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The original form of the Fuglede-Putnam theorem states that the operator equation AX = XB implies A*X = XB* when A and B are normal. In our previous paper we have relaxed the normality in the hypotheses on A and B as follows: if A and B* are subnormal and if X is an operator such that AX -XB, then A*X -XB*. We shall show asymptotic versions of this generalized Fuglede-Putnam theorem; these results are also extensions of results of Moore and Rogers.doi:10.1090/s0002-9939-1980-0572310-6 fatcat:ki7reap2nbbqxp6ikktd72f4qy