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View This Volume's Front and Back Matter http://dx.doi.org/10.1090/surv/036 MATHEMATICAL SURVEYS AND MONOGRAPHS SERIES LIST Volume 1 The problem of moments, 16 Symplectic groups, J. A. Shohat and J. D. Tamarkin O. Timothy O'Meara 2 The theory of rings, 17 Approximation by polynomials N. Jacobson with integral coefficients, 3 Geometry of polynomials, Le Baron O. Ferguson M. Marden 18 Essentials of Brownian motion 4 The theory of valuations, and diffusion, Frank B. Knight O. F. G. Schilling 19 Contributions to the theory of 5 The kernel function and transcendental numbers, Gregory conformal mapping, V. Chudnovsky S. Bergman 20 Partially ordered abelian groups 6 Introduction to the theory of with interpolation, Kenneth R. algebraic functions of one Goodearl variable, C. C. Chevalley 21 The Bieberbach conjecture: 7.1 The algebraic theory of Proceedings of the symposium on semigroups, Volume I, A. H. the occasion of the proof, Albert Clifford and G. B. Preston Baernstein, David Drasin, Peter Duren, and Albert Marden, 7.2 The algebraic theory of Editors semigroups, Volume II, A. H. Clifford and G. B. Preston 22 Noncommutative harmonic analysis, Michael E. Taylor 8 Discontinuous groups and automorphic functions, 23 Introduction to various aspects of J. Lehner degree theory in Banach spaces, E. H. Rothe 9 Linear approximation, Arthur Sard 24 Noetherian rings and their applications, Lance W. Small, 10 An introduction to the analytic Editor theory of numbers, R. Ayoub 25 Asymptotic behavior of dissipative 11 Fixed points and topological systems, Jack K. Hale degree in nonlinear analysis, J. Cronin 26 Operator theory and arithmetic in H°°9 Hari Bercovici 12 Uniform spaces, J. R. Isbell 27 Basic hypergeometric series and 13 Topics in operator theory, applications, Nathan J. Fine A. Brown, R. G. Douglas, C. Pearcy, D. Sarason, A. L. 28 Direct and inverse scattering on Shields; C. Pearcy, Editor the lines, Richard Beals, Percy Deift, and Carlos Tomei 14 Geometric asymptotics, V. Guillemin and S. Sternberg 29 Amenability, Alan L. T. Paterson 15 Vector measures, J. Diestel and 30 The Markoff and Lagrange J. J. Uhl, Jr. spectra, Thomas W. Cusick and Mary E. Flahive MATHEMATICAL SURVEYS AND MONOGRAPHS SERIES LIST Volume 31 Representation theory and harmonic analysis on semisimple Lie groups, Paul J. Sally, Jr. and David A. Vogan, Jr., Editors 32 An introduction to CR structures, Howard Jacobowitz 33 Spectral theory and analytic geometry over non-Archimedean fields, Vladimir G. Berkovich 34 Inverse source problems, Victor Isakov 35 Algebraic geometry for scientists and engineers, Shreeram S. Abhyankar The Theory of Subnormal Operators MATHEMATICAL Surveys and Monographs Volume 36 The Theory of Subnormal Operators John B. Conway £ff |^^H \& American Mathematical Society ^v.. *[f[llT]lD ijB Providence, Rhode Island 1980 Mathematics Subject Classification (1985 Revision). Primary 47B20; Secondary 32A35. Library of Congress Cataloging-in-Publication Data Conway, John B. The theory of subnormal operators/John B. Conway. p. cm.—(Mathematical surveys and monographs, ISSN 0076-5376; no. 36) Includes bibliographical references and indexes. ISBN 0-8218-1536-9 (acid-free) 1. Subnormal operators. I. Title. II. Series. QA329.2.C667 19910 90-26659 515'.7246—dc20 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publi­ cation (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Ser­ vices, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The owner consents to copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that a fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. When paying this fee please use the code 0076-6376/91 to refer to this publication. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright ©1991 by the American Mathematical Society. All rights reserved. Printed in the United States of America 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. © This publication was typeset using AMS-T^X, the American Mathematical Society's T^X macro system. 109 8765432 1 95 94 93 92 91 For Ann, my favorite collaborator, and Bligh, our best theorem. Contents Preface xiii Chapter I. Preliminaries 1 § 1 Trace class and Hilbert-Schmidt operators 1 §2 Some topologies on 3B(&) 10 §3 Isometries 13 §4 The unilateral shift of multiplicity 1 18 §5 Direct sums 24 §6 Shifts of higher multiplicity 26 Chapter II. Subnormal Operators: The Elementary Theory 27 § 1 Definition and examples 27 §2 Pure operators and the minimal normal extension 37 §3 Quasinormal operators 43 §4 Hyponormal operators 46 §5 Cyclic subnormal operators 50 §6 Weighted shifts 53 §7 Bounded point evaluations 61 §8 Bergman operators 66 §9 Spectral sets 75 §10 The commutant of a subnormal operator 79 §11 The restriction algebra and the functional calculus 85 §12 The C*-algebra generated by a subnormal operator 88 §13 Unitary equivalence, similarity, and quasisimilarity 93 Chapter III. Function Theory On The Unit Circle 99 § 1 Cesaro sums 99 §2 Convolution on the circle 103 §3 Harmonic functions on the disk 105 §4 Fatou's Theorem 108 §5 Subharmonic functions 114 §6 Hardy spaces 117 §7 The Nevanlinna class 120 §8 Factorization of functions in Nevanlinna class 125 x CONTENTS §9 The disk algebra 132 §10 The invariant subspace lattice of the unilateral shift 135 §11 Weak-star closed ideals in H°° 140 §12 Szego's Theorem 141 § 13 Analytic Toeplitz operators 145 Chapter IV. Hyponormal Operators 149 § 1 The real part of a hyponormal operator 150 §2 The Berger-Shaw Theorem 152 §3 The area of the spectrum of a hyponormal operator 156 §4 The self-commutator of the Bergman operator 156 §5 The decomposition of essentially normal operators 158 Chapter V. Uniform Rational Approximation 163 §1 Function algebras: examples and elementary properties 163 §2 Distributions and some results from analysis 166 §3 The Cauchy transform 173 §4 Invariant subspaces for subnormal operators 181 §5 Vitushkin localization operators 184 §6 T-invariant algebras 187 §7 The Shilov boundary 190 §8 Representing measures 193 §9 Harmonic measure 196 §10 Hardy spaces for an arbitrary region 203 §11 Peak points 211 §12 Capacity 216 § 13 Some applications of analytic capacity 222 § 14 Dirichlet algebras 226 § 15 Gleason parts 234 §16 The Wermer Embedding Theorem 239 § 17 Bands of measures 244 § 18 Annihilating measures 250 § 19 Mergelyan's Theorem 253 §20 The double dual of a T-invariant algebra 254 §21 The Lautzenheiser-Mlak Theorem 260 §22 Davie's Theorem 266 Chapter VI. Weak-Star Rational Approximation 277 §1 Weak-star closed subalgebras of L°°(^) 277 §2 The envelope 282 §3 Chaumat's Theorem 287 §4 H°°(dK) for a Dirichlet algebra 291 §5 Dirichlet algebras revisited 296 §6 The weak-star closure of a Dirichlet algebra 298 §7 The weak-star closure of the polynomials 301 CONTENTS xi Chapter VII. Some Structure Theory For Subnormal Operators 309 § 1 A decomposition of subnormal operators 309 §2 The minimal normal extension problem for subnormal operators 315 §3 Spectral mapping theorems 324 §4 Spectral mapping theorems for the essential spectrum 333 §5 A factorization theorem 341 §6 An infinite factorization theorem 352 §7 Full analytic subspaces 357 §8 Reflexivity for subnormal operators 361 §9 Filling in the holes of the spectrum of a normal operator 365 § 10 Quasisimilarity revisited 368 Chapter VIII. Bounded Point Evaluations 375 §1 A coloring scheme 375 §2 A sufficient condition for the existence of bounded point evaluations 384 §3 Heavy barriers exist 389 §4 Bounded point evaluations exist 397 §5 Thomson's Theorem 399 §6 Some applications of Thomson's Theorem 407 Epilogue 409 Bibliography 411 Index 431 List of symbols 435 Preface This book is a successor to Conway [1981a]. In addition to reflecting the great strides in the development of subnormal operator theory since that first set of notes, the present work is oriented toward rational functions rather than polynomials. This necessitates additional function theory and a knowledge of analytic capacity, both of which are developed here. There are, however, some crucial estimates and applications of capacity to which I have only made references. I began the preface of Conway [1981a] by asking the question, "What is mathematics?" I went on to explain that my definition of mathematics is that it is a collection of examples as opposed to a body of theorems. This definition remains and is still reflected in the nature of this book. I believe mathematics is a collection of examples irrespective of the particular area and that the good theorems are those that explain, classify, and interpret large classes of examples. Paul R. Halmos introduced the concept of a subnormal operator at the same time that he defined hyponormal operators. He was led to do so by a study of the properties of the unilateral shift, the most understood non- normal operator. In a certain sense, subnormal operators were introduced too soon because the theory of function algebras and rational approximation was also in its infancy and could not be properly used to examine this class of operators.
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