Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators
Let A1 and A2 be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H1 and H2, respectively. For k≥2, let (i1,…,im) be a fixed sequence with i1,…,im∈{1,…,k} and assume that at least one of the terms in (i1,…,im) appears exactly once. Define the generalized Jordan...
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doaj-08cfefebad0148f8926d3e56465f40272020-11-24T23:54:03ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/192040192040Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint OperatorsWen Zhang0Jinchuan Hou1Department of Mathematics, Shanxi University, Taiyuan, Shanxi 030006, ChinaDepartment of Mathematics, Shanxi University, Taiyuan, Shanxi 030006, ChinaLet A1 and A2 be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H1 and H2, respectively. For k≥2, let (i1,…,im) be a fixed sequence with i1,…,im∈{1,…,k} and assume that at least one of the terms in (i1,…,im) appears exactly once. Define the generalized Jordan product T1∘T2∘⋯∘Tk=Ti1Ti2⋯Tim+Tim⋯Ti2Ti1 on elements in Ai. Let Φ:A1→A2 be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that Φ satisfies that σπ(Φ(A1)∘⋯∘Φ(Ak))=σπ(A1∘⋯∘Ak) for all A1,…,Ak, where σπ(A) stands for the peripheral spectrum of A, if and only if there exist a scalar c∈{-1,1} and a unitary operator U:H1→H2 such that Φ(A)=cUAU* for all A∈A1, or Φ(A)=cUAtU* for all A∈A1, where At is the transpose of A for an arbitrarily fixed orthonormal basis of H1. Moreover, c=1 whenever m is odd.http://dx.doi.org/10.1155/2014/192040 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Wen Zhang Jinchuan Hou |
spellingShingle |
Wen Zhang Jinchuan Hou Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators Abstract and Applied Analysis |
author_facet |
Wen Zhang Jinchuan Hou |
author_sort |
Wen Zhang |
title |
Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators |
title_short |
Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators |
title_full |
Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators |
title_fullStr |
Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators |
title_full_unstemmed |
Maps Preserving Peripheral Spectrum of Generalized Jordan Products of Self-Adjoint Operators |
title_sort |
maps preserving peripheral spectrum of generalized jordan products of self-adjoint operators |
publisher |
Hindawi Limited |
series |
Abstract and Applied Analysis |
issn |
1085-3375 1687-0409 |
publishDate |
2014-01-01 |
description |
Let A1 and A2 be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H1 and H2, respectively. For k≥2, let (i1,…,im) be a fixed sequence with i1,…,im∈{1,…,k} and assume that at least one of the terms in (i1,…,im) appears exactly once. Define the generalized Jordan product T1∘T2∘⋯∘Tk=Ti1Ti2⋯Tim+Tim⋯Ti2Ti1 on elements in Ai. Let Φ:A1→A2 be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that Φ satisfies that σπ(Φ(A1)∘⋯∘Φ(Ak))=σπ(A1∘⋯∘Ak) for all A1,…,Ak, where σπ(A) stands for the peripheral spectrum of A, if and only if there exist a scalar c∈{-1,1} and a unitary operator U:H1→H2 such that Φ(A)=cUAU* for all A∈A1, or Φ(A)=cUAtU* for all A∈A1, where At is the transpose of A for an arbitrarily fixed orthonormal basis of H1. Moreover, c=1 whenever m is odd. |
url |
http://dx.doi.org/10.1155/2014/192040 |
work_keys_str_mv |
AT wenzhang mapspreservingperipheralspectrumofgeneralizedjordanproductsofselfadjointoperators AT jinchuanhou mapspreservingperipheralspectrumofgeneralizedjordanproductsofselfadjointoperators |
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1725467570505515008 |