Solutions to operator equations on Hilbert [formula omitted]-modules

Abstract

Let A be a C ∗ -algebra, E , F and G be Hilbert A -modules, T ∈ L A ( E , F ) , and T ′ ∈ L A ( 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.