Solutions to operator equations on Hilbert \(C^*\)-modules
type: Item
instance of: scholarly article
title: Solutions to operator equations on Hilbert \(C^*\)-modules
zbMATH Open document ID: 1175.47014
author: Jing Yu
author: Xiaochun Fang
author: Hong Liang Yao
published in: Linear Algebra and its Applications
publication date: 2009-10-14T00:00:00Z
review text: Let \(E, F, G\) be Hilbert \(C^*\)-modules, \(T \in L(E, F)\), and \(T' \in L(G, F)\), where \(L(E, F)\) denotes the set of all adjointable maps from \(E\) to \(F\). The authors generalize the Douglas theorem [see \textit{R.\,G.\thinspace Douglas}, Proc.\ Am.\ Math.\ Soc.\ 17, 413--415 (1966; Zbl 0146.12503)] to Hilbert \(C^*\)-modules by showing that if \(\overline{R(T^*)}\) is orthogonally complemented, then the following statements are equivalent: (i) \(T'{T'}^*\leq \lambda TT^{*}\) for some \(\lambda>0\); (ii) there exists \(\mu > 0\) such that \(\|{T'}^*{z}\| \leq \mu\|T^*z\|\) for all \(z\in E\); (iii) there exists \(D\in L(G,E)\) such that \(T'=TD\), i.e., \(TX=T'\) has a solution; (iv) \(R(T') \subseteq R(T)\), where \(R(T)\) is the range of \(T\). They get the forms of general solutions of the equation \(TX = T'\) and the system of two equations \(TX = T'\) and \(XS = S'\), and give some sufficient and necessary conditions for the existence of solutions, and the existence of Hermitian solutions and positive solutions (in the case where \(G = E\)).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A62
Mathematics Subject Classification ID: 46L08
zbMATH DE Number: 5614892
zbMATH Keywords: Hilbert \(C^{*}\)-module
zbMATH Keywords: solution
zbMATH Keywords: Hermitian solution
zbMATH Keywords: positive solution
MaRDI profile type: Publication
full work available at URL: j.laa.2009.07.009
OpenAlex ID: W2032389887
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DOI: 10.1016/J.LAA.2009.07.009
description: scientific article; zbMATH DE number 5614892