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

Wen Zhang and Jinchuan Hou

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

Abstract: 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.

Date: 2014
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https://doi.org/10.1155/2014/192040

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:192040

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