Douglas' + Sebestyén's lemmas = a tool for solving an operator equation problem
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)]).
scientific article; zbMATH DE number 7138425
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