Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator II

Abstract

Let $T, S \in \mathcal{L}(H)$ be commuting operators on a Hilbert space $H$, where $T$ is a $w$-hyponormal operator satisfying $\ker T \subseteq \ker T^*$, and $S$ is normal. Consider the elementary operator $\psi_{T, S}$ defined by $\psi_{T, S}(X) = TX T^* - SX S^*,\;X\in \mathcal{L}(H).$ It is shown that
(1) For all $X\in \mathcal{L}(H),\;T X T^* = S X S^*$ implies $T^* X T = S^* X S,$ i.e., $\ker \psi_{T, S} \subseteq \ker \psi_{T^*, S^*}$; and
(2) For all $X, Y \in \mathcal{L}(H)$ with $Y \in \ker\left(\psi_{T,
S}\right)$ we have $$ \left\| T X T^* - S X S^*+Y\right\| \geq
\left\|Y\right\|,$$
which means the range of $\psi_{T, S}$ is orthogonal to its kernel, if and only if $\ker T \cap \ker S=\{0\}$.
These results are further extended to the elementary operator $\Psi \in \mathcal{L}(\mathcal{L}(H))$ defined by $\Psi(X) = AXB - CXD$, where $A, B^* \in \mathcal{L}(H)$ are $w$-hyponormal operators satisfying $\ker A \subseteq \ker A^*$ and $\ker B^* \subseteq \ker B$, and $C, D \in \mathcal{L}(H)$ are normal operators that commute with $A$ and $B$
respectively. This extension broadens classical results by Turn\v{s}ek, as well as the Putnam-Fuglede property, and the theorem of Weiss, by involving four operators, not all of which are normal.