## Abstract and Applied Analysis

### 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 $c\in \{-1,1\}$ and a unitary operator $U:{H}_{1}\to {H}_{2}$ such that $\mathrm{\Phi }(A)=cUA{U}^{\mathrm{\ast}}$ for all $A\in {\mathcal{A}}_{1}$, or $\mathrm{\Phi }(A)=cU{A}^{t}{U}^{\mathrm{\ast}}$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 192040, 8 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425049838

Digital Object Identifier
doi:10.1155/2014/192040

Mathematical Reviews number (MathSciNet)
MR3273902

Zentralblatt MATH identifier
07021907

#### Citation

Zhang, Wen; Hou, Jinchuan. Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators. Abstr. Appl. Anal. 2014 (2014), Article ID 192040, 8 pages. doi:10.1155/2014/192040. https://projecteuclid.org/euclid.aaa/1425049838