Some equivalent conditions of stable perturbation of operators in Hilbert spaces
type: Item
instance of: scholarly article
title: Some equivalent conditions of stable perturbation of operators in Hilbert spaces
zbMATH Open document ID: 1060.47018
DOI: 10.1016/S0096-3003(02)00810-X
author: Yifeng Xue
author: Guoliang Chen
published in: Applied Mathematics and Computation
publication date: 2003-11-25T00:00:00Z
review text: A number of results are presented concerning the perturbation of a bounded operator between Hilbert spaces \(H_1\) and \(H_2\) which has a Moore-Penrose (MP) generalized inverse. Let \(T_0\in B(H_1,H_2)\) with MP inverse \(T^+_0\) and let \(T= T_0+\delta T\in B(H_1, H_2)\). If \(R(T)\cap R(T_0)^\perp= 0\) then \(T\) is said to be a stable perturbation of \(T_0\). Five equivalent conditions are given that \(T\) be a stable perturbation of \(T_0\) given that \(\| T^+_0\|\,\|\delta T\|< 1\). If \(R(T)\cap R(T_0)^\perp= 0\), \(\| T^+_0\|\,\|\delta T\|< 1\) then \(T^+\) satisfies \[ \| T^+- T^+_0\|\leq {1+\sqrt{5}\over 2} \| T^+\|\,\| T^+_0\|\,\|\delta T\|. \] The authors find the optimal solution to the perturbed orthogonal projection problem: Find \(y\in H_1\) such that \(\| p+\delta p- y\|\) is minimized, subject to the condition \[ \| Ty- b-\delta b\|= \min_{z\in H}\,\| Tz- b-\delta b\|, \] \(p,\delta p\in H _1\), \(b,\delta b\in H_2\).
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 2009138
zbMATH Keywords: stable perturbation of operators
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
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full work available at URL: s0096-3003(02)00810-x
OpenAlex ID: W1999243221
description: scientific article; zbMATH DE number 2009138