**Graduate Studies in Mathematics**

Volume: 44;
2002;
308 pp;
Softcover

MSC: Primary 47; 30; 46; 32;

**Print ISBN: 978-1-4704-6855-2
Product Code: GSM/44.S**

List Price: $64.00

AMS Member Price: $51.20

MAA Member Price: $57.60

**Electronic ISBN: 978-1-4704-2095-6
Product Code: GSM/44.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

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# Pick Interpolation and Hilbert Function Spaces

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*Jim Agler; John E. McCarthy*

The book first rigorously develops the theory of reproducing kernel Hilbert
spaces. The authors then discuss the Pick problem of finding the function of
smallest \(H^\infty\) norm that has specified values at a finite number
of points in the disk. Their viewpoint is to consider \(H^\infty\) as the
multiplier algebra of the Hardy space and to use Hilbert space techniques to
solve the problem. This approach generalizes to a wide collection of
spaces.

The authors then consider the interpolation problem in the space of bounded
analytic functions on the bidisk and give a complete description of the
solution. They then consider very general interpolation problems. The book
includes developments of all the theory that is needed, including operator
model theory, the Arveson extension theorem, and the hereditary functional
calculus.

#### Readership

Graduate students and research mathematicians interested in operator theory, function spaces, and analysis.

#### Reviews & Endorsements

Written in a clear, straightforward style, at a level to make it accessible to someone—a mid-level graduate student, say—who wishes to study the material in detail for the first time … contains exercises … as well as … open questions. It brings the reader up to the current ‘state of the art’ and so will be a valuable resource for the specialist … would be an excellent basis for a graduate seminar or topics course.

-- Mathematical Reviews

Material is wonderfully presented, and the book serves as a lovely introduction to the subject. It is written by two authorities in the field, and helps grad students get entry into an exciting, modern, and very active research area.

-- Palle Jorgensen

#### Table of Contents

# Table of Contents

## Pick Interpolation and Hilbert Function Spaces

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents ix10 free
- Preface xv16 free
- Chapter 0. Prerequisites and Notation 122 free
- Chapter 1. Introduction 728
- Chapter 2. Kernels and Function Spaces 1536
- Chapter 3. Hardy Spaces 3556
- Chapter 4. P[sup(2)](μ) 4970
- Chapter 5. Pick Redux 5576
- Chapter 6. Qualitative Properties of the Solution of the Pick Problem in H[sup(∞)](D) 7192
- Chapter 7. Characterizing Kernels with the Complete Pick Property 79100
- Chapter 8. The Universal Pick Kernel 97118
- §8.1. The universal kernel 97118
- §8.2. The realization formula for the universal kernel 101122
- §8.3. Qualitative properties of solutions of the Pick problem for complete Pick kernels 105126
- §8.4. The Toeplitz-corona theorem 111132
- §8.5. Beurling theorems 114135
- §8.6. Holomorphic complete Pick spaces 117138
- §8.7. The Nevanlinna problem 118139
- §8.8. Uniqueness of kernels with the Pick property 123144
- §8.9. Historical notes 124145

- Chapter 9. Interpolating Sequences 125146
- §9.1. Interpolating sequences for H[sup(∞)](D) 125146
- §9.2. Grammians, Carleson measures and Riesz systems 126147
- §9.3. Interpolating sequences and the Pick property 133154
- §9.4. Zero sets 135156
- §9.5. Grammians bounded above and below 140161
- §9.6. Carleson's interpolation theorem 145166
- §9.7. Historical notes 148169

- Chapter 10. Model Theory I: Isometries 151172
- §10.1. Dilations and extensions 151172
- §10.2. The S[sub(z)].- Nagy dilation theorem 153174
- §10.3. The structure of isometries 156177
- §10.4. Von Neumann's inequality 158179
- §10.6. The commutant lifting theorem 162183
- §10.5. Andô's theorem 160181
- §10.7. Three or more contractions 163184
- §10.8. Historical notes 165186

- Chapter 11. The Bidisk 167188
- §11.1. The realization formula – scalar case 168189
- §11.2. The realization formula – matrix case 173194
- §11.3. The Pick theorem for the bidisk 180201
- §11.4. Toeplitz-corona for the bidisk 182203
- §11.5. The Nevanlinna problem for the bidisk 185206
- §11.6. A two point example 187208
- §11.7. Interpolating sequences 190211
- §11.8. The polydisk 192213
- §11.9. Open problems 192213
- §11.10. Exercises 193214
- §11.11. Historical notes 193214

- Chapter 12. The Extremal Three Point Problem on D[sup(2)] 195216
- §12.1. The two point problem 195216
- §12.2. The non-degenerate extremal three point problem: the strictly 2-dimensional case 197218
- §12.3. Finding Γ and Δ in the strictly 2-dimensional case 204225
- §12.4. The non-degenerate extremal three point problem: the not strictly 2-dimensional case 206227
- §12.5. Problems 209230
- §12.6. Historical notes 209230

- Chapter 13. Collections of Kernels 211232
- Chapter 14. Model Theory II: Function Spaces 237258
- Chapter 15. Localization 263284
- Appendix A. Schur Products 273294
- Appendix B. Parrott's Lemma 277298
- Appendix C. Riesz Interpolation 281302
- Appendix D. The Spectral Theorem for Normal m-Tuples 287308
- Bibliography 293314
- Index 303324
- BackCover BackCover1330