On the Hermitian solutions to a system of adjointable operator equations (Q445851)

The authors of the paper under review obtain necessary and sufficient conditions for the existence of a Hermitian solution to the simultaneous system of equations: \(A_1X_1=C_1\), \(X_1B_1=D_1\), \(A_2X_2=C_2\), \(X_2B_2=D_2\) and \(A_3X_1A_3^*+A_4X_2A_4^*=C_5\) for adjointable operators between Hilbert \(C^*\)-modules. In the case that these equations have a solution, a general Hermitian solution is given. Characterizations for the existence of a unique Hermitian solution to the simultaneous system consisting of the first two equations, together with \(A_3X_1A_3^*=C_5\), are presented. Illustrative examples are provided. The work reported here extends results known in the literature for matrices and for operators over Hilbert spaces.