Solutions to operator equations on Hilbert C∗-modules

Let A be a C∗-algebra, E,F and G be Hilbert A-modules, T∈LA(E,F), and T′∈LA(G,F). We generalize the Douglas theorem about the operator equation TX=T′ from Hilbert space to Hilbert C∗-module. To the equation TX=T′ and to the system of two equations TX=T′ and XS=S′, we get the forms of general solutions (in the case that there exists a solution), and give some sufficient and necessary conditions for the existence of solutions, and the existence of hermitian solutions and positive solutions (in the case G=E). In addition, the forms of general hermitian solution and general positive solution (in the case that there exists a solution and G=E) to the equation TX=T′ are given too.