Maps preserving peripheral spectrum of generalized Jordan products of operators

Abstract

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 $(i_1,...,i_m)$ 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 $T_1\circ T_2\circ \cdots \circ T_k=T_{i_1}{T}_{i_2}\cdots T_{i_m}+T_{i_m}\cdots T_{i_2}{T}_{i_1}$ on elements in ${\mathcal{A}}_i$. 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 $\Phi :{\mathcal{A}}_1\to {\mathcal{A}}_2$ be a map with range containing all operators of rank at most three. It is shown that $\Phi$ satisfies that ${\sigma }_{\pi }(\Phi (A_1)\circ \cdots \circ \Phi (A_k))={\sigma }_{\pi }(A_1\circ \cdots \circ A_k)$ for all $A_1,\dots ,A_k$, where ${\sigma }_{\pi }(A)$ stands for the peripheral spectrum of $A$, if and only if $\Phi$ is a Jordan isomorphism multiplied by an $m$th root of unity.

arXiv comment24 pages. arXiv admin note: substantial text overlap with arXiv:1004.3832

References in corpus (1)

• "Maps preserving the spectrum of generalized Jordan product of operators", and its "Addendum"