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 $T_{ 1}\circ_{ \ast} T_{ 2}=T_{ 1}T_{ 2}^{ \ast}-T_{ 2}T_{ 1}$.
Categories: math.SP