Pick interpolation and Hilbert function spaces / Jim Agler, John E. McCarthy.

Format:

Book

Author/Creator:

Agler, Jim.

Contributor:

McCarthy, John E. (John Edward), 1964-

Alumni and Friends Memorial Book Fund.

Series:

Graduate studies in mathematics 1065-7339 ; v. 44.

Graduate studies in mathematics, 1065-7339 ; v. 44

Language:

English

Subjects (All):

Hilbert space.

Interpolation.

Functions of complex variables.

Physical Description:

xix, 308 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2002]

Contents:

Chapter 0. Prerequisites and Notation 1

0.2. Blaschke products 2

0.3. Pseudo-hyperbolic metric 3

0.4. Hilbert spaces 3

0.5. The weak-star topology on B(H) 4

0.6. Rank one operators 4

0.7. Analytic functions of several variables 5

1.1. The Pick problem 7

1.2. H[superscript [infinity] as an operator algebra 9

1.3. The operator theory approach to the Pick problem 10

1.4. The Pick property 10

1.5. Model theory 11

1.6. Collections of kernels 12

Chapter 2. Kernels and Function Spaces 15

2.1. Hilbert function spaces 15

2.2. Kernels 17

2.3. Multipliers 21

2.4. Tensor products 22

2.5. Vector-valued Hilbert function spaces 23

2.6. Rescaling kernels 25

2.7. Factoring kernels 28

2.8. Operator-valued kernels 30

2.9. Historical notes 33

Chapter 3. Hardy Spaces 35

3.1. H[superscript p] spaces 35

3.2. Fatou's theorem 36

3.3. H[superscript p] spaces again 41

3.4. H[superscript [infinity] (D) as a multiplier algebra 43

3.5. Inner functions 45

3.6. Historical notes 48

Chapter 4. P[superscript 2] ([mu]) 49

4.1. Other spaces with H[superscript [infinity] (D) as the multiplier algebra 49

4.2. Vector-valued P[superscript 2] ([mu]) spaces 52

Chapter 5. Pick Redux 55

5.1. Necessity of positivity of the Pick matrix 55

5.2. The Szego kernel has the Pick property 59

5.3. The Caratheodory problem 65

5.4. Uniqueness of the Szego kernel 67

5.5. Historical notes 69

Chapter 6. Qualitative Properties of the Solution of the Pick Problem in H[superscript [infinity] (D) 71

6.1. A formula for the solution 71

6.2. The realization formula for H[superscript [infinity] (D) 73

6.3. Another formula for the solution 76

6.4. The Nevanlinna problem 78

Chapter 7. Characterizing Kernels with the Complete Pick Property 79

7.1. Characterization of the complete Pick property 79

7.2. Another characterization of the complete Pick property 87

7.3. Holomorphic spaces with the complete Pick property 88

7.4. The Sobolev space 91

7.5. The M[subscript s x t] Pick property 94

7.6. Historical notes 95

Chapter 8. The Universal Pick Kernel 97

8.1. The universal kernel 97

8.2. The realization formula for the universal kernel 101

8.3. Qualitative properties of solutions of the Pick problem for complete Pick kernels 105

8.4. The Toeplitz-corona theorem 111

8.5. Beurling theorems 114

8.6. Holomorphic complete Pick spaces 117

8.7. The Nevanlinna problem 118

8.8. Uniqueness of kernels with the Pick property 123

8.9. Historical notes 124

Chapter 9. Interpolating Sequences 125

9.1. Interpolating sequences for H[superscript [infinity] (D) 125

9.2. Grammians, Carleson measures and Riesz systems 126

9.3. Interpolating sequences and the Pick property 133

9.4. Zero sets 135

9.5. Grammians bounded above and below 140

9.6. Carleson's interpolation theorem 145

9.7. Historical notes 148

Chapter 10. Model Theory I: Isometries 151

10.1 Dilations and extensions 151

10.2. The Sz.-Nagy dilation theorem 153

10.3. The structure of isometries 156

10.4. Von Neumann's inequality 158

10.5. Ando's theorem 160

10.6. The commutant lifting theorem 162

10.7. Three or more contractions 163

10.8. Historical notes 165

Chapter 11. The Bidisk 167

11.1. The realization formula - scalar case 168

11.2. The realization formula - matrix case 173

11.3. The Pick theorem for the bidisk 180

11.4. Toeplitz-corona for the bidisk 182

11.5. The Nevanlinna problem for the bidisk 185

11.6. A two point example 187

11.7. Interpolating sequences 190

11.8. The polydisk 192

11.9. Open problems 192

11.11. Historical notes 193

Chapter 12. The Extremal Three Point Problem on D[superscript 2] 195

12.1. The two point problem 195

12.2. The non-degenerate extremal three point problem: the strictly 2-dimensional case 197

12.3. Finding [Gamma] and [Delta] in the strictly 2-dimensional case 204

12.4. The non-degenerate extremal three point problem: the not strictly 2-dimensional case 206

12.5. Problems 209

12.6. Historical notes 209

Chapter 13. Collections of Kernels 211

13.1. An abstract theory 211

13.2. Uniform algebras: the Cole-Lewis-Wermer approach 214

13.3. Finitely connected domains 219

13.4. When does a collection of kernels have the Pick property? 222

13.5. Historical notes 235

Chapter 14. Model Theory II: Function Spaces 237

14.1. Theorems of Stinespring and Arveson 237

14.2. Hereditary functional calculus 243

14.3. Co-analytic extensions 247

14.4. The Taylor spectrum 252

14.5. Co-analytic models for m-tuples 255

14.6. Von Neumann inequalities 260

14.7. Historical notes 260

Chapter 15. Localization 263

15.1. Localization of extensions 263

15.2. Kernels with the M[subscript s x s] Pick property 266

15.3. Localization of dilations 270

15.4. Historical notes 272

Appendix A. Schur Products 273

Appendix B. Parrott's Lemma 277

Appendix C. Riesz Interpolation 281

Appendix D. The Spectral Theorem for Normal m-Tuples 287

D.1. Normal tuples 287

D.2. Commuting isometries 290.

Notes:

Includes bibliographical references (pages 293-301) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0821828983

OCLC:

48474257