Topics in Operator Theory

Transcription

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2 Topics in Operator Theory

3 Mathematical Surveys and Monographs Number 13 Topics in Operator Theory Edited by Carl Pearcy American Mathematical Society Providence, Rhode Island

4 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 Mathematics Subject Classification. Primary 47-XX. Library of Congress Cataloging in Publication Data Pearcy, C 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' ISBN X ISSN Copyright 1974 by the American Mathematical Society. All rights reserved. Printed in the United States of America. Second printing, with addendum, 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:

5 Preface TABLE OF CONTENTS 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 Unitary operators The bilateral shift Maximal subalgebras The Volterra operator 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 Elementary properties, Weighted sequence spaces The commutant The spectrum Analytic structure Hyponormal and subnormal shifts, Algebras generated by shifts Strictly cyclic shifts Invariant subspaces Cyclic vectors Notes and comments 116 vii

6 TABLE OF CONTENTS Addendum (1978) 122a Bibliography 123 Supplemental Bibliography (1978) 128a III. A version of multiplicity theory 129 by Arlen Brown Introduction The standard spectral measure Cyclic subspaces Multiplicity : The problem The algebra of finite measures a-ideals of measures Cycles and measures a-ideals and subspaces Multiplicity : The solution Uniform multiplicity one The separable case The /, 2 -space of a a-ideal 156 Bibliography 160 IV. Canonical models 1"! by R. G. Douglas Introduction Canonical models and unitary dilations Unitary operators and invariant subspaces Absolute continuity and the functional calculus Characteristic operator functions Contractions in class C 0 and Jordan models 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

7 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 progress 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 summarized 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 impossible 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 operators 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 operators. 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= {a n } =0 be a bounded sequence of complex numbers (the weight sequence). It is easy to see that there exists a unique bounded operator T a on H such that T Ql e n = a n e n+\ for every nonnegative integer n, and that the adjoint operator T* is determined by the equations T*e 0 = 0 and T*e n = a n _ l e n _ ], n > 0. The operators T a and T* are typical examples of forward and backward weighted vii

8 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 = {j3 w } =_«,. In this case, the unique bounded operator TQ on H that satisfies the equation Tpf n = 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 comprehensive 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 problems. 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 problem treated arises as follows. In general, one wants to know when two operators T { and T 2 acting on Hilbert spaces K x and K 2, respectively, are unitarily equivalent (i.e., when there is an inner product preserving isomorphism *p of K { onto K 2 such that $T x $~ l = T 2 ). 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 operators an indexed family of objects {0 K (T)} X^A with the property that two operators T x and T 2 in the class are unitarily equivalent if and only if O x (T x ) = O x (T 2 ) for all indices \ in A. Such a collection {0 K (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 problem 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 experts alike. The fourth article in this volume, by R. G. Douglas, is concerned with canonical models for operators. The central underlying idea here is that given any contraction operator T on H (i.e., any operator T satifying 7, <1), there is viii

9 a canonical construction that associates with T an operator M T that is unitarily equivalent to T, called its "canonical model.'* Thus, one may study T by studying M T 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 It would indeed be surprising if further study and refinement of this technique does not lead to additional progress on the invariant subspace problem. ANN ARBOR, MICHIGAN DECEMBER 1973 CARL PEARCY

10 INDEX

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

12 Hardy spaces, 18, 28 Herglotz theorem, 27 Hurwitz theorem, 111 hyperinvariant subspace, 102, ,221,223,224,225,226 hyponormal shift, 83, 87, 117 inner function, 21, 23,110, 111, 178 invariant subspace, 102, 109, 118, 164 invariant subspace problem, 3, 221, 225 irreducible invariant subspace, 14 irreducible operator, 4 isometric Banach spaces, 64,79,93 isomorphic Banach spaces, 64 isomorphic lattices, 102 Jordan model, 207 kernel function, 24 L 2 (/J),58 r <P).6i lattice, attainable, 5 complete, 140 invariant subspace, 4 of measures, 139 Laurent polynomial, 79 Laurent series, 58 Lavrentiev's theorem, 227 Lebesgue decomposition, 139 Lebesgue decomposition (relative to a ideal), 141 Lomonosov technique, 221, 222 M 2,58 M 9 62 matrix (with respect to an orthogonal basis), 61,62 maximal subalgebra, 31, 63 maximum ideal space, 78,94, 110, measure algebra, 8 measure ring, 138 minimal, 166 minimal function, m T, 188 multiplicity of a stack, 147, 149 uniform, 152 multiplicity function, 137, 150,153, 155 multiplicity theory, 131,136 multiplication operator, 6, 11, 58, 62, 172 multiplicity ix T, 208 multiplicity function r?(x), 173 normal, 83 numerical range, 72 numerical radius, 72, 73 operator, multiplication, 6, 11, 16, 37 normal, 6 position, 134 translation, 30 unitary, 14 Volterra, 5, 33, 37 orthogonal basis, 57, 62 power bounded, 55 power dilation, 169 power series, 57 principal ideal (of measures), 138, 139, 140, 142, 145, 153, 154 pure invariant subspace, 174 purely contractive, 192 r - quasi-analytic, 103, 109 quasi-invertible, 198 quasi nilpotent operator, 104, 225 quasi similar, 200, 225 Radon-Nikodym Theorem, 142 range function, 174 reducing subspace, 4, 6, 116, 164 reflexive algebra, 10, 31 reflexive operator, 104 regular factorization, 203 reproducing kernel, 73 Riesz Theorem,

13 a-ideal (of measures), 139 ff. scalar spectral measure, 173 Schauder-Tychonoff Theorem, 222, 223, 225 Schur, 117 semi-invariant subspace, 169 separating (family linear functional), 60 shift, bilateral, 20, 29,31, 176 unilateral, 3, 16, 20, 177 weighted, 5, 51 similar shifts, 54, 60, 87, 108, singular function, 27 spatially isomorphic lattices, 102 spectral measure, 132 ff. of a normal operator, 133 standard, 134, 136,137,142, , 147, 151,154,157. unitary equivalence of, , 146, 150 spectral radius, 66 spectral theorem, 133 spectrum, 66 stack (of subspaces), 147 ff. strictly cyclic (algebra, operator, vect 78,79,92, 118 strong operator topology, 76, 88, 90, 93, 226 strongly strictly cyclic, 98, 118 subnormal shift, 83, 117 subspace, cyclic, 135 ff. support of inner function, 196 T M, 168 ~, 188 Titchmarsh theorem, 36 trace class, 117 transitive algebra, 221, 222, 224, 226 unicellular shift, 104 unilateral shift, 3, 16, 20, 177 unitary dilation, 14, 165 unitary function, 175 unitary semigroup, 19, 29 unitarily equivalent shifts, 51, 53, 59 upper semicontinuous, 66 vector, cyclic, 109, 135, 154 von Neumann's inequality, 75, 81, 82, 167 wandering subspace, 16, 175 wandering vector, 16 weak operator topology, 64, 76, 79, 83, ), 86, 88,

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