Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator
Given Hilbert space commuting operators $T, S \in \mcl(H)$, such that $T$ is $w$-hyponormal with $\ker T \subseteq \ker T^*$ and $S$ is normal operator. Let $\phi_{T, S} \in \mcl(\mcl(H))$ be the elementary operator defined by $\phi_{T, S} (X) = T X S^*-S X T^*$. In this paper, we show firstly that (1) $\ker \phi_{T, S} \subset \ker \phi_{T^*, S^*}$. (2) The range of $\phi_{T, S}$ is orthogonal to the kernel of $\phi_{T, S}$ ( $ \mcr(\phi_{T, S}) \perp \ker \phi_{T, S} $ ) if and only if $\ker T \cap \ker S=\{0\}$. Secondly, we will extend these results to the elementary operator $\Phi \in \mcl(\mcl(H))$ defined by $\;\Phi(X)=A X D-C X B$ where $[A, C]=[B, D]= 0$. Related orthogonality results for the elementary operator $\Phi$ are also given.