A remark on generalised Putnam-Fuglede theorems

Given $A, B \in B (H)$, the algebra of operators on a Hilbert space $H$, define $\delta_{A, B} : B(H) \rightarrow B(H)$ and $\triangle_{A, B} : B(H) \rightarrow B(H)$ by $\delta_{A, B}(X) = AX-XB$ and $\triangle_{A, B}(X) = AXB - X$. Let $P_1$ and $P_2$ be two classes of operators strictly larger than the class of normal operators. Define $(P_1, P_2) \in PF (\delta)$ (resp., $PF (\triangle))$ if $ker \delta _{A,B} \subset ker \delta_{A^*, B^*}$ (resp., $ker \triangle_{A, B} \subset ker \triangle_{A^*, B^*})$ for all $A \in P_1$and $B^* \in P_2$. This note shows that the equivalence $(P_1, P_2) \in PF (\delta) \Longleftrightarrow (P_1, P_2) \in PF(\triangle)$ holds for a number of the commonly considered classes of operators.