Solvability of \(AXB - CXD = E\) in the operators algebra \(B(H)\) (Q694356)

This short note is a continuation of the paper published by the author and \textit{S. Mecheri} [Lobachevskii J. Math. 30, No. 3, 224--228 (2009; Zbl 1254.47015)]. It is proved that the equation \(AXB-CXD=E\), where \(A, B, C, D\in B(H)\) are normal operators on a separable (infinite-dimensional) complex Hilbert space \(H\) such that \(A\) and \(D\) commute and the pairs \((A,D)\) and \((B,C)\) possess the Fuglede-Putnam property, has a solution in \(B(H)\) if and only if, for every \(\lambda \in {\mathbb C}\), the block operator matrix which has \(A-\lambda I\) and \(D-\lambda I\) on the diagonal is equivalent to the block operator matrix which has \(A-\lambda I\) and \(D-\lambda I\) on the diagonal and \(E\) in the upper right corner.