Extensions of a theorem of Fuglede and Putnam

Abstract

The operator equation A X = X B AX = XB implies A ∗ X = X B ∗ {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 ∗ {B^ \ast } be hyponormal, or that B be invertible with ‖ A ‖ ‖ B − 1 ‖ ⩽ 1 \left \| A \right \|\left \| {{B^{ - 1}}} \right \| \leqslant 1 .