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
(0 p oo) where we will focus primarily on
characterizing the backward shift invariant subspaces of
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
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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