The solutions to some operator equations Original Research Article

In this paper, we study the solvability of the operator equation AXB*-BX*A*=C in the general setting of the adjointable operators between Hilbert C*-modules. Based on the generalized inverses of the associated operators, we propose the necessary and sufficient conditions for the existence of a solution to this equation, and obtain the general expression of the solution in the solvable case. We apply the results to the study of the real positive and positive solutions to the operator equation AXB=C. In the case that the underlying space is finite-dimensional or the range of B is contained in that of A*, we propose new necessary and sufficient conditions for the existence of a positive solution to the operator equation AXB=C, and derive new formula in each case for the general positive solution to this operator equation.