On the solutions of operator equation CAX=C=XACCAX=C=XAC

In this paper, we find the explicit solutions of the equation CAX=C=XACCAX=C=XAC for linear bounded operators on Hilbert spaces, where the unknown operator XX is called the inverse of AA along CC. This solution is expressed in terms of the inner inverses of the operators ACAC and CACA. From these formulas we derive necessary and sufficient conditions for the solution to be unique.