Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators

Abstract

Let ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces $H_1$ and $H_2$, respectively. For $k\ge 2$, let $(i_1,\dots ,i_m)$ be a fixed sequence with $i_1,\dots ,i_m\in$ $\{1,\dots ,k\}$ 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 ${\mathcal{A}}_i$. Let $\mathrm{\Phi }:{\mathcal{A}}_1\to {\mathcal{A}}_2$ be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that $\mathrm{\Phi }$ satisfies that ${\sigma }_{\pi }(\mathrm{\Phi }(A_1)\circ \cdots \circ \mathrm{\Phi }(A_k))={\sigma }_{\pi }(A_1\circ \cdots \circ A_k)$ for all $A_1,\dots ,A_k$, where ${\sigma }_{\pi }(A)$ stands for the peripheral spectrum of $A$, if and only if there exist a scalar $c\in \{-\mathrm{1,1}\}$ and a unitary operator $U:H_1\to H_2$ such that $\mathrm{\Phi }(A)=cUA{U}^{\mathrm{*}}$ for all $A\in {\mathcal{A}}_1$, or $\mathrm{\Phi }(A)=cU{A}^t{U}^{\mathrm{*}}$ for all $A\in {\mathcal{A}}_1$, where $A^t$ is the transpose of $A$ for an arbitrarily fixed orthonormal basis of $H_1$. Moreover, $c=1$ whenever $m$ is odd.