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.