A remark on generalised Putnam-Fuglede theorems

Abstract:

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.