The familiar Fuglede-Putnam Theorem is as follows (see (5), (11) and (12)): If A and B are normal operators and if X is an operator such that AX = XB, then A X = XB . In this paper, the hypothesis on A and B can be relaxed by using a Hilbert-Schmidt operator X: Let A and B be class A operators such that AX = XB for a Hilbert Schmidt operators X. Then A X = XB . As a consequence of this result, we obtain that the range of the generalized derivation induced by this class of opera- tors is orthogonal to its kernel. Some properties of log-hyponormal operators are also given.