Characteristic Functions and Models of Nonself-Adjoint Operators

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).

Klappentext

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.

Inhalt

1. Regular Extensions of Hermitian Operators.- 1. Principal Concepts.- 2. Auxiliary Operators and Their Application.- 3. Dissipative Operators.- 2. Characteristic Functions.- 1. Characteristic Matrix Functions of Regular Extensions.- 2. A Criterion of Unitary Equivalence.- 3. Determination of Characteristic Matrix Functions.- 4. Examples.- 5. Properties of Characteristic Matrix-Functions.- 6. Theorem on Multiplication of Characteristic Matrix Functions.- 7. Factorization of Characteristic Matrix Functions.- 8. Characteristic Operator Functions.- 9. Characteristic Functions, Brief Survey.- 3. Models of Nonself-Adjoint Operators.- 1. Preliminary Remarks.- 2. Kr-Operators with Real Spectrum.- 3. Kr -Operators with Spectrum Concentrated at Infinity.- 4. Kr -Operators with Discrete Spectrum.- 5. General Case.- 6. Models of Linear Operators (A Brief Survey).- 4. Dilations of Linear Operators.- 1. A Brief Survey.- 2. Self-Adjoint Dilation of a Dissipative Operator (First Method).- 3. Symmetric and Self-Adjoint Dilations of Dissipative Operators (Second Method).- 4. J-Unitary and J-Selfadjoint Dilations.- 5. Coordinate-Free Form of Dilations.- Appendix 1.Contractions, Triangulation of Contractions.- 1. General Properties of Contractions.- 2. Canonical Representation. Completely Nonunitary Contractions.- 3. Triangulation of Contractions.- 4. Triangulation of Completely Nonunitary Contractions.- Appendix 2. The Structure of J-Nonexpanding Operators.- 4. Remark.- Appendix 3. Lax-Phillips Abstract Scattering Scheme in Pontryagin Spaces.- 1. Pontryagin Spaces. Terminology and Auxiliary Statements.- 2. Outgoing and Incoming Subspaces.- 3. Scattering Operator. Definition and Simple Properties.- 5. Properties of Scattering Matrix.- 7. Individual Case.- 8. Continuous Case.- 9. Remark.- References.- Author Index.- Notation.