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

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 $T\bullet S=TS+S{T}^{\ast }$ and $[T\circ S{]}_{\ast }=TS-S{T}^{\ast }$ the Jordan $\ast$-product and the skew Lie product of $T$ and $S$, respectively. Fix $\epsilon >0$ and $T\in B(H)$, let ${\sigma }_{\epsilon }(T)$ denote the $\epsilon$-pseudo spectrum of $T$. In this paper, we describe bijective maps $\phi$ on $B(H)$ which satisfy \begin{align*} σ_{\varepsilon}([T_{1}\bullet T_{2},T_{3}]{\ast})=σ{\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 $\phi :B(H)\to B(H)$ that satisfy \begin{align*} σ{\varepsilon}(T_{1}\diamond T_{2}\circ_{\ast} T_{3})=σ_{\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 $T_1{\circ }_{\ast }{T}_2=T_1{T}_2^{\ast }-T_2{T}_1$.