Gil' Michael I. Operator Functions and Operator Equations

World Scientific Publishing, 2018. — 250 p. — ISBN: 978-981-3221-26-0.This book is devoted to norm estimates for operator-valued functions of one and two operator arguments, as well as to their applications to spectrum perturbations of operators and to linear operator equations, i.e. to equations whose solutions are linear operators. Linear operator equations arise in both mathematical theory and engineering practice. The norm estimates suggested in the book have applications to the theories of ordinary differential, difference, functional-differential and integro-differential equations, as well as to the theories of integral operators and analytic functions. This book provides new tools for specialists in matrix theory and functional analysis. A significant part of the book covers the theory of triangular representations of operators that was developed by L de Branges, M S Brodskii, I C Gohberg, M G Krein, M S Livsic and other mathematicians.Preliminaries
Representations of Solutions to Operator Equations
Functions of Finite Matrices
Solution Estimates for Polynomial Matrix Equations
Two-sided Matrix Sylvester Equations
Bounds for Condition Numbers of Diagonalizable Matrices
Functions of a Compact Operator in a Hilbert Space
Triangular Representations of Non-selfadjoint Operators
Resolvents of Bounded Non-selfadjoint Operators
Regular Functions of a Bounded Non-selfadjoint Operator
Functions of an Unbounded Operator
Similarity Condition Numbers of Unbounded Diagonalizable Operators
Commutators and Perturbations of Operator Functions
Functions of Two Non-commuting Operators in Hilbert Spaces