Maps preserving peripheral spectrum of generalized products of operators

Abstract

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be standard operator algebras on complex Banach spaces $X_1$ 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 product $T_1* T_2*\cdots* T_k=T_{i_1}T_{i_2}\cdots T_{i_m}$ on elements in $\mathcal{A}_i$. Let $\Phi:\mathcal{A}_1\rightarrow\mathcal{A}_2$ be a map with the range containing all operators of rank at most two. We show that $\Phi$ satisfies that $\sigma_\pi(\Phi(A_1)*\cdots*\Phi(A_k))=\sigma_\pi(A_1*\cdots* A_k)$ for all $A_1,\ldots, A_k$, where $\sigma_\pi(A)$ stands for the peripheral spectrum of $A$, if and only if $\Phi$ is an isomorphism or an anti-isomorphism multiplied by an $m$th root of unity, and the latter case occurs only if the generalized product is quasi-semi Jordan. If $X_1=H$ and $X_2=K$ are complex Hilbert spaces, we characterize also maps preserving the peripheral spectrum of the skew generalized products, and prove that such maps are of the form $A\mapsto cUAU^*$ or $A\mapsto cUA^tU^*$, where $U\in\mathcal{B}(H,K)$ is a unitary operator, $c\in\{1,-1\}$.