**Mathematical Surveys and Monographs**

Volume: 125;
2006;
272 pp;

MSC: Primary 30; 32; 47; 46;

**Electronic ISBN: 978-1-4704-1352-1
Product Code: SURV/125.E**

List Price: $81.00

AMS Member Price: $64.80

MAA Member Price: $72.90

#### Supplemental Materials

# The Cauchy Transform

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*Joseph A. Cima; Alec L. Matheson; William T. Ross*

The Cauchy transform of a measure on the circle is a subject of both classical and current interest with a sizable literature. This book is a thorough, well-documented, and readable survey of this literature and includes full proofs of the main results of the subject. This book also covers more recent perturbation theory as covered by Clark, Poltoratski, and Aleksandrov and contains an in-depth treatment of Clark measures.

#### Reviews & Endorsements

Proofs are presented very carefully, so that the whole material becomes accessible to any graduate student interested in analysis. Experienced researchers will also be thankful to the authors for having presented them this material for the first time. We recommend this survey to any person interested in function theory and operator theory on the disk.

-- Zentralblatt MATH

The book is user-friendly it is clearly written, at an agreeable pace. Many of the deepest results of the theory are afforded complete treatments. Scattered throughout are mentions of related results, with references to the literature. The authors are thorough and meticulous in their citation of references.

-- Mathematical Reviews

...contains a wealth of interesting results, presented in an attractive and readable style...

-- Bulletin of the LMS

Combining both classical and recent result, the book presents a great interest for students, teachers and researchers interested mainly in functional analysis methods in complex analysis. The topics are presented in an elegant manner, with many comments, detours and historical references. The result is a fine book that deserves to be on the bookshelf of each analyst.

-- Studia Universitatis Babes-Bolyai, Mathematica

#### Table of Contents

# Table of Contents

## The Cauchy Transform

- Contents v6 free
- Preface ix10 free
- Overview 112 free
- Chapter 1. Preliminaries 1122
- 1.1. Basic notation 1122
- 1.2. Lebesgue spaces 1122
- 1.3. Borel measures 1425
- 1.4. Some elementary functional analysis 1728
- 1.5. Some operator theory 2031
- 1.6. Functional analysis on the space of measures 2233
- 1.7. Non-tangential limits and angular derivatives 2536
- 1.8. Poisson and conjugate Poisson integrals 3041
- 1.9. The classical Hardy spaces 3243
- 1.10. Weak-type spaces 3546
- 1.11. Interpolation and Carleson's theorem 3647
- 1.12. Some integral estimates 3950

- Chapter 2. The Cauchy transform as a function 4152
- Chapter 3. The Cauchy transform as an operator 6172
- Chapter 4. Topologies on the space of Cauchy transforms 8394
- Chapter 5. Which functions are Cauchy integrals? 99110
- Chapter 6. Multipliers and divisors 115126
- Chapter 7. The distribution function for Cauchy transforms 163174
- Chapter 8. The backward shift on H[sup(2)] 179190
- 8.1. Beurling's theorem 179190
- 8.2. A theorem of Douglas, Shapiro, and Shields 180191
- 8.3. Spectral properties 184195
- 8.4. Kernel functions 185196
- 8.5. A density theorem 186197
- 8.6. A theorem of Ahern and Clark 192203
- 8.7. A basis for backward shift invariant subspaces 192203
- 8.8. The compression of the shift 194205
- 8.9. Rank-one unitary perturbations 196207

- Chapter 9. Clark measures 201212
- 9.1. Some basic facts about Clark measures 201212
- 9.2. Angular derivatives and point masses 208219
- 9.3. Aleksandrov's disintegration theorem 211222
- 9.4. Extensions of the disintegration theorem 212223
- 9.5. Clark's theorem on perturbations 218229
- 9.6. Some remarks on pure point spectra 221232
- 9.7. Poltoratski's distribution theorem 222233

- Chapter 10. The normalized Cauchy transform 227238
- Chapter 11. Other operators on the Cauchy transforms 249260
- List of Symbols 255266
- Bibliography 257268
- Index 267278