Characteristic Functions and Models of Nonself-Adjoint Operators: 349 (Mathematics and Its Applications) - Hardcover

Over the last decades, the study of nonself-adjoint or nonunitary operators has been mainly based on the method of characteristic functions and on methods of model construction or dilatation for corresponding operator classes. The characteristic function is a mathematical object (a matrix or an operator) associated with a class of nonself-adjoint (or nonunitary) operators that describes the spectral properties of the operators from this class. It may happen that characteristic functions are simpler than the corresponding operators; in this case one can significantly simplify the problem under investigation for these operators. For given characteristic function of an operator A, we construct, in explicit form, an operator that serves as a model A of the operator A in a certain linear space (to some extent this resembles the construction of diagonal and triangular matrices' unitary equivalent or similar, to certain matrix classes). The study of this model operator may give much information about the original operator (its spectrum, the completeness of the system of root subspaces, etc.). In this book, we consider various classes of linear (generally speaking, unbounded) operators, construct and study their characteristic functions and models. We also present a detailed study of contractiol)s and dissipative operators (in particular, from the viewpoint of their triangulation).

This work aims to give a systematic presentation of methods used in the spectral theory of non-selfadjoint, generally unbounded, operators. Subjects treated include the wide class of both selfadjoint and non-selfadjoint extensions of Hermitian operators; characteristic functions of a regular extension; the construction of some operator models for different classes of non-selfadjoint operators; the construction of the selfadjoint dilation of an arbitrary dissipative operator and J-unitary and J-selfadjoint dilations of linear operators; the abstract Lax Phillips scheme in scattering theory for spaces with indefinite metric (Pontryagin spaces); the relation between scattering matrix and the characteristic function of J-nonexpansive operators; a structure of J-nonexpansive operators; and contractions and their triangulation. Audience: This book will be of interest to post-graduate students and researchers in the field of functional analysis.