Let H be a Hilbert space and \(C^ 2(H)\) the class of all Hilbert-Schmidt operators on H. For fixed bounded operators A and B on H let \(\delta_{A,B}\) and \(\tau_{A,B}\) be the operators on \(C^ 2(H)\) defined by \(\delta_{A,B}(X)=AX-XB\) and \(\tau_{A,B}(X)=AXB\) respectively. It is shown that \(\delta_{A,B}\) is a subnormal operator if and only if A and \(B^*\) are subnormal operators. If \(A\neq 0\) and \(B\neq 0\), the same characterization holds for the subnormality of \(\tau_{A,B}\). The quasinormal operators of the form \(\delta_{A,B}\) and \(\tau_{A,B}\) are also characterized.