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.
Contents: - 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
Readership: Undergraduate and graduate students, and researchers in matrix theory, functional analysis and their applications to differential and difference equations.
Keywords:Operator Functions;Spectrum Perturbations of Operators;Operator Equations;Generalized Sylvester EquationsReview:
Key Features:- Presents norm estimates for functions of two operator arguments and their applications
- Gives a systematic exposition of solution estimates of linear operator equations
- Suggests bounds for the similarity condition numbers of diagonalizable operators
- The published results concerning the functions of one operator argument in this book are considerably simplified and supplemented
