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.