Solutions to some solvable modular operator equations

Abstract

We find explicit solution of the operator equation $TXT^* -SXS^*=A$ in
the general setting of the adjointable operators between Hilbert
$C^*$-modules, using some block operator matrices. Furthermore, we obtain solutions to the solvable
operator equation $TXR-SY Q = A $ over Hilbert $C^*$-modules, when both
${\rm ran}(T)+{\rm ran}(S)$ and ${\rm ran}(R^*)+{\rm ran}(Q^*)$ are
closed.