**Mathematical Surveys and Monographs**

Volume: 36;
1991;
436 pp;
Hardcover

MSC: Primary 47;
Secondary 32; 46

Print ISBN: 978-0-8218-1536-6

Product Code: SURV/36

List Price: $154.00

Individual Member Price: $123.20

**Electronic ISBN: 978-1-4704-1263-0
Product Code: SURV/36.E**

List Price: $154.00

Individual Member Price: $123.20

# The Theory of Subnormal Operators

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*John B. Conway*

“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. The progress in the theory of subnormal operators that has come about during the last several years grew out of applying the results of rational approximation.”

—from the Preface

This book is the successor to the author's 1981 book on the same subject.
In addition to reflecting the great strides in the development of subnormal
operator theory since the first book, the present work is oriented toward
rational functions rather than polynomials. Although the book is a research
monograph, it has many of the traits of a textbook, including exercises.

The book requires background in function theory and functional analysis, but
is otherwise fairly self-contained. The first few chapters cover the basics
about subnormal operator theory and present a study of analytic functions on
the unit disk. Other topics included are: some results on hyponormal
operators, an exposition of rational approximation interspersed with
applications to operator theory, a study of weak-star rational approximation, a
set of results that can be termed structure theorems for subnormal operators,
and a proof that analytic bounded point evaluations exist.

#### Table of Contents

# Table of Contents

## The Theory of Subnormal Operators

- Contents ix10 free
- Preface xiii14 free
- Chapter I. Preliminaries 118 free
- Chapter II. Subnormal Operators: The Elementary Theory 2744
- §1 Definition and examples 2744
- §2 Pure operators and the minimal normal extension 3754
- §3 Quasinormal operators 4360
- §4 Hyponormal operators 4663
- §5 Cyclic subnormal operators 5067
- §6 Weighted shifts 5370
- §7 Bounded point evaluations 6178
- §8 Bergman operators 6683
- §9 Spectral sets 7592
- §10 The commutant of a subnormal operator 7996
- §11 The restriction algebra and the functional calculus 85102
- §12 The C*-algebra generated by a subnormal operator 88105
- §13 Unitary equivalence, similarity, and quasisimilarity 93110

- Chapter III. Function Theory On The Unit Circle 99116
- §1 Cesaro sums 99116
- §2 Convolution on the circle 103120
- §3 Harmonic functions on the disk 105122
- §4 Fatou's Theorem 108125
- §5 Subharmonic functions 114131
- §6 Hardy spaces 117134
- §7 The Nevanlinna class 120137
- §8 Factorization of functions in Nevanlinna class 125142
- §9 The disk algebra 132149
- §10 The invariant subspace lattice of the unilateral shift 135152
- §11 Weak-star closed ideals in H[sup(∞)] 140157
- §12 Szegö's Theorem 141158
- §13 Analytic Toeplitz operators 145162

- Chapter IV. Hyponormal Operators 149166
- Chapter V. Uniform Rational Approximation 163180
- §1 Function algebras: examples and elementary properties 163180
- §2 Distributions and some results from analysis 166183
- §3 The Cauchy transform 173190
- §4 Invariant subspaces for subnormal operators 181198
- §5 Vitushkin localization operators 184201
- §6 T-invariant algebras 187204
- §7 The Shilov boundary 190207
- §8 Representing measures 193210
- §9 Harmonic measure 196213
- §10 Hardy spaces for an arbitrary region 203220
- §11 Peak points 211228
- §12 Capacity 216233
- §13 Some applications of analytic capacity 222239
- §14 Dirichlet algebras 226243
- §15 Gleason parts 234251
- §16 The Wermer Embedding Theorem 239256
- §17 Bands of measures 244261
- §18 Annihilating measures 250267
- §19 Mergelyan's Theorem 253270
- §20 The double dual of a T-invariant algebra 254271
- §21 The Lautzenheiser-Mlak Theorem 260277
- §22 Davie's Theorem 266283

- Chapter VI. Weak-Star Rational Approximation 277294
- Chapter VII. Some Structure Theory For Subnormal Operators 309326
- §1 A decomposition of subnormal operators 309326
- §2 The minimal normal extension problem for subnormal operators 315332
- §3 Spectral mapping theorems 324341
- §4 Spectral mapping theorems for the essential spectrum 333350
- §5 A factorization theorem 341358
- §6 An infinite factorization theorem 352369
- §7 Full analytic subspaces 357374
- §8 Reflexivity for subnormal operators 361378
- §9 Filling in the holes of the spectrum of a normal operator 365382
- §10 Quasisimilarity revisited 368385

- Chapter VIII. Bounded Point Evaluations 375392
- Epilogue 409426
- Bibliography 411428
- Index 431448
- List of symbols 435452

#### Reviews

As we have come to expect of the author, the standard of exposition is extremely high.

-- Bulletin of the London Mathematical Society

The authors pedagogical, almost conversational style makes this book a pleasure to read; even experts in subnormal operator theory will want this book as a reference … the author has developed in this text the theory of subnormal operators with clarity and humor. Anyone interested in operator theory or related areas of functional analysis will find this book a valuable reference.

-- Mathematical Reviews