Maps preserving the $\varepsilon$-Pseudo Spectrum of some product of operators

Abstract

Let $B(H)$ be the algebra of all bounded linear operators on infinite-dimensional complex Hilbert space $H$. For $T, S \in B(H)$ denote by $T\bullet S=TS+ST^{\ast}$ and $[T\circ S]_{\ast}=TS-ST^{\ast}$ the Jordan $\ast$-product and the skew Lie product of $T$ and $S$, respectively. Fix $\varepsilon > 0$ and $T \in B(H)$, let $\sigma_{\varepsilon}(T)$ denote the $\varepsilon$-pseudo spectrum of $T$. In this paper, we describe bijective maps $\varphi$ on $B(H)$ which satisfy \begin{align*} \sigma_{\varepsilon}([T_{1}\bullet T_{2},T_{3}]_{\ast})=\sigma_{\varepsilon}([\varphi(T_{1})\bullet \varphi(T_{2}),\varphi(T_{3})]_{\ast}), \end{align*} for all $T_{1}, T_{2}, T_{3} \in B(H)$. We also characterize bijective maps $\varphi: B(H) \rightarrow B(H)$ that satisfy \begin{align*} \sigma_{\varepsilon}(T_{1}\diamond T_{2}\circ_{\ast} T_{3})=\sigma_{\varepsilon}(\varphi(T_{1})\diamond \varphi(T_{2})\circ_{\ast} \varphi(T_{3})), \end{align*} for all $T_{1}, T_{2}, T_{3} \in B(H)$, where $T_{1}\diamond T_{2}=T_{1}T_{2}^{\ast}+T_{2}^{\ast}T_{1}$ and $T_{1}\circ_{\ast} T_{2}=T_{1}T_{2}^{\ast}-T_{2}T_{1}$.