A generalization of a classic theorem in the perturbation theory for linear operators
type: Item
instance of: scholarly article
title: A generalization of a classic theorem in the perturbation theory for linear operators
zbMATH Open document ID: 0944.47009
author: Yanyan Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-01-17T00:00:00Z
review text: Let \(T\) be a bounded linear operator of a Banach space \(X\) into another \(Y\) such that the null space \(N(T)\) is a complementary subspace of \(X\) and the range \(R(T)\) a complementary subspace of \(Y\). This note considers the equation \(Tx= y\) for given \(y\) and its perturbed one \((T+\delta T)\widetilde x= y+\delta y\), to estimate the norm of the difference between the solutions \(x\) and \(\widetilde x\). The argument used is essentially based on the Neumann lemma to the effect that for a bounded linear operator \(S\) on a Banach space, \(I+S\) has a bounded inverse if \(\|S\|< 1\).
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 1389239
zbMATH Keywords: null space
zbMATH Keywords: complementary subspace
zbMATH Keywords: range
zbMATH Keywords: Neumann lemma
reviewed by: Takashi Ichinose
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6558
OpenAlex ID: W2005722356
cites work: Q3207128
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cites work: Q3312749
cites work: Inner, outer, and generalized inverses in banach and hilbert spaces
DOI: 10.1006/JMAA.1999.6558
description: scientific article; zbMATH DE number 1389239