Nearly Invariant Subspaces and their Applications

This book offers a comprehensive introduction to nearly invariant subspaces, a subject of active contemporary research within functional analysis. Written for graduate students in mathematical analysis and suitable as a reference for experienced researchers, the book surveys the historical development of nearly invariant subspaces from their origins in the study of kernels of Toeplitz operators and invariant subspaces of shift operators. It presents recent advances, including applications to the invariant subspace problem, to truncated Toeplitz operators, and to strongly continuous semigroups of operators. Although mostly concerned with operators on Hardy spaces, the book includes a discussion of the subject in the context of Bergman and Dirichlet spaces too. The book begins with a chapter recalling basic results in analysis and function theory, and each chapter contains a selection of accessible exercises to supplement the text.