Topics in Operator Theory Mathematical Surveys and Monographs

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http://dx.doi.org/10.1090/surv/013

Number 13

Edited by Carl Pearcy

American Mathematical Society Providence, Rhode Island The articles in this volume were originally commissioned for the MAA STUDIES series, but were transferred by mutual agreement of the Association and the Society to MATHEMATICAL SURVEYS.

2000 Mathematics Subject Classification. Primary 47-XX.

Library of Congress Cataloging in Publication Data

Pearcy, C. 1935- Topics in operator theory.

(Mathematical surveys, no. 13) Includes bibliographies. 1. Linear operators. I. Title. II. Series: American Mathematical Society. Mathematical surveys. no. 13. QA329.2.P4 S15'.72 74-8254 ISBN 0-8218-1513-X ISSN 0076-5376

© Copyright 1974 by the American Mathematical Society. All rights reserved. Printed in the United States of America. Second printing, with addendum, 1979. The American Mathematical Society retains all rights except those granted to the United States Government. © The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 06 05 04 03 02 01 TABLE OF CONTENTS

Preface vii I. Invariant subspaces 1 by Donald Sarason 1. Introduction 3 2. Some immediate observations 4 3. Reducing subspaces of normal operators 6 4. Invariant subspaces and operator algebras 10 5. Unitary operators 14 6. The bilateral shift 20 7. Maximal subalgebras 31 8. The Volterra operator 33 9. The Volterra operator plus multiplication by x 37 Addendum (1978) 44 Bibliography 45 II. Weighted shift operators and analytic function theory 49 by Allen L. Shields 1. Introduction 51 2. Elementary properties , 51 3. Weighted sequence spaces 57 4. The commutant 61 5. The spectrum 66 6. Analytic structure 73 7. Hyponormal and subnormal shifts , 83 8. Algebras generated by shifts 88 9. Strictly cyclic shifts 92 10. Invariant subspaces 102 11. Cyclic vectors 109 12. Notes and comments 116 TABLE OF CONTENTS

Addendum (1978) 122a Bibliography 123 Supplemental Bibliography (1978) 128a III. A version of multiplicity theory 129 by Arlen Brown Introduction 131 1. The standard spectral measure 133 2. Cyclic subspaces 135 3. Multiplicity : The problem 136 4. The algebra of finite measures 137 5. a-ideals of measures 139 6. Cycles and measures 142 7. a-ideals and subspaces 144 8. Multiplicity : The solution 147 9. Uniform multiplicity one 152 10. The separable case 153

11. The /,2-space of a a-ideal 156 Bibliography 160 IV. Canonical models 1"! by R. G. Douglas Introduction 163 1. Canonical models and unitary dilations 164 2. Unitary operators and invariant subspaces 170 3. Absolute continuity and the functional calculus 179 4. Characteristic operator functions 191

5. Contractions in class C0 and Jordan models 205 6. Related and more general models 212 Bibliography 215 V. A survey of the Lomonosov technique in the theory of invariant subspaces 219 by Carl Pearcy and Allen L. Shields Addendum (1978) 228 Bibliography 229 Index 231 PREFACE

The articles in this volume are concerned with various aspects of the theory of bounded linear operators on Hilbert space. This area of mathematical research is presently experiencing a period of intense excitement, due, no doubt, to the fact that during the past year several remarkable advances have been made on hard problems in the field. One particular problem on which considerable prog ress has been made is the "invariant subspace problem." This is the question whether every (bounded, linear) operator T on a separable, infinite-dimensional, complex Hilbert space H maps some (closed) subspace different from (0) and H into itself. It is therefore highly appropriate that the first and last of the five expository articles in this volume deal with invariant subspaces. The main theme of the first article, by Donald Sarason, may be summa rized as follows. If T is a bounded linear operator on H, then the collection of all subspaces M of H such that T maps M into itself forms a complete lattice, denoted by Lat (7), under the inclusion ordering. It is presently impos sible to determine, in general, the lattice-theoretic structure of Lat (T). (This is not surprising, since we cannot even say whether the possibility Lat (7) = {(0), H } is realizable.) There are a few operators 7, however, for which the structure of Lat (7*) is known, and Sarason's article is devoted to a discussion of such opera tors and their associated invariant subspace lattices. The presentation is extremely lucid, and one of the interesting features of the exposition is the interplay between operator theory and classical analysis that is so often found in Sarason's work. The second article in this volume is concerned with weighted shift opera tors. Interest focuses on this class of operators because it is a particularly simple class to define. Let {e„}^=0 be an orthonormal basis for tf, and let a= {an}„=0 be a bounded sequence of complex numbers (the weight sequence). It is easy to see that there exists a unique bounded operator Ta on H such that TQlen = anen+\ for every nonnegative integer n, and that the adjoint operator T* is determined by the equations T*e0 = 0 and T*en = an_len_], n > 0. The operators Ta and T* are typical examples of forward and backward weighted

vii unilateral shifts, respectively. A different type of weighted shift is obtained by considering an orthonormal bases {f^-..^ for H indexed by the set of all integers, and an associated (bounded) weight sequence j3 = {j3w }£=_«,. In this case, the unique bounded operator TQ on H that satisfies the equation Tpfn = j3„/n + , for all integers n is a typical forward weighted bilateral shift, and a typical backward weighted bilateral shift can be defined analogously. Allen Shields has taken essentially all of the information presently known about weighted shift operators (with scalar weights) and incorporated it into his com prehensive article. A central theme of the exposition is the interplay between weighted shift operators and analytic function theory, and, as an added bonus for the reader, the article contains a list of thirty-two interesting research prob lems. The third article in this volume is an exposition by Arlen Brown of the theory of spectral multiplicity for normal operators on Hilbert space. The prob lem treated arises as follows. In general, one wants to know when two operators

T{ and T2 acting on Hilbert spaces Kx and K2, respectively, are unitarily equivalent (i.e., when there is an inner product preserving isomorphism *p of l K{ onto K2 such that $Tx$~ = T2). Unitary equivalence is the analog for operators of the concept of isomorphism for groups, rings, etc. The problem is usually attacked by trying to attach to each operator T in a fixed class of oper ators an indexed family of objects {0K(T)}X^A with the property that two op erators Tx and T2 in the class are unitarily equivalent if and only if Ox(Tx) =

Ox(T2) for all indices \ in A. Such a collection {0K(T)}K^A is called a complete set of unitary invariants for the class of operators. There are a few classes of operators for which a "reasonable" complete set of unitary invariants is known; in this connection see the bibliography of Brown's article. The theory of spectral multiplicity furnishes a reasonable complete set of unitary invariants for normal operators, and this set of invariants can be obtained in several different ways. One may consider the (commutative) C*-algebra or von Neumann algebra generated by a given normal operator, and study the unitary equivalence problem for such algebras. In this approach, the solution of the unitary equivalence prob lem for normal operators becomes a corollary of the solution of the problem for commutative C -algebras or von Neumann algebras. One may also proceed more directly and focus attention on the concept of a spectral measure. It is this last approach that Brown follows, and his clear presentation of this circle of ideas should lead to a better understanding of multiplicity theory by beginners and ex perts alike. The fourth article in this volume, by R. G. Douglas, is concerned with canon ical models for operators. The central underlying idea here is that given any con traction operator T on H (i.e., any operator T satifying ||7,||<1), there is

viii a canonical construction that associates with T an operator MT that is unitarily equivalent to T, called its "canonical model.'* Thus, one may study T by study ing MT instead, and this is one of the themes of the book Harmonic analysis of operators on Hilbert space, by Sz.-Nagy and Foias. Douglas, who has contributed significantly to the geometrization of the theory of canonical models, exposes in his article various important components of this theory, and thereby gives the reader much insight into its successes and failures. The final article in this volume, written by Allen Shields and myself, is a survey of some invariant-subspace theorems that resulted from the brilliant and elegant method of proof introduced by Victor Lomonosov early in 1973. It would indeed be surprising if further study and refinement of this technique does not lead to addi tional progress on the invariant subspace problem.

CARL PEARCY ANN ARBOR, MICHIGAN DECEMBER 1973 INDEX absolutely continuous unitary operator, conjugate space, 64 179 convolution, 34 algebra, 88, 121 Corona conjecture, 78 analytic measure, 17 cycle: see subspace, cyclic analytic projection, 91 cycle, generalized, 158 approximate point spectrum, 69, 79, 83 cyclic vectors, 92,109 Aronszajn-Smith technique, 228 defect operators, 182 Bergman space 86, 102, 111, 115, 120 dilation, 165 Beurling, A., 3,21,63, 102, 111 direct integral, 171 bilateral shift, 20, 29, 31, 51, 176 direct limit, 156-157 bilateral weighted shift, 181 Dirichlet space, 79,102,115,120,121 Blaschke factor, 24 double commutant theorem, 10 Blaschke product eigenvalues, 70, 109, 115 finite, 25 eigenvectors, 70, 71, 74, 80 infinite, 25 equivalence (of measures), 134, 138 Boolean algebra, 139 Boolean ring, 139 Fatou's theorem, 22, 27 bounded point evaluation, 73, 76, 79, Fejer kernel, 90, 117 80,85, 109, 110 fiber, 78 Burnside's Theorem, 227 finite dimensional weighted shift, 56,110 Fourier transformation, 29, 34 Cesaro means, 90 Fuglede's theorem, 10, 11 characteristic operator function, 192 full range invariant subspace, 178 circular symmetry, 52, 66, 67, 72, 74, 75 full subspace, 164 class C operators, 180 functional calculus, 185 class C0 operators, 188 2 coefficient multipliers, 91 H ,20 commutant, 6, 61, 91,117, 221, 224, 226 //°°, 22 P compact, 52 H ,2S 2 complete wandering subspace, 176 # (0), 58 completely nonunitary, 168 H°°(Pl 61 compression, 165 Hardy-Littlewood maximal theorem, 114

233 Hardy spaces, 18, 28 minimal function, mT, 188 Herglotz theorem, 27 multiplicity Hurwitz theorem, 111 of a stack, 147, 149 hyperinvariant subspace, 102, 109 uniform, 152 200,221,223,224,225,226 multiplicity function, 137, 150,153, 155 hyponormal shift, 83, 87, 117 multiplicity theory, 131,136 inner function, 21, 23,110, 111, 178 multiplication operator, 6, 11, 58, 62, 172 invariant subspace, 102, 109, 118, 164 multiplicity ixT, 208 invariant subspace problem, 3, 221, 225 multiplicity function r?(X), 173 irreducible invariant subspace, 14 normal, 83 irreducible operator, 4 numerical range, 72 isometric Banach spaces, 64,79,93 numerical radius, 72, 73 isomorphic Banach spaces, 64 operator, isomorphic lattices, 102 multiplication, 6, 11, 16, 37 Jordan model, 207 normal, 6 kernel function, 24 position, 134 translation, 30 2 L (/J),58 unitary, 14 r°matrix (with respect to an orthogonal range function, 174 basis), 61,62 reducing subspace, 4, 6, 116, 164 maximal subalgebra, 31, 63 reflexive algebra, 10, 31 maximum ideal space, 78,94, 110, reflexive operator, 104 measure algebra, 8 regular factorization, 203 measure ring, 138 reproducing kernel, 73 minimal, 166 Riesz Theorem, 17

234 a-ideal (of measures), 139 ff. strong operator topology, 76, 88, 90, 93, 226 scalar spectral measure, 173 strongly strictly cyclic, 98, 118 Schauder-Tychonoff Theorem, 222, subnormal shift, 83, 117 223, 225 subspace, cyclic, 135 ff. Schur, 117 support of inner function, 196 semi-invariant subspace, 169 separating (family linear functional), TM, 168 60 ~, 188 shift, Titchmarsh theorem, 36 bilateral, 20, 29,31, 176 trace class, 117 unilateral, 3, 16, 20, 177 transitive algebra, 221, 222, 224, 226 weighted, 5, 51 unicellular shift, 104 similar shifts, 54, 60, 87, 108, unilateral shift, 3, 16, 20, 177 singular function, 27 unitary dilation, 14, 165 spatially isomorphic lattices, 102 unitary function, 175 spectral measure, 132 ff. unitary semigroup, 19, 29 of a normal operator, 133 unitarily equivalent shifts, 51, 53, 59 standard, 134, 136,137,142,145 upper semicontinuous, 66 146, 147, 151,154,157. vector, cyclic, 109, 135, 154 unitary equivalence of, 133-134, von Neumann's inequality, 75, 81, 82, 146, 150 167 spectral radius, 66 spectral theorem, 133 wandering subspace, 16, 175 spectrum, 66 wandering vector, 16 stack (of subspaces), 147 ff. weak operator topology, 64, 76, 79, 83, strictly cyclic (algebra, operator, vect ), 86, 88, 93. 78,79,92, 118

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