**Mathematical Surveys and Monographs**

Volume: 100;
2004;
318 pp;
Hardcover

MSC: Primary 30; 46;

Print ISBN: 978-0-8218-0810-8

Product Code: SURV/100

List Price: $92.00

Individual Member Price: $73.60

**Electronic ISBN: 978-1-4704-1327-9
Product Code: SURV/100.E**

List Price: $92.00

Individual Member Price: $73.60

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#### Supplemental Materials

# Bergman Spaces

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*Peter Duren; Alexander Schuster*

Over the last ten years, the theory of Bergman spaces has undergone a remarkable metamorphosis. In a series of major advances, central problems once considered intractable were solved, and a rich theory emerged. Although progress continues, the time seems ripe for a full and unified account of the subject, weaving the old and new results together. This thorough exposition provides just that.

The subject of Bergman spaces is a masterful blend of complex function theory with functional analysis and operator theory. It has much in common with Hardy spaces, but involves new elements such as hyperbolic geometry, reproducing kernels, and biharmonic Green functions.

In this book, the authors develop background material and provide a self-contained introduction to a broad range of topics, including recent advances on interpolation and sampling, contractive zero-divisors, and invariant subspaces. The book is accessible to researchers and advanced graduate students who have studied basic complex function theory, measure theory, and functional analysis.

#### Table of Contents

# Table of Contents

## Bergman Spaces

- Contents v6 free
- Preface ix10 free
- Chapter 0. Overview 112 free
- Chapter 1. The Bergman Kernel Function 718
- Chapter 2. Linear Space Properties 2536
- 2.1. Hardy spaces 2536
- 2.2. Strict and uniform convexity 2839
- 2.3. The Bergman projection 3041
- 2.4. Dual spaces 3546
- 2.5. The pseudohyperbolic metric 3849
- 2.6. The Bloch space 4354
- 2.7. Harmonic conjugates 5465
- 2.8. Linear isometries 5667
- 2.9. Function multipliers 5970
- 2.10. Carleson measures 6172
- 2.11. Uniformly discrete sequences 6778

- Chapter 3. Analytic Properties 7384
- Chapter 4. Zero-Sets 93104
- Chapter 5. Contractive Zero-Divisors 119130
- Chapter 6. Sampling and Interpolation 153164
- 6.1. Definitions and motivations 153164
- 6.2. Interpolation in Hardy spaces 157168
- 6.3. A family of sampling and interpolation sequences 159170
- 6.4. Some explicit examples 168179
- 6.5. The density theorems 171182
- 6.6. Direct calculation of densities 176187
- 6.7. Sharpened forms of Horowitz' theorems 182193
- 6.8. Sufficient conditions with the pseudohyperbolic metric 187198
- 6.9. Duality relations 192203

- Chapter 7. Proofs of Sampling and Interpolation Theorems 197208
- 7.1. Perturbation of sampling sequences 197208
- 7.2. Necessity of the sampling condition 204215
- 7.3. Sampling in the growth space 207218
- 7.4. Sufficiency of the sampling condition in A[sup(p)] 216227
- 7.5. Sufficiency of the interpolation condition 219230
- 7.6. Necessity of the interpolation condition 234245
- 7.7. Weak interpolation 240251

- Chapter 8. Invariant Subspaces 245256
- Chapter 9. Structure of Invariant Subspaces 271282
- References 297308
- Index 313324 free

#### Readership

Advanced graduate students and research mathematicians interested in complex function theory and operator theory.