Stable perturbations of operators and related topics
type: Item
instance of: scholarly article
zbMATH Open document ID: 1241.47001
author: Yifeng Xue
publication date: 2012-04-11T00:00:00Z
full work available at URL: 9789814383608.html
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 6022909
zbMATH Keywords: linear operators
zbMATH Keywords: generalised inverses
zbMATH Keywords: perturbation theory
MaRDI profile type: Publication
title: Stable perturbations of operators and related topics
review text: The present book consists of six chapters, two appendices, a bibliography, and an index.NEWLINENEWLINEChapter 1 contains background material from functional analysis (including Banach algebras and \(C^*\)-algebras) given partly without proofs, but with references.NEWLINENEWLINEChapter 2 deals with stable perturbation of closed densely defined linear operators. By a stable perturbation of a linear closed operator \(A\) by a linear bounded operator \(B\), the author means that \(\operatorname{Ran}(A+B)\cap \operatorname{Ker}A^+=\{0\}\), where \(A^+\) denotes the generalized inverse of \(A\), \(\operatorname{Ran}(A)\) denotes the range of \(A\), and \(\operatorname{Ker}A\) denotes the null-space of \(A\).NEWLINENEWLINEChapters 3 and 4 deal with the Moore-Penrose (MP), Drazin (D), and \(T-S\) generalized inverses and their stable perturbations.NEWLINENEWLINEChapter 5, Miscellaneous Applications, deals with the generalized Bott-Duffin inverse, some applications of MP inverses to the theory of frames, generalized Cowen-Douglas mappings, condition numbers for MP and D inverses, and with a local linearization of \(C^1\)-smooth nonlinear maps.NEWLINENEWLINEChapter 6, Some Related Topics, deals with the perturbation of the reduced minimum modulus, the density of MP invertible elements in some \(C^*\)-algebras, and other topics. The reduced minimum modulus of a linear operator \(T\) is denoted by \(\gamma(T)\) and is defined by the formula \(\gamma(T)=\inf_{x\in D(T), \operatorname{dist}(x, \operatorname{Ker}T)=1}\| Tx\|\).NEWLINENEWLINEAppendix A contains some results from topology and analysis, and Appendix~B contains some results from the theory of \(C^*\)-algebras.NEWLINENEWLINEThe book has some weak points which negatively affect its suitability as a textbook for students: (a) some definitions are not given (for example, the definition of the space \(X^*\) in Proposition 1.1.4 on p.\,3, is not given; the term ``unital Banach algebra'', p.\,30, is not defined); (b) some th…
reviewed by: A. G. Ramm
description: scientific article; zbMATH DE number 6022909