A Hilbert-Schmidt norm inequality associated with the Fuglede-Putnam theorem

Abstract

The familiar FugledemPutnam theorem asserts that AX=XB implies A ' X = XB* when A and B are normal. We prove that let A and B* be hyponormal operators and let C be hyponormal commuting with A* and also let D* be a hyponormal operator commuting with B respectively, then for every Hilbert--Schmidt operator X, the Hilbert--Schmidt norm of AXD+CXB is greater than or equal to the Hilbert--Schmidt norm of A*XD*+C*XB*. In particular, AXD=CXB implies A*XD*=C*XB*. If we strengthen the hyponormality conditions on A, B*, C and D* to quasinormality, we can relax Hilbert--Schmidt operator of the hypothesis on X to be every operator in B(H) and still retain the inequality under hypotheses that C commutes with A and satisfies an operator equation and also D* commutes withB* and satisfies another similar operator equation respectively.