Solutions to operator equations on Hilbert \(C^*\)-modules
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 \(G = E\)).
scientific article; zbMATH DE number 5614892