The backward shift on the Hardy space / Joseph A. Cima, William T. Ross.

Format:

Book

Author/Creator:

Cima, Joseph A., 1933-

Contributor:

Ross, William T., 1964-

Series:

Mathematical surveys and monographs ; no. 79.

Mathematical surveys and monographs, 0076-5376 ; v. 79

Language:

English

Subjects (All):

Hardy spaces.

Physical Description:

xi, 199 pages : illustrations ; 27 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2000]

Contents:

Chapter 2. Classical boundary value results 9

2.1. Limits 9

2.2. Pseudocontinuations 13

Chapter 3. The Hardy space of the disk 17

3.2. H[superscript p] and boundary values 17

3.3. Fourier analysis and H[superscript p] theory 21

3.4. The Cauchy transform 23

3.5. Duality 28

3.6. The Nevanlinna class 39

Chapter 4. The Hardy spaces of the upper-half plane 45

4.1. Motivation 45

4.2. Basic definitions 47

4.3. Poisson and conjugate Poisson integrals 49

4.4. Maximal functions 52

4.5. The Hilbert transform 54

4.6. Some examples 55

4.7. The harmonic Hardy space 60

4.8. Distributions 61

4.9. The atomic decomposition 72

4.10. Distributions and H[superscript p] 75

4.11. The space H[superscript p] (C\R) 76

Chapter 5. The backward shift on H[superscript p] for p [set membership] (1, [infinity]) 81

5.1. The case p > 1 81

5.2. The first and most straightforward proof 82

5.3. The second proof - using Fatou's jump theorem 85

5.4. Application: Bergman spaces 87

5.5. Application: spectral properties 94

5.6. The third proof - using the Nevanlinna theory 97

5.7. Application: VMOA, BMOA, and L[superscript 1] [square root]H[superscript 1 subscript 0] 99

5.8. The case p = 1 101

5.9. Cyclic vectors 105

5.10. Duality 109

5.11. The commutant 109

5.12. Compactness of the inclusion operator 111

Chapter 6. The backward shift on H[superscript p] for p [set membership] (0, 1) 115

6.2. The parameters 120

6.3. A reduction 133

6.4. Rational approximation 136

6.5. Spectral properties 185

6.6. Cyclic vectors 186

6.7. Duality 187

6.8. The commutant 188.

Notes:

Includes bibliographical references (pages 191-194) and index.

ISBN:

0821820834

OCLC:

43607449