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 $k\geq2$, let $(i_1,...,i_m)$ be a sequence with terms chosen from $\{1,\ldots,k\}$ and assume that at least one of the terms in $(i_1,\ldots,i_m)$ appears exactly once. Define the generalized Jordan product $T_1\circ T_2\circ\cdots\circ T_k=T_{i_1} T_{i_2}\cdots T_{i_m}+T_{i_m}\cdots T_{i_2} T_{i_1}$ on elements in $\mathcal{A}_i$. This includes the usual Jordan product $A_1A_2+A_2A_1$, and the Jordan triple $A_1A_2A_3+A_3A_2A_1$. Let $Φ:\mathcal{A}_1\rightarrow\mathcal{A}_2$ be a map with range containing all operators of rank at most three. It is shown that $Φ$ satisfies that $σ_π(Φ(A_1)\circ\cdots\circΦ(A_k))=σ_π(A_1\circ\cdots\circ A_k)$ for all $A_1, \ldots, A_k$, where $σ_π(A)$ stands for the peripheral spectrum of $A$, if and only if $Φ$ is a Jordan isomorphism multiplied by an $m$th root of unity.

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