Perturbation analysis for the reduced minimum modulus of bounded linear operator in Banach spaces.
type: Item
instance of: scholarly article
title: Perturbation analysis for the reduced minimum modulus of bounded linear operator in Banach spaces.
zbMATH Open document ID: 1055.47001
DOI: 10.1016/S0096-3003(02)00434-4
author: Chao Zhu
author: Jing Cai
author: Guoliang Chen
published in: Applied Mathematics and Computation
publication date: 2003-11-25T00:00:00Z
review text: Let \(X\), \(Y\) be Banach spaces (both over \(\mathbb{C}\)) and let \(T: X\to Y\) be a bounded linear operator with the generalized inverse \(T^+\). Consider a bounded perturbation \(\overline T= T+\delta T\) with the property \(\| T^+\|\,\|\delta T\|< {1\over 2}\). It is shown in the paper under review that the reduced minimum modulus \(\gamma\) of \(T\) defined as \[ \gamma(T)= \inf\{\| T\|: \text{dist}(x,\text{ker\,}T)= 1,\,x\in X\} \] under appropriate conditions is upper semicontinuous (continuous). This implies a kind of stability of these perturbations.
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 2008948
zbMATH Keywords: generalized inverse
zbMATH Keywords: reduced minimum modulus
zbMATH Keywords: stable perturbation of operators
reviewed by: Danuta Przeworska-Rolewicz
MaRDI profile type: Publication
cites work: Q4061081
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description: scientific article; zbMATH DE number 2008948