**Graduate Studies in Mathematics**

Volume: 21;
2000;
372 pp;
Hardcover

MSC: Primary 47;

Print ISBN: 978-0-8218-2065-0

Product Code: GSM/21

List Price: $60.00

Individual Member Price: $48.00

**Electronic ISBN: 978-1-4704-2076-5
Product Code: GSM/21.E**

List Price: $60.00

Individual Member Price: $48.00

# A Course in Operator Theory

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*John B. Conway*

Operator theory is a significant part of many important areas of modern
mathematics: functional analysis, differential equations, index theory,
representation theory, mathematical physics, and more. This text covers the
central themes of operator theory, presented with the excellent clarity and
style that readers have come to associate with Conway's writing.

Early chapters introduce and review material on C*-algebras, normal
operators, compact operators and non-normal operators. The topics include the
spectral theorem, the functional calculus and the Fredholm index. Also, some
deep connections between operator theory and analytic functions are presented.

Later chapters cover more advanced topics, such as representations of
C*-algebras, compact perturbations and von Neumann algebras. Major results,
such as the Sz.-Nagy Dilation Theorem, the Weyl-von Neumann-Berg Theorem and
the classification of von Neumann algebras, are covered, as is a treatment of
Fredholm theory. These advanced topics are at the heart of current research.

The last chapter gives an introduction to reflexive subspaces, i.e., subspaces
of operators that are determined by their invariant subspaces. These, along
with hyperreflexive spaces, are one of the more successful episodes in the
modern study of asymmetric algebras.

Professor Conway's authoritative treatment makes this a compelling and rigorous
course text, suitable for graduate students who have had a standard course in
functional analysis.

#### Readership

Graduate students and research mathematicians interested in operator theory.

#### Reviews & Endorsements

John B. Conway belongs to the best authors of basic textbooks … The present book continues this tradition of clear and elegant way of presentation. … this book can be highly recommended for students of operator theory as well as to experts in the field who will find many interesting ideas there.

-- Mathematica Bohemica

Conway's book adds a complementary volume of study for those just becoming acquainted with the field … shares … a style which is relaxed, yet concise … recommend it to anyone wishing to gain a better understanding of operator theory.

-- Bulletin of the London Mathematical Society

This is an excellent course in operator theory and operator algebras … leads the reader to deep new results and modern research topics … the author has done more than just write a good book—he has managed to reveal the unspeakable charm of the subject, which is indeed the ‘source of happiness’ for operator theorists.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## A Course in Operator Theory

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents ix10 free
- Preface xiii14 free
- Chapter 1. Introduction to C*-Algebras 118 free
- §1. Definition and examples 118
- §2. Abelian C*-algebras and the Functional Calculus 724
- §3. The positive elements in a C*-algebra 1229
- §4. Approximate identities 1734
- §5. Ideals in a C*-algebra 2138
- §6. Representations of a C*-algebra 2441
- §7. Positive linear functionals and the GNS construction 2946

- Chapter 2. Normal Operators 3754
- Chapter 3. Compact Operators 7188
- Chapter 4. Some Non-Normal Operators 105122
- §22. Algebras and lattices 105122
- §23. Isometries 111128
- §24. Unilateral and bilateral shifts 118135
- §25. Some results on Hardy spaces 126143
- §26. The functional calculus for the unilateral shift 132149
- §27. Weighted shifts 136153
- §28. The Volterra operator 143160
- §29. Bergman operators 147164
- §30. Subnormal operators 157174
- §31. Essentially normal operators 170187

- Chapter 5. More on C*-Algebras 181198
- Chapter 6. Compact Perturbations 207224
- Chapter 7. Introduction to Von Neumann Algebras 241258
- §43. Elementary properties and examples 242259
- §44. The Kaplansky Density Theorem 250267
- §45. The Pedersen Up-Down Theorem 253270
- §46. Normal homomorphisms and ideals 258275
- §47. Equivalence of projections 265282
- §48. Classification of projections 270287
- §49. Properties of projections 278295
- §50. The structure of Type I algebras 282299
- § 51. The classification of Type I algebras 289306
- §52. Operator-valued measurable functions 294311
- §53. Some structure theory for continuous algebras 301318
- §54. Weak-star continuous linear functionals revisited 305322
- §55. The center-valued trace 311328

- Chapter 8. Reflexivity 319336
- Bibliography 355372
- Index 367384
- List of Symbols 371388
- Back Cover Back Cover1390