\(\Gamma\)-inverses of bounded linear operators
type: Item
instance of: scholarly article
title: \(\Gamma\)-inverses of bounded linear operators
zbMATH Open document ID: 1306.47006
author: Xiaochun Fang
author: Xiaoming Xu
author: Hongke Du
publication date: 2014-06-03T00:00:00Z
review text: Let \(\mathcal H\) be a Hilbert space and \(A,P,Q\) be bounded linear operators on \(\mathcal H\). If there exists a bounded linear operator \(X\) on \(\mathcal H\) such that \(APXQA=A\), \(XQAPX=X\), \((QAPX)^*=QAPX\) and \((XQAP)^*=XQAP\), then \(X\) is called the \(\Gamma\)-inverse of \(A\) associated with \(P\) and \(Q\) and is denoted by \(A^{\dagger}_{P,Q}\). If \(P=Q=I\), then \(A^{\dagger}_{P,Q}\) reduces to the Moore-Penrose inverse \(A^{\dagger}\). The main results of the authors are a necessary and sufficient condition for the existence of \(A^{\dagger}_{P,Q}\) and an explicit formula for it, when it exists.
reviewed by: K. C. Sivakumar
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 6299407
zbMATH Keywords: generalized inverse
zbMATH Keywords: \(\Gamma\)-inverse
zbMATH Keywords: Moore-Penrose inverse
MaRDI profile type: Publication
full work available at URL: s10114-013-2552-y
OpenAlex ID: W2398531669
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DOI: 10.1007/S10114-013-2552-Y
published in: Acta Mathematica Sinica, English Series
description: scientific article