Douglas' + Sebestyén's lemmas = a tool for solving an operator equation problem
type: Item
instance of: scholarly article
title: Douglas' + Sebestyén's lemmas = a tool for solving an operator equation problem
zbMATH Open document ID: 1440.47033
author: Qing-Wen Wang
author: Qingxiang Xu
author: Dragana Cvetković-Ilić
published in: Journal of Mathematical Analysis and Applications
publication date: 2019-11-28T00:00:00Z
review text: The matrix equation \(AXB=C\) is related to some important subjects such as the Moore-Penrose inverse. This equation in various spaces of operators has been investigated by several authors; see [\textit{M. L. Arias} and \textit{M. C. Gonzalez}, Linear Algebra Appl. 433, No. 6, 1194--1202 (2010; Zbl 1200.47023)] and [\textit{D. S. Cvetković-Ilić}, J. Aust. Math. Soc. 84, No. 1, 63--72 (2008; Zbl 1157.15012)] and references therein. In particular, the existence of a positive solution of \(AXB=C\) is of special interest. When we are dealing with operators on Hilbert \(C^*\)-modules, some specific conditions such as regularity and range inclusion \(R(B)\subseteq R(A^*)\) seem to be necessary. The paper under review is a nice article in which the authors present some equivalences of the existence of a positive solution of the above operator equation in the context of Hilbert space operators, without any additional range assumptions, by utilizing two lemmas of \textit{Z. Sebestyén} [Acta Sci. Math. 46, 299--301 (1983; Zbl 0551.47005)] and \textit{R. G. Douglas} [Proc. Am. Math. Soc. 17, 413--415 (1966; Zbl 0146.12503)] (see also the survey article of \textit{M. S. Moslehian} et al. [Banach J. Math. Anal. 13, No. 3, 726--743 (2019; Zbl 1476.47006)] and the paper of \textit{V. Manuilov} et al. [Proc. Am. Math. Soc. 148, No. 3, 1139--1151 (2020; Zbl 1462.47012)]).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 47A62
zbMATH DE Number: 7138425
zbMATH Keywords: positive solution
zbMATH Keywords: operator equation
zbMATH Keywords: Douglas' lemma
MaRDI profile type: Publication
full work available at URL: j.jmaa.2019.123599
OpenAlex ID: W2979951043
Wikidata QID: Q124813258
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DOI: 10.1016/J.JMAA.2019.123599
description: scientific article; zbMATH DE number 7138425