Extension of the Notion of D-Symmetric Operators Using the Aluthge Transform

Abstract

Let $A \in \mathcal{L}(H)$ be a bounded operator on a Hilbert space $H$, and let $\delta_A$ denote the inner derivation defined by $\delta_A(X) = AX - XA$.
$A$ is called $D$-symmetric if
$
\overline{\mathcal{R}(\delta_{A^*})} = \overline{\mathcal{R}(\delta_A)},
$
where $\overline{\mathcal{R}(\delta_A)}$ is the norm closure of the range of $\delta_A$. Motivated by the results of \cite{C1,C2}, we introduce $\widetilde{D}$-symmetric operators for which
$$
\mathcal{R}(\delta_{\widetilde{A}}) \subset \overline{\mathcal{R}(\delta_A)},
$$
where $\widetilde{A}$ denotes the Aluthge transform of $A$. We show that this class contains quasinormal operators, isometries, co-isometries, cyclic subnormal operators, and all $D$-symmetric operators, and we characterize $\widetilde{D}$-symmetric operators via the range of the associated derivation in the Calkin algebra. For invertible operators, an additional characterization is obtained using suitable linear functionals, and $A$ is $D$-symmetric if and only if $\widetilde{A}$ is $D$-symmetric.