**Mathematical Surveys and Monographs**

Volume: 79;
2000;
199 pp;
Hardcover

MSC: Primary 47;
Secondary 46

**Print ISBN: 978-0-8218-2083-4
Product Code: SURV/79**

List Price: $64.00

AMS Member Price: $51.20

MAA Member Price: $57.60

**Electronic ISBN: 978-1-4704-1306-4
Product Code: SURV/79.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

#### Supplemental Materials

# The Backward Shift on the Hardy Space

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*Joseph A. Cima; William T. Ross*

Shift operators on Hilbert spaces of analytic functions play an important
role in the study of bounded linear operators on Hilbert spaces since they
often serve as models for various classes of linear operators. For
example, “parts” of direct sums of the backward shift operator on the
classical Hardy space \(H^2\) model certain types of
contraction operators and potentially have connections to understanding the
invariant subspaces of a general linear operator.

This book is a thorough treatment of the characterization of the
backward shift invariant subspaces of the well-known Hardy spaces
\(H^{p}\). The characterization of the backward shift invariant
subspaces of \(H^{p}\) for \(1 < p < \infty\) was done in
a 1970 paper of R. Douglas, H. S. Shapiro, and A. Shields, and the case
\(0 < p \le 1\) was done in a 1979 paper of A. B. Aleksandrov
which is not well known in the West. This material is pulled together
in this single volume and includes all the necessary background
material needed to understand (especially for the \(0 < p < 1\)
case) the proofs of these results.

Several proofs of the Douglas-Shapiro-Shields result are provided so
readers can get acquainted with different operator theory and theory
techniques: applications of these proofs are also provided for
understanding the backward shift operator on various other spaces of
analytic functions. The results are thoroughly examined. Other features
of the volume include a description of applications to the spectral
properties of the backward shift operator and a treatment of some general
real-variable techniques that are not taught in standard graduate seminars.
The book includes references to works by Duren, Garnett, and Stein for
proofs and a bibliography for further exploration in the areas of operator
theory and functional analysis.

#### Readership

Advanced graduate students with a background in basic functional analysis, complex analysis and the basics of the theory of Hardy spaces; professional mathematicians interested in operator theory and functional analysis.

#### Reviews & Endorsements

The book has been carefully written and contains a wealth of information … It will probably appeal most to those with an interest in the interplay between operator theory and modern function theory.

-- Bulletin of the LMS

#### Table of Contents

# Table of Contents

## The Backward Shift on the Hardy Space

- Contents vii8 free
- Preface ix10 free
- Numbering and notation xi12 free
- Chapter 1. Overview 114 free
- Chapter 2. Classical boundary value results 922
- Chapter 3. The Hardy space of the disk 1730
- Chapter 4. The Hardy spaces of the upper-half plane 4558
- 4.1. Motivation 4558
- 4.2. Basic definitions 4760
- 4.3. Poisson and conjugate Poisson integrals 4962
- 4.4. Maximal functions 5265
- 4.5. The Hilbert transform 5467
- 4.6. Some examples 5568
- 4.7. The harmonic Hardy space 6073
- 4.8. Distributions 6174
- 4.9. The atomic decomposition 7285
- 4.10. Distributions and H[sup(p)] 7588
- 4.11. The space H[sup(p)](C\R) 7689

- Chapter 5. The backward shift on H[sup(p)] for p ∈ [1, ∞) 8194
- 5.1. The case p > 1 8194
- 5.2. The first and most straightforward proof 8295
- 5.3. The second proof - using Fatou's jump theorem 8598
- 5.4. Application: Bergman spaces 87100
- 5.5. Application: spectral properties 94107
- 5.6. The third proof - using the Nevanlinna theory 97110
- 5.7. Application: VMOA, BMOA, and L[sup(1)]/H[sup(1)][sub(0)] 99112
- 5.8. The case p = 1 101114
- 5.9. Cyclic vectors 105118
- 5.10. Duality 109122
- 5.11. The commutant 109122
- 5.12. Compactness of the inclusion operator 111124

- Chapter 6. The backward shift on H[sup(p)] for p ∈ (0,1) 115128
- Bibliography 191204
- Index 195208