The solutions to the operator equation $TXS^* -SX^*T^*=A$ in Hilbert $C^*$-modules

Abstract

In this paper, we find explicit solution to the operator equation $TXS^* -SX^*T^*=A$ in the general setting of the adjointable operators between Hilbert $C^*$-modules, when $T,S$ have closed ranges and $S$ is a self adjoint operator.