Solvability of the system of operator equations \(AX=C, XB=D\) in Hilbert \(C{{}^*}\)-modules (Q2660419)

The system of equations \(AX=C,\,\,\, XB=D\) has been widely studied for matrices, Hilbert space operators, closed range operators on Hilbert \(C^*\)-modules. In this paper, the authors investigate the system above in the setting of Hilbert \(C^*\)-modules, in the case when \(\overline{{\mathcal R}(A^*)}\) and \(\overline{{\mathcal R}(B)}\) are orthogonally complemented. Using results given in [\textit{V. Manuilov} and \textit{M. S. Moslehian}, Proc. Am. Math. Soc. 148, No. 3, 1139--1151 (2020; Zbl 1462.47012)], they derive necessary and sufficient conditions for the solvability and the existence of Hermitian and positive solutions of the system \(AX=C\), \(XB=D\).