Common hermitian and positive solutions to the adjointable operator equations AX=C, XB=D

Let A be a C∗-algebra. For any Hilbert A-modules H and K, let L(H,K) be the set of adjointable operators from H to K. Let H,K,L be Hilbert A-modules, A,C∈L(H,K) and B,D∈L(L,H). In this paper, we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions X∈L(H) to the equations , and obtain the formulae for the general forms of these solutions. Some results, known for finite matrices and Hilbert space operators, are extended to the adjointable operators acting on Hilbert C∗-modules.