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 R(C) R(A) holds and CC tCA for some t > 0. In addition, we deal with the solvability of the operator equation (P + Q)1=2X = P, where P and Q are projections. We provide a tricky counterexample to show that there exist a C-algebra A, a Hilbert A-module H and projections P and Q on H such that the operator equation (P + Q)1=2X = P has no solution. Moreover, we give a perturbation result related to the latter equation.

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