Maps preserving peripheral spectrum of generalized Jordan products of operators

Let $X_1$ and $X_2$ be complex Banach spaces with dimension at least three, ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ be standard operator algebras on $X_1$ and $X_2$, respectively. For $k\ge 2$, let $(i_1,\mathrm{.}\mathrm{.}\mathrm{.},i_m)$ be a sequence with terms chosen from $\{1,\dots ,k\}$ and assume that at least one of the terms in $(i_1,\dots ,i_m)$ appears exactly once. Define the generalized Jordan product $T_1\circ T_2\circ\cdots\circ T_k=T_{i_1} T_{i_2}\cdots T_{i_m}+T_{i_m}\cdots T_{i_2} T_{i_1}$onelementsin$$\mathcal{A}$i$. This includes the usual Jordan product $A_1{A}_2+A_2{A}_1$, and the Jordan triple $A_1{A}_2{A}_3+A_3{A}_2{A}_1$. Let $\mathrm{\Phi }:{\mathcal{A}}_1\to {\mathcal{A}}_2$ be a map with range containing all operators of rank at most three. It is shown that $\mathrm{\Phi }$ satisfies that $\sigma\pi(\Phi(A_1)\circ\cdots\circ\Phi(A_k))=\sigma_\pi(A_1\circ\cdots\circ A_k)$forall$A_1, \ldots, A_k$,where$\sigma_\pi(A)$ stands for the peripheral spectrum of $A$, if and only if $\mathrm{\Phi }$ is a Jordan isomorphism multiplied by an $m$th root of unity.