Abstract
The familiar Fuglede-Putnam theorem asserts thatAX=XBimpliesA*X=XB*whenAandBare normal. We prove thatAandB*be hyponormal operators and letCbe a hyponormal commuting withA*and also letD*be a hyponormal operator commuting withBrespectively, then for every Hilbert-Schmidt operatorX, the Hilbert-Schmidt norm ofAXD+CXBis greater than or equal to the Hilbert-Schmidt norm ofA*XD*+C*XB*. In particular,AXD=CXBimpliesA*XD*=C*XB*. If we strengthen the hyponormality conditions onA, B*, CandD*to quasinormality, we can relax Hilbert-Schmidt operator of the hypothesis onXto be every operator and still retain the inequality under some suitable hypotheses.
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