Douglas factorization theorem revisited

Inspired by the Douglas factorization theorem, we investigate the solvability of the operator equation AX=C in the framework of Hilbert C^*-modules. Utilizing partial isometries, we present its general solution when A is a semi-regular operator. For such an operator A, we show that the equation AX=C has a positive solution if and only if the range inclusion {\mathcal R}(C) \subseteq {\mathcal R}(A) holds and CC^*\le t\, CA^* for some t>0. In addition, we deal with the solvability of the operator equation (P+Q)^{1/2}X=P, where P and Q are projections. We provide a tricky counterexample to show that there exist a C^*-algebra \mathfrak{A}, a Hilbert \mathfrak{A}-module \mathscr {H}, and projections P and Q on \mathscr {H} such that the operator equation (P+Q)^{1/2}X=P has no solution. Moreover, we give a perturbation result related to the latter equation.