Perturbation analysis for the operator equation \(Tx=b\) in Banach spaces
type: Item
instance of: scholarly article
title: Perturbation analysis for the operator equation \(Tx=b\) in Banach spaces
zbMATH Open document ID: 0903.47004
author: Yifeng Xue
author: Guoliang Chen
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-01-07T00:00:00Z
review text: The authors provide a perturbation analysis for the generalized inverse \(T^+\) of a bounded linear operator \(T\), with closed range, acting between complex Banach spaces. For perturbations \(\overline{T}\) with \(\delta T= \overline{T}-T\) satisfying \(\| T^+\| \|\delta T\|<1\) and \(R(\overline{T})\cap N(T^+)= \{0\}\) (or \(N(\overline{T})\) and \(N(T)\) finite-dimensional), it is shown that \(\overline{T}^+= (I+ T^+\delta T)^{-1} T^+\) and \(\| \overline{T}^+\|\leq \| T^+\| (1-\| T^+\| \|\delta T\|)\). This generalizes a result of \textit{M. Z. Nashed} [``Generalized inverses and applications'', New York (1976; Zbl 0324.00009)]. The results are applied to a perturbation analysis of the equation \(Tx=b\).
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 1053554
zbMATH Keywords: operator equation
zbMATH Keywords: perturbation analysis
zbMATH Keywords: generalized inverse
reviewed by: Charles W. Groetsch
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5482
OpenAlex ID: W2054418615
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DOI: 10.1006/JMAA.1997.5482
description: scientific article; zbMATH DE number 1053554