On the perturbation of the reduced minimum modulus of bounded linear operators.
type: Item
instance of: scholarly article
title: On the perturbation of the reduced minimum modulus of bounded linear operators.
zbMATH Open document ID: 1047.47010
DOI: 10.1016/S0096-3003(02)00195-9
published in: Applied Mathematics and Computation
publication date: 2003-09-09T00:00:00Z
review text: In the Banach space \(L(X,Y)\) of all bounded linear operators between two Banach spaces \(X\) and \(Y\), the notion of the reduced minimum modulus \(\gamma(T), T\in L(X,Y),\) is introduced by the formula \(\gamma(T)=\inf_{x\in X}\frac{Tx}{d(x,N(T))}\). Here \(N(T)\) is the zero subspace of \(T\), \(d(x,N(T))\) is the distance between the element \(x\) and the set \(N(T)\) and 0/0 is defined to be \(\infty\). The author obtains some perturbation results for the reduced minimum modulus \(\gamma(T)\) for Hilbert and Banach spaces. These results imply perturbation bounds for generalized inverses and least squares problems in Hilbert spaces. Reviewer's remarks: The obtained results correct some mistakes in the works of \textit{G.-L. Chen} and \textit{Y.-M. Wei} [Chin. J. Contemp. Math. 19, 245--252 (1998; Zbl 1015.47001); Appl. Math. Comput. 121, 177--183 (2001; Zbl 1038.47006)].
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 1978064
zbMATH Keywords: Banach space
zbMATH Keywords: operators
zbMATH Keywords: reduced minimum modulus
zbMATH Keywords: perturbation
author: J. Ding
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
cites work: Q3207128
cites work: Perturbation analysis of the least squares solution in Hilbert spaces
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cites work: Perturbation of generalized inverses of linear operators in Hilbert spaces
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cites work: Q5526484
cites work: Q3312749
cites work: Perturbation of least squares problem in Hilbert spaces.
full work available at URL: s0096-3003(02)00195-9
OpenAlex ID: W2000035160
description: scientific article; zbMATH DE number 1978064