The equation $AX =XB$ implies $A^{*}X =XB^{*}$ when $A$ and 
$B$ are normal operators
is known as the familiar Fuglede-Putnam theorem. In this paper, the 
hypothesis on $A$ and $B$ can be relaxed by using a Hilbert-Schmidt operator 
$X$: Let $A$ be a $(p,k$)-quasihyponormal  operator and $B^{*}$ be an
invertible $(p,k$)-quasihyponormal operator 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 operators is orthogonal to its kernel.