7 and 16.11]). Corresponding to the classes ΓR [X ], ΓN [X ], and Γ [X ], define the following subsets of C (actually, subsets of σ(T )). Definition A.28. For each operator T ∈ B[X ], consider the following sets.   ϑR (T ) = λ ∈ C : λI − T ∈ B[X ]\ΓR [X ]   = λ ∈ C : R(λI − T )− is not complemented ,   ϑN (T ) = λ ∈ C : λI − T ∈ B[X ]\ΓN [X ]   = λ ∈ C : N (λI − T ) is not complemented , ϑ(T ) = ϑR (T ) ∪ ϑN (T )   = λ ∈ C : λI − T ∈ B[X ]\ΓR [X ] ∩ ΓN [X ]   = λ ∈ C : λI − T ∈ B[X ]\Γ [X ]   = λ ∈ C : R(λI − T )− or N (λI − T ) is not complemented . According to Theorem A.22, left-right essential spectra and upper-lower semi-Fredholm spectra are related as follows.

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Corollary A.29. Let T be an operator on a Banach space X . σe (T ) = σe+ (T ) ∪ ϑR (T )  = λ ∈ C : R(λI − T ) is not closed, or dim N (λI − T ) = ∞,  or R(λI − T ) is not a complemented subspace . σre (T ) = σe− (T ) ∪ ϑN (T )  = λ ∈ C : R(λI − T ) is not closed, or dim X /R(λI − T ) = ∞,  or N (λI − T ) is not a complemented subspace . Proof. Straightforward by Definitions A.26, A.27, A.28 and Theorem A.22:   / F [X ] σe (T ) = λ ∈ C : λI − T ∈   / (Φ+ [X ] ∩ ΓR [X ]) = λ ∈ C : λI − T ∈   / ΓR [X ] / Φ+ [X ] or λI − T ∈ = λ ∈ C : λI − T ∈ = σe+ (T ) ∪ ϑR (T ),   σre (T ) = λ ∈ C : λI − T ∈ / Fr [X ]   / (Φ− [X ] ∩ ΓN [X ]) = λ ∈ C : λI − T ∈   / Φ− [X ] or λI − T ∈ / ΓN [X ] = λ ∈ C : λI − T ∈ = σe− (T ) ∪ ϑN (T ).



The next proposition is the complement of Proposition A.16. Proposition A.30. The following inclusions hold for every T ∈ B[X ]. (a) ϑN (T ) ⊆ σe+ (T ). (b) ϑR (T ) ⊆ σe− (T ). (c) ϑ(T ) ⊆ σe (T ). Proof. Since Φ+ [X ] ⊆ ΓN [X ] and Φ− [X ] ⊆ ΓR [X ] (Proposition A.16), we get B[X ]\ΓN [X ] ⊆ B[X ]\Φ+ [X ] and B[X ]\ΓR [X ] ⊆ B[X ]\Φ− [X ]. Then (Defini tions A.27 and A.28) ϑN (T ) ⊆ σe+ (T ) and ϑR (T ) ⊆ σe− (T ). Remark A.31. Additional Properties. These are consequences of Corollary A.29 and Proposition A.30, and constitute the complement of Remark A.24. (a) σe (T ) ∩ σre (T ) = (σe+ (T ) ∩ σe− (T )) ∪ ϑ(T ), σe (T )\σre (T ) = (σe+ (T )\σe− (T ))\ϑN (T ) = (σe+ (T )\σe− (T ))\ϑ(T ), σre (T )\σe (T ) = (σe− (T )\σe+ (T ))\ϑR (T ) = (σe− (T )\σe+ (T ))\ϑ(T ). (b) If σe− (T ) = σe+ (T ), then σe (T ) = σre (T ) = σe− (T ) = σe+ (T ). (c) If ϑR (T ) ⊆ σe+ (T ) and ϑN (T ) ⊆ σe− (T ) then σe+ (T ) = σe (T ) and σe− (T ) = σre (T ).

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In particular, (d) If ϑR (T ) = ϑN (T ), then σe+ (T ) = σe (T ) and σe− (T ) = σre (T ). In this case ϑR (T ) and ϑN (T ) play no role in Corollary A.29. Actually, as we will see next, if ϑR (T ) and ϑN (T ) are open subsets of the complex plane, then they play no role in Corollary A.29 either. Corollary A.32. Take T ∈ B[X ] on a Banach space X . (a) If ΓR [X ] is open (closed ) in B[X ], then ϑR (T ) is closed (open) in C . If ΓN [X ] is open (closed ) in B[X ], then ϑN (T ) is closed (open) in C . (b) If ϑR (T ) is open in C , then σe+ (T ) = σe (T ). If ϑN (T ) is open in C , then σe− (T ) = σre (T ). Proof. Let T be an operator on a Banach space X . (a) Item (a) follows by the complementary roles played between ΓR [X ] and ϑR (T ), and between ΓN [X ] and ϑN (T ). Indeed, as ϑR (T ) is bounded (Proposition A.30), take an arbitrary λ ∈ C \ϑR (T ). Thus λI − T ∈ ΓR [X ] (Definition A.28). Suppose ΓR [X ] is nonempty and open in B[X ]. Thus there exists ελ > 0 for which, if A ∈ B[X ] is such that A − (λI − T ) < ελ , then A ∈ ΓR [X ]. So if ζ ∈ C is such that |ζ − λ| < ελ , then (ζI − T ) − (λI − T ) = |ζ − λ| < ελ , and hence ζI − T ∈ ΓR (T ), which means ζ ∈ C \ϑR (T ). Therefore C \ϑR (T ) is open (i.e., ϑR (T ) is closed). Now suppose ϑR (T ) = ∅ (otherwise ϑR (T ) is trivially open). Take an arbitrary λ ∈ ϑR (T ). Thus λI − T ∈ B[X ]\ΓR [X ] (Definition A.28). Suppose ΓR [X ] is closed in B[X ]. Then B[X ]\ΓR [X ] is nonempty and open in B[X ]. So there is an ελ > 0 for which, if A ∈ B[X ] is such that A − (λI − T ) < ελ, then A ∈ B[X ]\ΓR [X ]. Hence if ζ  ∈ C is such that |ζ  − λ | < ελ, then (ζ  I − T ) − (λ I − T ) = |ζ  − λ | < ελ, which implies (ζ  I − T ) ∈ B[X ]\ΓR (T ), and so ζ  ∈ ϑR (T ). Thus ϑR (T ) is open. Similarly (applying the same argument), if ΓN [X ] is open (closed) in B[X ], then C \ϑN (T ) is open (closed) in C . (b) By Corollary A.29, σe (T ) = σe+ (T ) ∪ ϑR (T ) = σe+ (T ) ∪ (ϑR (T )\σe+ (T )). Suppose σe (T ) = σe+ (T ). So ϑR (T ) ⊆ σe+ (T ), which means ϑR (T )\σe+ (T ) = ∅. Since σe (T ) and σe+ (T ) are closed, ϑR (T )\σe+ (T ) is not open. If ϑR (T ) is open, then ϑR (T )\σe+ (T ) = ϑR (T ) ∩ (C \σe+ (T )) is open (because σe+ (T ) is closed), which is a contradiction. Thus ϑR (T ) is not open. Similarly, since σre (T ) = σe− (T ) ∪ ϑN (T ) (Corollary A.29) and σre (T ) and σe− (T ) are closed, the same argument ensures ϑN (T ) is not open whenever σre (T ) = σe− (T ). This concludes a contrapositive proof for item (b).  Remark. If ΓR [X ] and ΓN [X ] are closed, then ϑR (T ) and ϑN (T ) are open (by Corollary A.32(a)), which implies (by Corollary A.32(b)) σe+ (T ) = σe (T )

and

σe− (T ) = σre (T ).

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The sets ΓR [X ] and ΓN [X ] are not necessarily closed. For instance, there may be sequences of operators Tn in ΓR [p ] converging uniformly to an operator T not in ΓR [p ] for 1 < p = 2 [25, Example 1], and so ΓR [X ] may not be closed. On the other hand, even though the sets ΓR [X ] and ΓN [X ] may not be open, they include the nonempty open sets Φ+ [X ] and Φ− [X ] respectively (Proposition A.16), which in turn include the open group G[X ]. Consider again the sets Z of all integers and Z = Z ∪ {−∞} ∪ {+∞} of all extended integers. Take an operator T ∈ B[X ] on a Banach space X . If R(T ) is a subspace of X (i.e., if R(T ) is closed in X ), and if one of dim N (T ) or codim R(T ) is finite, then the Fredholm index of T is defined in Z by ind (T ) = dim N (T ) − codim R(T ). Thus ind (T ) is well defined if and only if T ∈ Φ+ [X ] ∪ Φ− [X ] (cf. Definition A.15). In particular, ind (T ) is well defined if T ∈ F [X ] ∪ Fr [X ] (cf. Remark A.24(a)). If ind (T ) is finite (i.e., if ind (T ) ∈ Z ), then T ∈ Φ+ [X ] ∩ Φ− [X ] = Φ[X ] = F[X ] = F [X ] ∩ Fr [X ] (i.e., then T is Fredholm). In particular, a Weyl operator is a Fredholm operator with null index, where   W[X ] = T ∈ Φ[X ] : ind (T ) = 0 denotes the class of all Weyl operators from B[X ]. Compare with Chapter 5 (see Remarks 5.3(f) and 5.7(b) in Section 5.1). With respect to the essential spectrum σe (T ) ∪ σre (T ) = σe (T ) = σe+ (T ) ∪ σe− (T ), consider the set   σw (T ) = λ ∈ C : λI − T ∈ B[X ]\W[X ]   = λ ∈ C : λI − T is not a Fredholm operator of index zero   = λ ∈ C : λI − T is not Fredholm or is Fredholm of index not zero   = λ ∈ C : either λ ∈ σe (T ) or λI − T ∈ Φ[X ] and ind (λI − T ) = 0 . This is the Weyl spectrum of T . Since λI − T ∈ Φ[X ] and ind (λI − T ) = 0 if and only if R(λI − T ) is closed and dim N (λI − T ) = dim N (λI − T ∗ ) with both dimensions finite (cf. Definition A.15 and Remark  A.2(b)), then following the argument in the proof of Theorem 5.16(b) we get λ ∈ C : λI − T ∈ Φ[X ] and ind (λI − T ) = 0 ⊆ σ(T ). Hence σw (T ) ⊆ σ(T ), and therefore σe (T ) ⊆ σw (T ) ⊆ σ(T ). Let σ0 (T ) denote the complement of σw (T ) in σ(T ), σ0 (T ) = σ(T )\σw (T )   = λ ∈ σ(T ) : λI − T ∈ W[X ]   = λ ∈ σ(T ) : λI − T is a Fredholm operator of index zero   = λ ∈ σ(T ) : λ ∈ σe (T ) and ind (λI − T ) = 0 . Moreover, for each nonzero (finite) integer k ∈ Z \{0}, set

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  σk (T ) = λ ∈ C : λI − T ∈ Φ[X ] and ind (λI − T ) = k = 0 , which are open subsets of σ(T ), being the holes of the essential spectrum σe+ (T ) ∪ σe− (T ) = σe (T ) = σe (T ) ∪ σre (T ) (again by the same argument as in the proof of Theorem 5.16). Also set σ+∞ (T ) = σe+ (T )\σe− (T )

and

σ−∞ (T ) = σe− (T )\σe+ (T ).

These open sets are holes of σe− (T ) and σe+ (T ), respectively (but are not holes of σe (T )), referred to as the pseudoholes of σe (T ). So for each nonzero extended integer k ∈ Z \{0} the sets σk (T ) are the holes of σ+ (T ) ∩ σ− (T ) (Theorem 5.16 and Fig. § 5.2). The next partition of the spectrum is the Spectral Picture analogous to that given in Corollary 5.18. For T ∈ B[X ],  σk (T ), σ(T ) = σe (T ) ∪ k∈Z   σe (T ) = σe+ (T ) ∩ σe− (T ) ∪ σ+∞ (T ) ∪ σ−∞ (T ), and the Weyl spectrum is the union of the essential spectrum and all its holes  σw (T ) = σ(T )\σ0 (T ) = σe (T ) ∪ σk (T ), k∈Z \{0}

as in Theorem 5.24 (Schechter Theorem). Remark. σe (T ), σw (T ) and σk (T ) for finite k are exactly the same sets as defined in Chapter 5. However, σ+∞ (T ) and σ−∞ (T ) as defined above are counterparts of the sets with same notation defined in Chapter 5. In fact, Remark A.31(a) says σe (T )\σre (T ) = (σe+ (T )\σe− (T ))\ϑ(T ) and σre (T )\σe (T ) = (σe− (T )\σe+ (T ))\ϑ(T ). If the Banach space X is such that Γ [X ] = B[X ] (and so ϑ(T ) = ∅; for instance, if X is a Hilbert space), then the differences coincide: σe (T )\σre (T ) = σe+ (T )\σe− (T ) and σre (T )\σe (T ) = σe− (T )\σe+ (T ). Corollary A.33. The essential spectrum has no pseudoholes if and only if upper and lower semi-Fredholm spectra coincide, σ+∞ (T ) = σ−∞ (T ) = ∅

⇐⇒ σe+ (T ) = σe− (T ),

which implies that right and left essential spectra coincide, (a) σe+ (T ) = σe− (T ) =⇒ σe (T ) = σre (T ). Conversely, (b) σe (T ) = σre (T ) ⇒ σ+∞ (T ) = ϑN (T )\σe−(T )

&

σ−∞ (T ) = ϑR (T )\σe+(T ).

Certain special cases are of particular interest. Suppose σe (T ) = σre (T ). (b1 ) If ϑR (T ) and ϑN (T ) are open, then σ+∞ (T ) = σ−∞ (T ) = ∅. (b2 ) If ϑR (T ) and ϑN (T ) are closed, then σ+∞ (T ) = σ−∞ (T ) = ∅. (b3 ) If ϑR (T ) ∪ ϑN (T ) ⊆ σe+ (T ) ∩ σe− (T ) then σ+∞ (T ) = σ−∞ (T ) = ∅. (b4 ) If ϑR (T ) = ϑN (T ), then σ+∞ (T ) = σ−∞ (T ) = ∅.

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215

(b5 ) If ϑR (T )∩σe+ (T ) = ϑN (T )∩σe− (T ) = ∅, then σ+∞ (T ) = σ−∞ (T ) = ∅. (b6 ) If ϑR (T ) = ϑN (T ) = ∅, then σ+∞ (T ) = σ−∞ (T ) = ∅. Proof. Let σ+∞ (T ) = σe+ (T )\σe− (T ) and σ−∞ (T ) = σe− (T )\σe+ (T ) be the pseudoholes of σe (T ), and consider the following assertions. (i) (ii) (iii) (iv)

σ+∞ (T ) = ∅ and σ−∞ (T ) = ∅ (i.e., σe (T ) has no pseudoholes). σe+ (T ) = σe− (T ). σe (T ) = σre (T ). σ+∞ (T ) = ϑN (T )\σe− (T ) and σ−∞ (T ) = ϑR (T )\σe+ (T ).

Assertions (i) and (ii) are equivalent by definition of σ+∞ (T ) and σ−∞ (T ). (a) Assertion (ii) implies (iii) according to Remark A.31(b). (b) If σe (T ) = σre (T ), then σe (T )\σre (T ) = σre (T )\σe (T ) = ∅. Thus by Remark A.31(a) and Proposition A.30(a,b) we get σe+ (T )\σe− (T ) ⊆ ϑN (T ) and σe− (T )\σe+ (T ) ⊆ ϑR (T ). Then using Proposition A.30(a,b) again we have σe+ (T )\σe− (T ) ⊆ ϑN (T )\σe− (T ) ⊆ σe+ (T )\σe− (T ) and σe− (T )\σe+ (T ) ⊆ ϑR (T )\σe+ (T ) ⊆ σe− (T )\σe+ (T ). Thus (iii) implies (iv). (b1 ) This is straightforward from Corollary A.32(b) — so it does not need item (b). Indeed, if σe (T ) = σre (T ), and if ϑR (T ) and ϑN (T ) are open, then σe+ (T ) = σe− (T ) by Corollary A.32(b) and so σ+∞ (T ) = σ−∞ (T ) = ∅. (b2 ) Assumption 1: σe (T ) = σre (T ). Thus σ+∞ (T ) = ϑN (T )\σe− (T ) by item (b). Recall: σ+∞ (T ) is open and bounded. Assumption 2: ϑN (T ) is closed. Suppose σ+∞ (T ) = ∅. If ϑN (T ) ∩ σe− (T ) = ∅, then σ+∞ (T ) = ϑN (T )\σe− (T ) ⊆ ϑN (T ) is not open (because ϑN (T ) and σe− (T ) are closed), which is a contradiction. On the other hand, if ϑN (T ) ∩ σe− (T ) = ∅, then σ+∞ (T ) = ϑN (T ), which is another contradiction (since σ+∞ (T ) is open and bounded and ϑN (T ) is closed). Therefore σ+∞ (T ) = ∅. Outcome: σe (T ) = σre (T ) and ϑN (T ) closed imply σ+∞ (T ) = ∅. Similarly (same argument by using (b) again), σe (T ) = σre (T ) and ϑR (T ) closed imply σ−∞ (T ) = ∅. (b3 ) By item (b) if σe (T ) = σre (T ) and if ϑR (T )∪ϑN (T ) ⊆ σe+ (T ) ∩ σe− (T ), then σ+∞ (T ) = ϑN (T )\σe− (T ) = ∅ and σ−∞ (T ) = ϑR (T )\σe+ (T ) = ∅. (b4 ) This follows from Remark A.31(d) — it does not need item (b) either. (b5 ) If σe (T ) = σre (T ) and if ϑR (T ) ∩ σe+ (T ) = ϑN (T ) ∩ σe− (T ) = ∅ then item (b) ensures σ+∞ (T ) = ϑN (T ) and σ−∞ (T ) = ϑR (T ). Since σ+∞ (T ) and σ−∞ (T ) are open sets, then ϑN (T ) and ϑR (T ) are open sets, and therefore σ+∞ (T ) = σ−∞ (T ) = ∅ according to item (b1 ). (b6 ) A trivial (or tautological) particular case of any of the above (bi ).



Remark. The identity σe+(T ) = σe−(T ) implies the identity σe (T ) = σre (T ) by Corollary A.33(a). The converse holds if ϑR (T ) and ϑN (T ) satisfy any of the

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assumptions in Corollary A.33(bi ). The converse, however, might fail if for instance ϑR (T ) and ϑN (T ) are nonclosed or nonopen. For example, take a compact nonempty subset D of C (e.g., the closed unit disk) and two proper subsets DR and DN of D , both are neither closed nor open, such that (D \DR )− = (D \DN )−. Consider the following configuration: σe (T ) = D , σre (T ) = D , ϑR (T ) = DR , ϑN (T ) = DN , σe+(T ) = (D \DR )−, σe− (T ) = (D \DN )−. If such a configuration is possible, then σe (T ) = σre (T ) and σe+(T ) = σe−(T ). Remark A.34. Let T be an arbitrary operator acting on a Banach space X . (a) No Holes nor Pseudoholes. The essential spectrum of T ∈ B[X ] has no holes if σk (T ) = ∅ for every finite nonzero integer k ∈ Z \{0}. Equivalently, σe (T ) has no holes if and only if σe (T ) = σw (T ) as in Theorem 5.24 (Schechter Theorem). The essential spectrum of T has no pseudoholes if σ+∞ (T ) = σ−∞ (T ) = ∅. Equivalently, σe (T ) has no pseudoholes if and only if σe+ (T ) = σe− (T ) by Corollary A.33(a) — which trivially implies σe+ (T ) = σe− (T ) = σe (T ) (since σe+ (T ) ∪ σe− (T ) = σe (T ) = σe (T ) ∪ σre (T )). Therefore σe (T ) has no holes and no pseudoholes if and only if σe+ (T ) = σe− (T ) = σe (T ) = σw (T ),

(∗)

σe (T ) = σre (T ) = σe (T ) = σw (T )

(∗∗)

which implies according to Corollary A.33(a). Although (∗) always implies (∗∗), the identities in (∗∗) will imply the identities in (∗) if any of the assumptions in Corollary A.33(bi ) are satisfied; in particular, (∗∗) implies (∗) if ϑR (T ) = ϑN (T ) = ∅ as in Corollary A.33(b6 ), which is the case whenever X is a Hilbert space. (b) Compact Operators. Take a compact operator K on an infinite-dimensional Banach space X . The Fredholm Alternative as in Remark 5.7(b,ii) says λ = 0

=⇒

λI − K ∈ W[X ].

Since σw (T ) = {λ ∈ C : λI − T ∈ W[X ]}, then σw (K) ⊆ {0}. Since σw (T ) = ∅ if dim X = ∞ (cf. Remark 5.27(a)), then σw (K) = {0}. Moreover, σe (T ) = ∅ if dim X = ∞ (cf. Remark 5.15(a)) and σe (T ) ⊆ σw (T ). Hence σe (K) = σw (K) = {0}. Since B∞[X ] is an ideal of B[X ], and since the identity is not compact in an infinite-dimensional space, we get by Definition A.20 (cf. Remark 5.7(b)) K ∈ F [X ] ∪ Fr [X ] = SF[X ].

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Thus 0 ∈ σe (K) ∩ σre (K) (Definition A.26). Since σe (T ) ∪ σre (T ) = σe (T ) (Definition A.26 again) and σe (K) = {0}, σe (K) = σre (K) = {0}. According to Corollary A.25(c) λI − K ∈ ΓN [X ] ∩ ΓR [X ] = Γ [X ]

for

λ = 0.

So by Definition A.28 ϑR (K) ∪ ϑN (K) = ϑ(K) ⊆ {0}. Hence ϑR (K) and ϑN (K) are both closed. Since σe (K) = σre (K), then σe+ (T ) = σe− (K) by Corollary A.33(b2 ). If ϑR (K) and ϑN (K) are both {0} or both ∅, then by Remark A.31(d) σe+ (K) = σe (K) and σe− (K) = σre (K). On the other hand if ϑR (K) = ∅ and ϑN (K) = {0}, then σe (K) = σe+ (K) and σre (K) = σe− (K) ∪ {0} by Corollary A.29. Similarly, if ϑR (K) = {0} and ϑN (K) = ∅, then σre (K) = σe− (K) and σe (K) = σe+ (K) ∪ {0}. Thus in any case σe (K) = σre (K) = σe+ (K) = σe− (K). Therefore σe (K) = σre (K) = σe+ (K) = σe− (K) = σe (K) = σw (K) = {0}. So the identities (∗) and (∗∗) of item (a) not only coincide but they do hold for a compact operator on an infinite-dimensional Banach space, and so compact operators have no holes and no pseudoholes. The class of operators with no holes and no pseudoholes (which properly includes the compact operators) is referred to as the class of biquasitriangular operators (see [85, Section 6]).

Suggested Readings Abramovich and Aliprantis [1] Aiena [3] Arveson [7] Barnes, Murphy, Smyth, and West [11] Caradus, Pfaffenberger, and Yood [26] Conway [30] Harte [60]

Heuser [65] Kato [72] Laursen and Neumann [87] Megginson [93] M¨ uller [95] Schechter [106] Taylor and Lay [111]

B A Glimpse at Multiplicity Theory

It has been written somewhere in the Web, “Multiplicity is a lot of something.” Indeed, this matches the common (colloquial) meaning, and perhaps might suggest that the term “multiplicity” has a multiplicity of meanings.

B.1 Meanings of Multiplicity The first time we met the term in this book was in Section 2.2 with regard to the multiplicity of an eigenvalue, defined as the dimension of the respective eigenspace. For an isolated eigenvalue this is sometimes called the geometric multiplicity of the isolated eigenvalue, while the algebraic multiplicity of an isolated point of the spectrum, as considered in Section 4.5, is the dimension of the range of the Riesz idempotent associated with the isolated point of the spectrum (see also Section 5.3). Related to these definitions, there is the notion of multiplicity of a point in a disjoint union, introduced in Section 3.3, which refers to how often an element is repeated in a disjoint union; in particular, in a disjoint union of spectra of direct summands of a normal operator. Multiplicity of a shift as in Section 2.7 is the common dimension of the shifted spaces. So there is a multiplicity function on the set of shifts (bilateral or unilateral) assigning to each shift a cardinal number: the shift multiplicity. Multiplicity of an arc was defined in Section 4.3 as a function assigning to each point in the unit interval a cardinal number called the multiplicity of a point in an arc indicating how often an arc transverses a given point. Remark. The term is also applied with respect to multiplicity of prime factors, to multiplicity of roots of polynomials (or of holomorphic — i.e., analytic — complex functions). More generally, there is the notion of multiplicity of zeroes and poles of meromorphic (i.e., analytic up to a discrete set of isolated points) or rational complex functions, among a multiplicity of other instances. In this appendix, however, we return to Hilbert spaces and will treat multiplicity from the spectral point of view, as recapitulated in the first paragraph © Springer Nature Switzerland AG 2020 C. S. Kubrusly, Spectral Theory of Bounded Linear Operators, https://doi.org/10.1007/978-3-030-33149-8

219

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B. A Glimpse at Multiplicity Theory

of this section. In particular, multiplicity is approached here in connection with the notion of spectral measure, with the central focus being placed upon the spectral theorems of Chapter 3, so that it is revealed as a concept intrinsically linked to normal operators. Indeed, this appendix can be viewed as an extension of (or a complement to) Chapter 3.

B.2 Complete Set of Unitary Invariants Let H and K be Hilbert spaces. Let U ∈ B[H, K] be a unitary transformation, which means an invertible isometry, or equivalently a surjective isometry, or (still equivalently) an isometry and a coisometry. So U is unitary if and only if U ∗ U = I (identity on H) and U U ∗ = I (identity on K), and so U −1 = U ∗. Since U ∗∗ = U and U ∗−1 = U −1∗, the transformation U ∗ ∈ B[K, H] is as unitary as U is. If there is a unitary U ∈ B[H, K] between Hilbert spaces H and K, then these are unitarily equivalent spaces and we write H ∼ = K (or K ∼ = = H), where ∼ is an equivalence relation on the collection of all Hilbert spaces. Two Hilbert spaces are unitarily equivalent if and only if they have the same dimension (i.e., same orthogonal or Hilbert dimension — see, e.g., [78, Theorem 5.49]). Take T ∈ B[H] and S ∈ B[K]. These are unitarily equivalent operators if there exists a unitary transformation U ∈ B[H, K] for which U T = S U, or equivalently T = U ∗S U, or (still equivalently) S = U T U ∗. In this case we ∼ S (or S ∼ write T = = T ), where ∼ = is now an equivalence relation on the collection of all Hilbert-space operators. Clearly, T ∼ = K. If two = S implies H ∼ operators are unitarily equivalent, then there is no possible criterion based only on the geometry of Hilbert space by which T and S can be distinguished from each other, and so two unitarily equivalent operators are abstractly the same operator. Given a Hilbert-space operator, consider the equivalence class of all Hilbert-space operators unitarily equivalent to it, which is referred to as a unitary equivalence class. If H is a Hilbert space and T ∈ B[H], then the unitary equivalence class of all Hilbert-space operators unitarily equivalent to T consists of all operators S ∈ B[K] such that S ∼ = H. = T on an arbitrary K ∼ Throughout this appendix ∼ = means unitarily equivalent. An invariant (property) for a collection of elements is a property common to all elements in the collection. In particular, a unitary invariant is a property common to all operators belonging to a unitary equivalence class (i.e., a property shared by all operators in a unitary equivalence class — which is invariant under unitary equivalence — universal to all operators unitarily equivalent to a given operator — trivial example: norm). A set of unitary invariants is a set of properties common to all operators in a unitary equivalence class. A complete set of unitary invariants is a set of properties (i.e., a set of unitary invariants) such that two operators belong to a unitary equivalence class if and only if they satisfy all properties in the set (i.e., if and only if they posses all

B.3 Multiplicity Function for Compact Operators

221

unitary invariants in the set). In other words, it is a set of properties sufficient for determining whether or not two operators are unitarily equivalent. Thus the notion of a complete set of unitary invariants is related (actually, it is the answer) to the following question. When are two Hilbert-space operators unitarily equivalent? More generally, when do a family of Hilbert-space operators belong to the same unitary equivalence class? The question may be faced by trying to attach to each operator in a fixed family of operators a set of attributes such that two operators in the family are unitarily equivalent if and only if the set of attributes attached to them is identical (or equivalent in some sense). Such a set of attributes, which is enough for determining unitary equivalence, is referred to as a complete set of unitary invariants for the family of operators. There are a few families of operators for which a reasonable complete set of unitary invariants is known. In this context the term reasonable means it is easier to verify if these attributes are satisfied by every operator in the family than to verify if the operators in the family are unitarily equivalent. The theory of spectral multiplicity yields a reasonable complete set of unitary invariants for the family of normal operators.

B.3 Multiplicity Function for Compact Operators The Spectral Theorem fully describes a compact normal operator T ∈ B[H] by breaking it down into its spectral decomposition  λ Eλ , T = λ∈σP (T )

where Eλ is the orthogonal projection with R(Eλ ) = N (λI − T ). The family {N (λI − T )}λ∈σP (T ) of eigenspaces is the heart of the spectral decomposition for compact normal operators because, besides forming an orthogonal family of subspaces, it spans the space (cf. proof of Theorem 3.3): 

− 

− R(Eλ ) = H. N (λI − T ) = λ∈σP (T )

λ∈σP (T )

If T is an operator on any Banach space, then N (λI − T ) = {0} if and only if λ ∈ σP (T ). If T is compact, then dim N (λI − T ) < ∞ whenever λ = 0, and it is just dim N (T ) that may not be finite. With this in mind, let N and N 0 be the sets of all positive and nonnegative integers, respectively, and set N 0 = N 0 ∪ {+∞}, the extended nonnegative integers. By the Fredholm Alternative (Corollary 1.20 and Theorem 2.18), if T is compact, then dim N (λI − T ) ∈ N for every λ ∈ σ(T )\{0} = σP (T )\{0} and dim N (λI − T ) ∈ N 0 for λ ∈ C \{0}. However, for the case of λ = 0, dim N (T ) may not be finite, and since the actual cardinality of any basis for N (T ) is immaterial in this context, we simply say dim N (T ) ∈ N 0 . Thus, in general, dim N (λI − T ) ∈ N 0 for every λ ∈ C .

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Definition B.1. The multiplicity mT (λ) of a point λ ∈ C for a compact operator T ∈ B[H] is the extended nonnegative integer mT (λ) = dim N (λI − T ). (Equivalently, mT (λ) = dim R(Eλ ).) The multiplicity function mT : C → N 0 for a compact operator T is the extended-nonnegative-integer-valued map on C assigning dim N (λI − T ) to each complex number λ ∈ C : λ → dim N (λI − T ). Theorem B.2. Two compact normal operators are unitarily equivalent if and only if they have the same multiplicity function. Proof. Take T ∈ B[H] and S ∈ B[K] on Hilbert spaces H and K. ∼ S, then dim N (λI − T ) = dim N (λI − S) for every λ ∈ C . Indeed, if (a) If T = W ∈ G[H, K] (i.e., an invertible transformation), then x ∈ N (λI − W T W −1 ) if and only if W T W −1 x = λx, which means T W −1 x = λ W −1 x, or equivalently W −1 x ∈ N (λI − T ). This ensures W (N (λI − T )) = N (λI − W T W −1 ). Since W is an isomorphism between the linear spaces H and K, dim N (λI − T ) = dim W (N (λI − T )) = dim N (λI − W T W −1 ). Thus T ∼ = S =⇒ mT = mS . (b) Now consider the family of all Hilbert-space compact normal operators. Let T ∈ B∞[H] and S ∈ B∞ [K] be normal operators on Hilbert spaces H and K. Conversely, suppose mT = mS (i.e., dim N (λI − T ) = dim N (λI − S) for every λ ∈ C ). The program is to show the existence of a unitary transformation in B[H, K] which makes T ∼ = S. Applying the Spectral Theorem for compact operators (Theorem 3.3) consider the spectral decompositions of T and S,   λk Ek and S = ζj Pj , T = k∈K

j∈J

where {λk }k∈K = σP (T ) and {ζj }j∈J = σP (S) are (nonempty countable) sets of all (distinct) eigenvalues of T and S, and {Ek }k∈K and {Pj }j∈J are B[H]valued and B[K]-valued resolutions of the identity. Without loss of generality, assume the countable index sets K and J are subsets of the nonnegative integers N 0 . The operators Ek ∈ B[H]and Pj ∈ B[K] are orthogonal projections onto the eigenspaces N (λk I − T ) = R(Ek ) ⊆ H and N (ζj I − S) = R(Pj) ⊆ K. Again with no loss of generality, assume 0 ∈ K ∩ J , set λ0 = ζ0 = 0, and let E0 and P0 be the projections onto N (λ0 I − T ) = N (T ) and N (ζ0 I − S) = N (S). (Note: without modifying the above sums, zero may or may not be originally included in them, whether or not zero lies in their point spectra.) Take an arbitrary k = 0. Since mT = mS (and 0 = λk ∈ σP (T )), 0 < dim N (λk I − T ) = dim N (λk I − S). By the unique spectral decomposition of T and S, for each λk there is a unique ζj such that ζj = λk . So there is an injective function θ : K \{0} → J \{0} for which ζθ(k) = λk . Similarly,

B.3 Multiplicity Function for Compact Operators

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0 < dim N (ζj I − S) = dim N (ζj I − T ), and so for each ζj there is a unique λk such that λk = ζj . Hence the function θ : K \{0} → J \{0} is bijective. Set θ(0) = 0 and extend the function θ over K onto J . The extended θ : K → J remains bijective (i.e., a permutation). Then #K = #J : the cardinalities of the index sets K and J coincide. Since the subspaces R(Ek ) = N (λk I − T ) and R(Pθ(k) ) = N (ζθ(k) I − S) have the same dimension, they are unitarily equivalent, and so there exists a unitary transformation Uk : R(Ek ) → R(Pθ(k) ). Claim. There exists a unitary U ∈ B[H, K] for which U T = S U . Proof. Since {Ek } and {Pθ(k) } are consider  − of the identity, − the  resolutions R(P and K = R(E ) ) . Let H orthogonal topological sums = k θ(k) k  −k   be defined follows. Take an arbitrary as ) −→ R(P ) U : k R(Ek θ(k) k  x = k xk ∈ k R(Ek ) with xk ∈ R(Ek ). Set U x = k Uk xk where  Uk xk lies inR(Pθ(k) ). This defines a bounded linear transformation U of k R(Ek ) onto k R(Pθ(k) ). It may be extended by continuity to a bounded linear transformation, onto K. As is readily verified,  U, over H−   also denoted − by  ∗its adjoint ∗ ) → R(P ) R(E given is by U y = U∗ : k θ(k) k k Uk yk for an  k  ) with y R(P ∈ R(P Thus ). U U ∗ x = x and arbitrary y = k yk ∈  k θ(k) θ(k) k  ∗ U U y = y for every x ∈ k R(Ek ) and y ∈ k R(Pθ(k) ). So when extended over H onto K the transformation U ∈ B[H, K] is unitary. Hence H and K are unitarily equivalent. Moreover, if xk ∈ R(Ek ), then T xk = λk xk , which implies U T xk = U λk xk = ζθ(k) U xk = SU xk because U xk lies in R(Pθ(k) ) and so SU xk = ζθ(k) U xk . Therefore     xk = SU xk = SU xk = SU x U T xk = UTx = UT k k k k   for  every x ∈− k R(Ek ). Extending by continuity from k R(Ek ) to H = ∼ k R(Ek ) , U T x = SU x for every x ∈ H. Thus mT = mS =⇒ T = S. (c) Outcome: T ∼ = S ⇐⇒ mT = mS .



Therefore if two compact normal operators have the same multiplicity function, then they have the same spectrum, and the multiplicity function alone is a complete set of unitary invariants. Remark. According to the Fredholm Alternative, zero is the only accumulation point of the spectrum of a compact operator T and the spectrum itself is countable, and so the set of all scalars λ for which the dimension of N (λI − T ) is not zero must be countable, and the dimension of N (λI − T ) is also finite provided λ is not zero (cf. Corollaries 1.20 and 2.20). Under these necessary constraints, every multiplicity function for a compact normal operator is possible (see, e.g., [30, Exercise II.8.5]). Although Theorem B.2 holds for compact normal operators, it does not hold for nonnormal, even compact, operators. The classical

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B. A Glimpse at Multiplicity Theory

example is the Volterra operator, which is compact, quasinilpotent (nonnormal with empty point spectrum), and so its multiplicity function is the zero function, but it is not unitarily equivalent to the null operator.

B.4 Multiplicity-Free Normal Operators Roughly speaking, a normal operator is free of multiplicity if just part (a) in the proof of the first version of the Spectral Theorem (Theorem 3.11) is enough for a full description of the unitary equivalence with a multiplication operator. This means it has a star-cyclic vector or, equivalently, it is a starcyclic operator. Star-cyclicity for normal operators, however, coincides with cyclicity (Corollary 3.14). Summing up: a normal operator is multiplicityfree if it is cyclic (equivalently, star-cyclic). A word on terminology: perhaps counterintuitively, being multiplicity-free does not mean “being of multiplicity zero” (but it means “being of multiplicity one”). So by the first version of the Spectral Theorem (cf. proof of Theorem 3.11 part (a)), T ∈ B[H] is a multiplicity-free normal operator if and only if T ∼ = Mϕ , where the normal operator Mϕ in B[L2 (σ(T ), μ)] is a multiplication operator, Mϕ f = ϕ f

for every

f ∈ L2 (σ(T ), μ),

with μ : Aσ(T ) → R being a positive finite measure on the σ-algebra Aσ(T ) of Borel subsets of σ(T ) = σ(Mϕ ), and ϕ : σ(T ) → σ(T ) is the identity map (i.e., ϕ(λ) = λ for λ ∈ σ(T )) and hence ϕ ∈ L∞ (σ(T ), μ) since σ(T ) is bounded. Accordingly, this multiplication operator Mϕ ∈ B[L2 (σ(T ), μ)] is again cyclic and normal and so multiplicity-free as well. Such a measure μ can be extended (same notation) to a σ-algebra AΩ of Borel subsets of any Borel set Ω including σ(Mϕ ) and so its support remains the same: support (μ) = σ(Mϕ ) ⊆ Ω ⊆ C ∈ AC where AΩ = AC ∩ ℘ (Ω) is a sub-σ-algebra of the σ-algebra AC of all Borel subsets of C . If in addition Ω is compact (so bounded), then the identity map ϕ on Ω is bounded (so ϕ ∈ L∞ (Ω, μ)) and support (μ) = ess Ω = σ(Mϕ ) ⊆ Ω ⊂ C (Proposition 3.B). Thus let μ and ν be positive finite measures with compact support on the σ-algebra AC of Borel subsets of C , consider their restrictions (denoted again by μ and ν) to a σ-algebra AΩ of Borel subsets of any nonempty compact (thus Borel) set Ω ⊂ C including the supports of μ and ν, let ϕ be the identity map on Ω, and consider two multiplication operators Mϕ ∈ B[L2 (Ω, μ)] and Nϕ ∈ B[L2 (Ω, ν)] so that (cf. paragraph following Proposition 3.B) σ(Mϕ ) = support (μ)

and

σ(Nϕ ) = support (ν).

B.4 Multiplicity-Free Normal Operators

225

Now recall from Sections 3.6 and 4.2: a measure μ is absolutely continuous with respect to a measure ν (both acting on the same σ-algebra AΩ ) if ´m ν(Λ) = 0 implies μ(Λ) = 0 for Λ ∈ AΩ (notation: μ  ν). The Radon–Nikody Theorem says: if μ and ν are σ-finite measures and if μ  ν, then there exists a unique (ν-almost $ everywhere unique) positive measurable function ψ on Ω for which μ(Λ) = Λ ψ dν for every Λ ∈ AΩ (see, e.g., [81, Theorem 7.8]). This unique function is called the Radon–Nikod´ym derivative of μ with respect to ν, denoted by ψ = dμ dν . Two measures μ and ν on the same σ-algebra are equivalent (notation μ ≡ ν or ν ≡ μ) if they are mutually absolutely continuous (i.e., μ  ν and ν  μ — one is absolutely continuous with respect to the other and vice versa), and so they share exactly the same sets of measure zero. In other words, μ ≡ ν means μ(Λ) = 0 if and only if ν(Λ) = 0 for Λ ∈ AΩ . If μ and ν have the same sets of measure zero, then by definition their supports coincide: μ ≡ ν trivially implies support (μ) = support (ν). But the converse fails: take the Lebesgue measure on [0, 1] and its sum with a Dirac measure on the singleton { 12 }. So under the previous assumptions (i.e., finite measures μ and ν on the same σ-algebra AΩ of subsets of the same compact set Ω, and multiplication operators by the identity map ϕ) we get μ≡ν

=⇒

σ(Mϕ ) = σ(Nϕ ).

Equivalent measures, however, go beyond such an identity of spectra. Thus as discussed above, let Mϕ ∈ B[L2 (Ω, μ)] and Nϕ ∈ B[L2 (Ω, ν)] be multiplicity-free multiplication operators, where μ and ν are regarded as finite positive measures with compact support on a σ-algebra AΩ of Borel subsets of any compact set ∅ = Ω ⊂ C for which support (μ) ∪ support (ν) ⊆ Ω, and ϕ is their common identity map on Ω. This is the setup for the next theorem. Theorem B.3.

Mϕ ∼ = Nϕ ⇐⇒ μ ≡ ν.

Proof. Recall: σ(Mϕ ) = support (μ) and σ(Nϕ ) = support (ν). (a) Suppose Mϕ ∼ = Nϕ . This implies σ(Mϕ ) = σ(Nϕ ). Thus set Ω = σ(Mϕ ) = σ(Nϕ ). Also let U : L2 (Ω, μ) → L2 (Ω, ν) be the unitary transformation for which UMϕ = Nϕ U. By a trivial induction UMϕk = Nϕk U and UMϕ∗j = Nϕ∗j U for any integers j, k ≥ 0. Let P (Ω) be the set of all polynomials p(· , ·) : Ω → C in λ and λ (i.e., n P (Ω) is a set of nall maps λ → p(λ, λ) taking each λ in Ω into p(λ, λ) = j,k=0 αj,k λj λ k = j,k=0 αj,k ϕ(λ)j ϕ(λ)k in C for an arbitrary nonnegative integer n and an arbitrary set of complex coefficients {αj,k }nj,k=0 ). n j ∗k For each p ∈ P (Ω) take the operators p(Mϕ , Mϕ∗ ) = j,k=0 αj,k Mϕ Mϕ in  n B[L2 (Ω, μ)] and p(Nϕ , Nϕ∗ ) = j,k=0 αj,k Nϕj Nϕ∗k in B[L2 (Ω, ν)]. Thus Up(Mϕ , Mϕ∗ ) = p(Nϕ , Nϕ∗ ) U. Since Mϕ f = ϕ f and Mϕ∗ f = ϕ f, and hence Mϕj Mϕ∗k f = ϕj ϕk f , we get pf = 2 $  n n j k ∗ 2 j k   jk αj,k λ λ f = [p(Mϕ , Mϕ )]f and also pf 2 = j,k αj,k λ λ f (λ) dμ ≤

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B. A Glimpse at Multiplicity Theory

diam(Ω)4n

n j,k

$ |αj,k |2 |f |2 dμ, for f ∈ L2 (Ω, μ) and p ∈ P (Ω). Therefore

p f = [ p(Mϕ , Mϕ∗ )] f ∈ L2 (Ω, μ)

and

p g = [ p(Nϕ , Nϕ∗ )] g ∈ L2 (Ω, ν)

for every f ∈ L2 (Ω, μ) and every g ∈ L2 (Ω, ν). Moreover, for f ∈ L2 (Ω, μ) U (p f ) = U [ p(Mϕ , Mϕ∗ )](f ) = [ p(Nϕ , Nϕ∗ )] U (f ) = p U (f ). $ 2 $Since U is2 an isometry, p f 2 = U (p f )22 = p U (f )2 , and so |p f | dμ = |p U (f )| dν. Then, by setting f = 1 ∈ L (Ω, μ), % % |p|2 dμ = |p|2 |U (1)|2 dν for every p ∈ P (Ω). Recall from the proof of Theorem 3.11 part (a): P (Ω)− = L2 (Ω, η) in the L2 -norm topology for every finite measure η defined on the σ-algebra AΩ of Borel subsets of a compact Ω ⊂ C . Then P (Ω)− = L2 (Ω, μ) ∩ L2 (Ω, ν). Thus (by continuity of the integral) the above identity holds for p ∈ P (Ω) replaced with the characteristic function χΛ ∈ L2 (Ω, μ) ∩ L2 (Ω, ν) for any Λ ∈ AΩ . So % % % 2 2 μ(Λ) = dμ = |U (1)| dν ≤ sup |U (1)| dν ≤ sup |U (1)|2 ν(Λ) Λ

Λ

Λ

(where $sup |U$(1)| = supλ∈Ω |U (1)(λ)| > 0 because U (1) = 0 since U (1)22 = 122 = dν = Ω dν = ν(Ω) > 0 — see, e.g., [81, Remark 10.1] ). Therefore   ν(Λ) = 0 =⇒ μ(Λ) = 0 =⇒ μ  ν. Similarly, since U is unitary, replace U by U −1 which is again an isometry, and apply the same argument to prove ν  μ. Outcome: Mϕ ∼ = Nϕ =⇒ μ ≡ ν. (b) Conversely, first suppose μ  ν. Then the measures μ and ν act on the same σ-algebra AΩ . Since these measures are finite (thus σ-finite), take the Radon–Nikod´ ym derivative ψ = dμ is a positive dν of μ with respect to ν (which $ $ 1 measurable C -valued function on Ω). If g ∈ L (Ω, μ), then g dμ = g dμ dν dν = $ gψ dν and so gψ ∈ L1 (Ω, ν). Thus for f ∈ L2 (Ω, μ) we get f 2 ∈ L1 (Ω, μ) and 1 hence ψf 2 ∈ L1 (Ω, ν). Then ψ 2 f ∈ L2 (Ω, ν) and % % % 1 1 ψ 2 f 22 = |ψ 2 f |2 dν = |f |2 ψ dν = |f |2 dμ = f 22 . Thus the multiplication operator Uψ 12 : L2 (Ω, μ) → L2 (Ω, ν) given by Uψ 12 f = 1 ψ 2 f for every f ∈ L2 (Ω, μ) is an isometry. If in addition ν  μ (and so μ ≡ ν) dν = ψ −1 (see, then the Radon–Nikod´ ym derivative of ν with respect to μ is dμ e.g., [81, Problem 7.9(d)] ). Using the same argument for any h ∈ L2 (Ω, ν), % % % % 1 1 dν h22 = |h|2 dν = |h|2 dμ dμ = |h|2 ψ −1 dμ = |ψ − 2 h|2 dμ = ψ − 2 h22 .

B.5 Multiplicity Function for Normal Operators

227

Then the multiplication operator Uψ -21 : L2 (Ω, ν) → L2 (Ω, μ) given for each 1 h ∈ L2 (Ω, ν) by Uψ -12 h = ψ − 2 h is an isometry as well. Since these isometries are the inverses of each other, Uψ 12 : L2 (Ω, μ) → L2 (Ω, ν) is a unitary transfor1 1 mation. Also, Uψ 12 Mϕ f = ψ 2 ϕf = ϕ ψ 2 f = Nϕ Uψ 12 f for every f in L2 (Ω, μ),  and so Uψ 12 Mϕ = Nϕ Uψ 12 . Outcome: μ ≡ ν =⇒ Mϕ ∼ = Nϕ . Corollary B.4. Two multiplicity-free normal operators are unitarily equivalent if and only if their scalar spectral measures are equivalent. Equivalently, if and only if their spectral measures are equivalent. Proof. Let T ∈ B[H] and S ∈ B[K] be multiplicity-free normal operators. So H and K are separable Hilbert spaces (Corollary 3.14). By Theorem 3.11 (in particular, by part (a) in the proof of Theorem 3.11) they are unitarily equivalent to multiplicity-free multiplication operators Mϕ ∈ L2 (Ω, μ) and Nϕ ∈ L2 (Ω, ν). That is, T ∼ = Mϕ and S ∼ = Nϕ . By transitivity T ∼ = S if and ∼ only if Mϕ = Nϕ . Thus σ(T ) = σ(S) = σ(Mϕ ) = σ(Nϕ ) and Ω denotes these coincident spectra. Then Mϕ and Nϕ have the same identity function ϕ on Ω. By Theorem B.3, Mϕ ∼ = Nϕ if and only if μ ≡ ν. Summing up: T ∼ =S

Mϕ ∼ = Nϕ

⇐⇒

⇐⇒

μ ≡ ν.

Let E $and P be the unique$ spectral measures of the spectral decompositions of T = σ(T ) λ dEλ and S = σ(S) λ dPλ as in Theorem 3.15. Let πe,e and ωf,f be scalar spectral measures for T and S which are positive finite measures equivalent to E and P, respectively (cf. Definition 4.4 and Lemma 4.7). That is, πe,e ≡ E

and

ωf,f ≡ P.

Thus, by the remark following Lemma 4.7 (since the Hilbert spaces are separable, and since in the multiplicity-free case π ˆe,e = πe,e on AΩ = Aσ(T ) ), we get πe,e = μ

and

ωf,f = ν.

All these measures act on the same σ-algebra AΩ = Aσ(T ) = Aσ(S) . Therefore μ≡ν

⇐⇒

πe,e ≡ ωf,f

⇐⇒

E ≡ P.



Spectral measure (up to equivalence) is a complete set of unitary invariants for multiplicity-free normal operators.

B.5 Multiplicity Function for Normal Operators Again, the hard work has already been done in the proof of the first version of the Spectral Theorem (Theorem 3.11), now referring to part (b) of that proof. Take a nonempty index set Γ . For each γ ∈ Γ let μγ be a positive finite measure with compact support acting on the σ-algebra AC of all Borel subsets of the complex plane C . Let Ωγ be a compact set including the support of μγ (support (μγ ) ⊆ Ωγ ). So the positive finite measure μγ can be regarded as

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B. A Glimpse at Multiplicity Theory

acting on the σ-algebra AΩγ of Borel subsets of the compact subset Ωγ of C . Let ϕγ be the identity map on the compact set Ωγ , so that ϕγ is bounded, and consider the multiplicity-free multiplication operator Mϕγ ∈ B[L2 (Ωγ , μγ )]. Thus σ(Mϕγ ) = support (μγ ) ⊆ Ωγ . Suppose the ordinary union of {Ωγ }γ∈Γ ˜ be any compact subset of C including the is a bounded subset of C . Let Ω ordinary union of {Ωγ }γ∈Γ . (In the proof of Theorem 3.11 part (b) we had ˜ = σ(T ).) Also let ϕ˜ : Ω ˜→Ω ˜ be the identity map on Ω ˜ (i.e., ϕ(λ) Ω = λ for ˜ ∞ ˜ ˜ ˜ is bounded. every λ ∈ Ω), which lies in L (Ω, μγ ) for every γ because Ω Consider the proof of Theorem 3.11 part (b). Since support (μγ ) ⊆ Ωγ for each γ, then we may replace each multiplicity-free multiplication operator Mϕγ ∈ B[L2 (Ωγ , μγ )] by the multiplicity-free multiplication operator ˜ μγ )], Mγ,ϕ˜ ∈ B[L2 (Ω, ˜ for every γ ∈ Γ, for which with a common identity map ϕ˜ on a common set Ω ˜ ⊂ C. σ(Mγ,ϕ˜ ) = support (μγ ) ⊆ Ωγ ⊆ Ω

Now consider the disjoint union Ω = γ∈Γ Ωγ . Let AΩ be the σ-algebra of Borel subsets of the disjoint union Ω and take the positive measure  μ on AΩ part (b) (i.e., proof of Theorem 3.11 as defined in the μ(Λ) = γ∈J μγ (Λγ )

for Λ = γ∈J Λγ ∈ AΩ with Λγ ∈ AΩγ and J ⊆ Γ countable). Take the identity map with multiplicity

ϕ : Ω → C as defined in the proof of Theorem 3.11 part -- = λ ∈ Ωβ ⊂ C for a unique β ∈ Γ and a (b) (i.e., if λ -- ∈ Ω = γ∈Γ Ωγ , then λ --) = ϕβ (λ) = λ ⊂ C ). Since the ordinary union of unique λ ∈ Ωβ , thus set ϕ(λ ˜ is bounded in C , then ϕ is a scalar-valued bounded {Ωγ }γ∈Γ (included in Ω) function on Ω. Consider the multiplication operator Mϕ ∈ B[L2 (Ω, μ)]. ˜ μγ )] with a common identity Take multiplication operators Mγ,ϕ˜ ∈ B[L2 (Ω, ˜ function ϕ˜ on Ω (which as explained above replace multiplication operators Mϕγ ∈ B[L2 (Ωγ , μγ )]), and consider the external orthogonal direct sum   2 ˜ ˜ ∈ B[ γ∈Γ Mγ,ϕ γ∈Γ L (Ω, μγ )]. So (cf. proof of Theorem 3.11 part (b) once again)  ∼ ˜ μγ ) and Mϕ = L2 (Ω, μ) ∼ = γ∈Γ L2 (Ω, ˜ γ∈Γ Mγ,ϕ   − −

= ϕ γ∈Γ σ(Mγ,ϕ˜ ) . with σ(Mϕ ) = ˜) γ∈Γ σ(Mγ,ϕ Definition B.5. Consider the above setup. The multiplicity mMϕ (λ) of a point λ ∈ C for the multiplication operator Mϕ is the cardinality of the inverse image ϕ−1 ({λ}) (denoted by λλ) of the singleton {λ} with respect to ϕ,  

mMϕ (λ) = #λλ = #ϕ−1 ({λ}) = # γ∈Γ {λ} ∩ Ωγ .

B.5 Multiplicity Function for Normal Operators

229

The multiplicity function for a multiplication operator Mϕ is a cardinalnumber-valued map mM on C assigning to each λ ∈ C its multiplicity mMϕ (λ), mMϕ : λ → mM (λ).

[Note: mMϕ (λ) = 0 for every λ ∈ R(ϕ) = ϕ(Ω) = γ∈Γ Ωγ (the ordinary union of {Ωγ }γ∈Γ ); in particular, if Ωγ = σ(Mγ,ϕ˜ ) so that ϕ(Ω)− = σ(Mϕ ) (cf. Proposition 2.F(d)), then mMϕ (λ) = 0 for every λ ∈ ρ(Mϕ ).] Let Mϕ ∈ B[L2 (Ω, μ)] and Nϕ ∈ B[L2 (Ω , ν)] be multiplication operators where Ω and Ω  are disjoint unions of compact subsets Ωγ and Ωγ of C whose ordinary unions are bounded, μ and ν are positive measures on the σ-algebras AΩ and AΩ  of Borel subsets of Ω and Ω , and ϕ and ϕ are identity maps with multiplicity on Ω and Ω  . Let mMϕ and mNϕ be their multiplicity functions. This is the setup of the next theorem. Theorem B.6.

Mϕ ∼ = Nϕ ⇐⇒ μ ≡ ν and mMϕ = mNϕ  . (In this case Ω = Ω  and so ϕ = ϕ.)

˜ and ϕ˜ were defined associated with Ω (over Γ ) and ϕ, let Ω ˜  and Proof. As Ω     ϕ˜ be analogously defined with respect to Ω (over Γ ) and ϕ . ∼ L2 (Ω , ν). Thus by transitivity (a) Suppose Mϕ ∼ = Nϕ . Then L2 (Ω, μ) =    2 ˜ 2 ˜ ∼ ∼  ˜. ˜ = γ∈Γ L (Ω, μγ ) = γ∈Γ  L (Ω , νγ ) and also γ∈Γ  Nγ,ϕ γ∈Γ Mγ,ϕ  2 2  ∼ Nγ,ϕ˜ and L (Ω, ˜ , νγ ) for every γ, and so Γ is ˜ μγ ) ∼ Therefore Mγ,ϕ˜ = = L (Ω in a one-to-one correspondence with Γ. Thus we may identify them and set ˜ and Ω ˜  are arbitrary compact subsets of C including the ordiΓ = Γ  . Since Ω ˜=Ω ˜  (including both), nary unions of {Ωγ }γ∈Γ and {Ωγ }γ∈Γ  , then we take Ω  and hence ϕ˜ = ϕ˜ . In this case Mγ,ϕ˜ ∼ = Nγ,ϕ˜ implies μγ ≡ νγ by Theorem B.3, and so μ ≡ ν by definition. Moreover, μγ ≡ νγ also implies support (μγ ) = support (νγ ). Thus set Ωγ = Ωγ such that Ωγ = σ(Mγ,ϕ˜ ) = support (μγ ) =  Since Ωγ = Ωγ , then ϕγ = ϕγ for every γ in support (νγ ) = σ(N γ,ϕ ˜ ) = Ωγ .

 Γ = Γ and Ω = γ∈Γ Ωγ = γ∈Γ  Ωγ = Ω  , and therefore ϕ = ϕ . Since ϕ = ϕ on Ω = Ω  we get ϕ−1 ({λ}) = ϕ−1 ({λ}) for every λ ∈ C (and so #ϕ−1 ({λ}) = #ϕ−1 ({λ}) trivially). Thus mMϕ = mNϕ by Definition B.5. (b) Conversely, if μ ≡ ν, then they act on the same σ-algebra and hence Ω = Ω . Since Ω = Ω  and mMϕ = mNϕ , then Ωγ = Ωγ for every γ and so we may regard Γ = Γ . Thus {Ωγ }γ∈Γ = {Ωγ }γ∈Γ  is a set of sets whose ordinary union is a priori bounded, and hence ϕ˜ = ϕ˜ (on the closure of their common bounded ordinary union). Moreover, Ωγ = Ωγ and μ ≡ ν imply μγ ≡ νγ for ≡ νγ and ϕ˜ = ϕ˜, we get Mγ,ϕ˜ ∼ each γ. Since μγ  = Nγ,ϕ˜ by Theorem B.3. ∼ N ϕ . ∼ ∼  Therefore Mϕ = γ∈Γ Mγ,ϕ˜ = γ∈Γ  Nγ,ϕ˜ = Let T ∈ B[H] be a normal operator on a Hilbert space H. Take $the unique spectral measure E:Aσ(T ) →B[H] of the spectral decomposition T = σ(T ) λ dEλ as in Theorem 3.15. By Theorem 3.11, T ∼ = Mϕ with Mϕ ∈ B[L2 (Ω, μ)] being a

230

B. A Glimpse at Multiplicity Theory

on the Hilbert space L2 (Ω, μ) where Ω multiplication (thus normal) operator

is a disjoint union included in γ∈Γ σ(T ), μ is a positive measure on AΩ , and ϕ : Ω → C is the identity map with multiplicity. Take an arbitrary λ ∈ σ(T ). As we had seen in the proof of Theorem 3.15 parts (a,b,c), ˆ −1 ({λ})) ∼ E({λ}) = E(ϕ = E  (λλ) = M χλλ with λλ = ϕ−1 ({λ}) ∈ Ω and M χλλ ∈ B[L2 (Ω, μ)]. Let μ|λλ : Aλλ → C be the restriction of the measure μ : AΩ → C to the sub-σ-algebra Aλλ = AΩ ∩ ℘ (λλ). So R(E({λ})) ∼ = R(M χλλ) = L2 (λλ , μ|λλ ), and hence

dim R(E({λ})) = dim L2 (λλ, μ|λλ ).

Definition B.7. Let E be the spectral measure on Aσ(T ) for the spectral decomposition of a normal operator T ∈ B[H] on a Hilbert space H. The multiplicity mT (λ) of a point λ ∈ C for T is defined as follows. If λ ∈ σ(T ), then mT (λ) = dim R(E({λ})) (since {λ} ∈ Aσ(T ) ) and so, according to Proposition 3.G, mT (λ) = dim R(E({λ})) = dim N (λI − T ). If λ ∈ ρ(T )), then mT (λ) = 0. The multiplicity function for T is a cardinalnumber-valued map mT on C assigning to each λ ∈ C its multiplicity, mT : λ → mT (λ). Corollary B.8. Two normal operators on separable Hilbert spaces are unitarily equivalent if and only if they have the same multiplicity function and their scalar spectral measures are equivalent (i.e., if and only if they have the same multiplicity function and their spectral measures are equivalent). Proof. Let T ∈ B[H] and S ∈ B[K] be normal operators. According to Theorem 3.11, they are unitarily equivalent to multiplication operators Mϕ ∈ L2 (Ω, μ) and Nϕ ∈ L2 (Ω  , ν). That is, T ∼ = S if = Mϕ and S ∼ = Nϕ . By transitivity T ∼ and only if Mϕ ∼ = Nϕ . By Theorem B.6 Mϕ ∼ = Nϕ if and only if μ ≡ ν and mMϕ = mNϕ , and in this case Ω = Ω  and ϕ = ϕ . Summing up: ∼S T =

⇐⇒

∼ N ϕ Mϕ =

⇐⇒

μ ≡ ν and mMϕ = mNϕ .

Now assuming T ∼ = S, let E and P be the $ spectral measures on $ Aσ(T ) = Aσ(S) of the spectral decompositions of T = σ(T ) λ dEλ and S = σ(S) λ dPλ as in Theorem 3.15, where by unitary equivalence σ(T ) = σ(Mϕ ) = σ(Nϕ ) = σ(S). Take any λ ∈ C . Set λλ = ϕ−1 ({λ}) and λλ = ϕ−1 ({λ}). If μ ≡ ν and mMϕ = mNϕ , then ϕ = ϕ on Ω = Ω  (cf. Theorem B.6) and so λλ =λλ. Thus since λλ = λλ and μ ≡ ν, then dim L2 (λλ, μ|λλ ) = dim L2 (λλ , ν|λλ) naturally. Conversely, if

B.5 Multiplicity Function for Normal Operators

231

μ ≡ ν and dim L2 (λλ, μ|λλ ) = dim L2 (λλ , ν|λλ), then #λλ = #λλ, which means ∼ L2 (Ω  , ν) are separable once H ∼ mMϕ = mNϕ . (Recall: L2 (Ω, μ) = = K are,  and this implies Γ and Γ countable, and so are λλ and λλ .) Therefore since dim R(E({λ})) = dim L2(λλ, μ|λλ) and dim R(P ({λ})) = dim L2(λλ, ν|λλ), we get μ ≡ ν and mMϕ = mNϕ

μ ≡ ν and mT = mS

⇐⇒

ˆ f,f be scalar by Definition B.7. Since H and K are separable, let π ˆe,e and ω ˆ and Pˆ on AΩ of the spectral measures equivalent to $the spectral measures E $ ˆλ and S = λ dPˆλ as in the proof of spectral decompositions of T = Ω λ dE Ω Theorem 3.15 part (b) — cf. Definition 4.4 and Lemma 4.7. That is, ˆ π ˆe,e ≡ E

and

ω ˆ f,f ≡ Pˆ .

By the remark following Lemma 4.7, π ˆe,e = μ

and

ˆ f,f = ν. ω

All these measures act on the same σ-algebra AΩ . Thus μ≡ν

⇐⇒

π ˆe,e ≡ ω ˆ f,f

⇐⇒

ˆ ≡ Pˆ . E

ˆ −1 (·)) and P (·) = Pˆ (ϕ−1 (·)) on Aσ(T ) = Aσ(S) with ϕ = ϕ Since E(·) = E(ϕ  on Ω = Ω as in the proof of Theorem 3.15 part (c), we finally get μ≡ν

⇐⇒

ˆ ≡ Pˆ E

⇐⇒

E ≡ P.



Multiplicity function and (equivalence of ) spectral measure form a complete set of unitary invariants for normal operators on separable spaces. Remark. By the argument in the proof of Corollary B.8 and under the separability assumption, Theorem B.6 can be restated as in Corollary B.8 with E $ measures of the spectral decompositions of Mϕ = $ and P being the spectral  = λ dE and N λ dPλ with σ(Mϕ ) = σ(Nϕ ) by unitary equivλ ϕ σ(Mϕ ) σ(Nϕ ) alence (as in Theorem 3.15). Theorem B.2 is also naturally translated along the lines of Corollary B.8 in the case of compact normal operators on separable Hilbert spaces, with R(Eλ ) = R(E({λ})) = N (λI − T ) and now with countable spectra (and so the appropriate measures are counting measures).

Suggested Readings Arveson [6] Brown [21] Conway [30]

Davidson [34] Dunford and Schwartz [46] Halmos [54]

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List of Symbols

∅, 1 N0, 1 N, 1 Z, 1 Q, 1 R, 1 C, 1 F, 1 {xn }, 1 x, y, 1 ×, 1 T , 1 B[X , Y ], 1 B[X ], 1 {0}, 2 I, 2 u −→ , 2 s −→ , 2 w −→ , 2 M−, 2 N (T ), 2 R(T ), 2 ⊕, 3 ∩, 3

T −1 , 3 G[X , Y ], 3 G[X ], 3 O, 5 ⊥, 5, 89 A⊥, 5 M⊥⊥, 5 ∼ =, 6, 23, 73, 191, 194, 220

, 6 span A, 6  A, 6 ∪, 6  , 7 D(T ), 9, 194 T ∗, 9, 193 T ∗∗, 9, 194 T |M , 10 ≤, 12