On the Δ-Symmetric Operators and Applications

Let $\mathcal{L}(\mathcal{H})$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $\mathcal{H}$ into itself. Given $A, B \in \mathcal{L}(\mathcal{H})$, we define the elementary operator $\Delta_{A, B}: \mathcal{L}(\mathcal{H}) \longrightarrow \mathcal{L}(\mathcal{H})$ by $\Delta_{A, B}(X)=A X B-X$. This note will initiate a study on the class of pairs $(A, B)$ such that $\overline{R\left(\Delta_{A, B}\right)}=\overline{R\left(\Delta_{\left.B^{*}, A^{*}\right)}\right.}$; i.e. $\overline{R\left(\Delta_{A, B}\right)}$ is self-adjoint. A technical application is also shown.