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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. 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 infinity was done in a 1970
paper of R. Douglas, H. S. Shapiro, and A. Shields, and the case
0p\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 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.
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