**Mathematical Surveys and Monographs**

Volume: 138;
2007;
348 pp;

MSC: Primary 47; 30; 46; 32;

**Electronic ISBN: 978-1-4704-1365-1
Product Code: SURV/138.E**

List Price: $97.00

AMS Member Price: $77.60

MAA Member Price: $87.30

#### Supplemental Materials

# Operator Theory in Function Spaces: Second Edition

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*Kehe Zhu*

This book covers Toeplitz operators, Hankel operators, and composition
operators on both the Bergman space and the Hardy space. The setting is the
unit disk and the main emphasis is on size estimates of these operators:
boundedness, compactness, and membership in the Schatten classes.

Most results concern the relationship between operator-theoretic properties
of these operators and function-theoretic properties of the inducing
symbols. Thus a good portion of the book is devoted to the study of analytic
function spaces such as the Bloch space, Besov spaces, and BMOA, whose
elements are to be used as symbols to induce the operators we study.

The book is intended for both research mathematicians and graduate students
in complex analysis and operator theory. The prerequisites are minimal; a
graduate course in each of real analysis, complex analysis, and functional
analysis should sufficiently prepare the reader for the book. Exercises and
bibliographical notes are provided at the end of each chapter. These notes
will point the reader to additional results and problems.

Kehe Zhu is a professor of mathematics at the State University of
New York at Albany. His previous books include Theory of Bergman
Spaces (Springer, 2000, with H. Hedenmalm and B. Korenblum) and
Spaces of Holomorphic Functions in the Unit Ball (Springer,
2005). His current research interests are holomorphic function spaces
and operators acting on them.

#### Reviews & Endorsements

...the present title definitely has become a classic in the area. This highly welcome updated, revised, and enlarged new edition of Zhu's book has every chance of maintaining the book's position as one of the standard references in the area.

-- Mathematical Reviews

Overall, providing current major developments in operator theoretic function theory in a clear and unified way, this new edition appears to be a significant contribution to the field and will be of value to both active researchers and advanced graduate students.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Operator Theory in Function Spaces: Second Edition

- Contents vii8 free
- Preface to the Second Edition xi12 free
- Preface to the First Edition xv16 free
- Chapter 1. Bounded Linear Operators 118 free
- Chapter 2. Interpolation of Banach Spaces 3350
- Chapter 3. Integral Operators on L[sup(p)] Spaces 4764
- Chapter 4. Bergman Spaces 6582
- Chapter 5. Bloch and Besov Spaces 101118
- Chapter 6. The B erezin Transform 133150
- Chapter 7. Toeplitz Operators on the Bergman Space 163180
- Chapter 8. Hankel Operators on the Bergman Space 207224
- 8.1. BMO in the Bergman Metric 207224
- 8.2. VMO in the Bergman Metric 219236
- 8.3. Bounded Hankel Operators 221238
- 8.4. Compact Hankel Opeators 225242
- 8.5. Schatten Class Hankel Operators 227244
- 8.6. Hankel Operators on the Unweighted Bergman Space 233250
- 8.7. Little Hankel Operators 244261
- 8.8. Notes 248265
- 8.9. Exercises 250267

- Chapter 9. Hardy Spaces and BMO 253270
- Chapter 10. Hankel Operators on the Hardy Space 285302
- Chapter 11. Composition Operators 303320
- Bibliography 325342
- Index 345362