The Backward Shift on the Hardy Space

215 Pages·2000·16.128 MB·English

by  Joseph A. Cima

http://dx.doi.org/10.1090/surv/079
Selected Titles in This Series
79 Joseph A. Cima and William T. Ross, The backward shift on the Hardy space, 2000
78 Boris A. Kupershmidt, KP or mKP: Noncommutative mathematics of Lagrangian,
Hamiltonian, and integrable systems, 2000
77 Fumio Hiai and Denes Petz, The semicircle law, free random variables and entropy,
2000
76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmiiller theory, 2000
75 Greg Hjorth, Classification and orbit equivalence relations, 2000
74 Daniel W. Stroock, An introduction to the analysis of paths on a Riemannian manifold,
2000
73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000
72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999
71 Lajos Pukanszky, Characters of connected Lie groups, 1999
70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems
and differential equations, 1999
69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, second edition,
1999
68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999
67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and
representations of reductive groups, second edition, 2000
66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999
65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra,
1999
64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential
equations: Six perspectives, 1999
63 Mark Hovey, Model categories, 1999
62 Vladimir I. Bogachev, Gaussian measures, 1998
61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic
algebra for ordinary differential and quasi-differential operators, 1999
60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace
C*-algebras, 1998
59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998
58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on
representation theory and Knizhnik-Zamolodchikov equations, 1998
57 Marc Levine, Mixed motives, 1998
56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum
groups: Part I, 1998
55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998
54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in
analysis, 1997
53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997
52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in
. domains with point singularities, 1997
51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial
differential equations, 1997
50 Jon Aaronson, An introduction to infinite ergodic theory, 1997
49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential
equations, 1997
48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997
(Continued in the back of this publication)
This page intentionally left blank
The Backward Shift
on the Hardy Space
This page intentionally left blank
Mathematical
Surveys
and
Monographs
Volume 79
The Backward Shift
on the Hardy Space
Joseph A. Cima
William T. Ross
American Mathematical Society
Editorial Board
Georgia Benkart Michael Loss
Peter Landweber Tudor Ratiu, Chair
2000 Mathematics Subject Classification. Primary 47B38;
Secondary 46E10, 46E15.
ABSTRACT. This book is a thorough treatment of the classification of the backward shift invariant
subspaces of the well-known Hardy spaces Hp. For 1 < p < oo, the characterization was done by
Douglas, Shapiro, and Shields. The case 0 < p < 1 was done by A. B. Aleksandrov in a paper
which was not translated into English and as a result is not readily available in the West. This
book puts all of these results, along with the necessary background material, under one roof.
Library of Congress Cataloging-in-Publication Data
Cima, Joseph A. , 1933-
The backward shift on the Hardy space / Joseph A. Cima, William T. Ross.
p. cm. (Mathematical surveys and monographs, ISSN 0076-5376; v. 79)
Includes bibliographical references and index.
ISBN 0-8218-2083-4 (alk. paper)
1. Hardy spaces. I. Ross, William T., 1964- . II. Title. III. Series: Mathematical surveys
and monographs; no. 79.
QA331.C53 2000
515'.94-dc21 00-028032
CIP
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Assistant to the Publisher, American Mathematical Society,
P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to
reprint-permissionOams.org.
© 2000 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at URL: http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00
Contents
Preface ix
Numbering and notation xi
Chapter 1. Overview 1
Chapter 2. Classical boundary value results 9
2.1. Limits 9
2.2. Pseudocontinuations 13
Chapter 3. The Hardy space of the disk 17
3.1. Introduction 17
3.2. Hp and boundary values 17
3.3. Fourier analysis and Hp 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 W 75
4.11. The space HP{C\R) 76
Chapter 5. The backward shift on Hp for p G [1, oc) 81
5.1. The case p > l 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 L1/~H^ 99
5.8. The case p = 1 101
5.9. Cyclic vectors 105
i CONTENTS
5.10. Duality 109
5.11. The commutant 109
5.12. Compactness of the inclusion operator 111
Chapter 6. The backward shift on HP for p e (0,1) 115
6.1. Introduction 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
Bibliography 191
Index 195
Preface
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 H2 model certain
types of contraction operators and potentially have connections to understanding
the invariant subspaces of a general linear operator.
In this book, we do not want to give a general treatment of the backward shift
on H2 and its connections to problems in operator theory. This has been done
quite thoroughly by Nikolskii in his book [65]. Instead, we wish to work in the
Banach (and F-space) setting of Hp (0 < p < oo) where we will focus primarily on
characterizing the backward shift invariant subspaces of Hp. When p G (1, oo), this
characterization problem was solved by R. Douglas, H. S. Shapiro, and A. Shields
in a well known paper [29] which employed the concept of a 'pseudo continuation'
developed earlier by Shapiro [84]. When p G (0,1), the characterization problem
is more difficult, due to some topological differences between the two settings p G
[l,oo) and p G (0,1), and was solved in a paper of A. B. Aleksandrov [3] which
was never translated from its original Russian and hence is not readily available in
the West. The Aleksandrov paper is also quite complicated and makes use of the
distribution theory and Coifman's atomic decomposition for the Hardy spaces of
the upper half plane, a topic we feel is not always at the fingertips of those schooled,
as we were, in classical function theory and operator theory. It is for these reasons
that we gather up these results, along with the necessary background material, and
put them all under one roof.
In developing the necessary background results, we do not wish to reproduce
the material in the books of Duren [31] or Garnett [39] (for a general treatment of
Hardy spaces) or Stein [95] (for a detailed treatment of harmonic analysis and real
variable Hp theory). Instead, we will only review this material and refer the inter
ested reader to the appropriate places in these texts for the proofs. The reader is
expected to have a reasonable background in functional analysis and function the
ory (including the basics of Hp theory), but might want to have Rudin's functional
analysis book [78], Duren's Hp book [31], and Stein's harmonic analysis book [95]
at the ready while reading this book. We will try to develop the more specialized
topics as we need them.
The authors wish to thank several people who helped us along the way. First,
we thank A. B. Aleksandrov, who, through many e-mails, helped us understand
the more difficult parts of his papers. Secondly, we thank Alec Matheson and Don
Sarason, who read a draft of this book and provided us with useful suggestions and
corrections. Thirdly, we thank Olga Troyanskaya, who translated the Aleksandrov
paper [3] from the original Russian. Finally, the second author wishes to thank
ix