Extensions of a theorem of Fuglede and Putnam

The operator equation $AX = XB$ implies ${A^ \ast }X = X{B^ \ast }$ when A and B are normal (theorem of Fuglede and Putnam). If X is of Hilbert-Schmidt class, the assumptions on A and B can be relaxed: it suffices that A and ${B^ \ast }$ be hyponormal, or that B be invertible with $\left\Vert A \right\Vert\left\Vert {{B^{ - 1}}} \right\Vert \leqslant 1$.