The common positive solution to adjointable operator equations with an application

We give necessary and sufficient conditions for the existence of the common positive solution to the equations A 1 X 1 = C 1 , X 1 B 1 = D 1 , A 2 X 2 = C 2 , X 2 B 2 = D 2 , A 3 X 1 A 3 ∗ + A 4 X 2 A 4 ∗ = C 5 for adjointable operators between Hilbert C ∗ -modules, and present an expression of the positive solution to the system when the solvability conditions are satisfied. As an application, we consider the reflexive positive semidefinite solution to the system of complex matrix equations A X = B , X C = D , E X E ∗ = F .