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

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