Abstract and Applied Analysis

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

Wen Zhang and Jinchuan Hou

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Abstract

Let A 1 and A 2   be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces H 1   and   H 2 , respectively. For   k 2 , let   ( i 1 , , i m ) be a fixed sequence with i 1 , , i m { 1 , , k } and assume that at least one of the terms in ( i 1 , , i m ) appears exactly once. Define the generalized Jordan product   T 1 T 2 T k = T i 1 T i 2 T i m + T i m T i 2 T i 1 on elements in   A i . Let Φ : A 1 A 2 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 σ π ( Φ ( A 1 ) Φ ( A k ) ) = σ π ( A 1 A k ) for all A 1 , , A k , where σ π ( A ) stands for the peripheral spectrum of A , if and only if there exist a scalar c { - 1,1 } and a unitary operator U : H 1 H 2 such that Φ ( A ) = c U A U * for all A A 1 , or Φ ( A ) = c U A t U * for all A 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

Permanent link to this document
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


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