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Mappings preserving spectra of products of matrices

Authors: Jor-Ting Chan, Chi-Kwong Li and Nung-Sing Sze
Journal: Proc. Amer. Math. Soc. 135 (2007), 977-986
MSC (2000): Primary 15A04, 15A18
Published electronically: October 4, 2006
MathSciNet review: 2262897
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Abstract: Let $ M_n$ be the set of $ n\times n$ complex matrices, and for every $ A\in M_n$, let $ \operatorname{Sp}(A)$ denote the spectrum of $ A$. For various types of products $ A_1* \cdots *A_k$ on $ M_n$, it is shown that a mapping $ \phi: M_n \rightarrow M_n$ satisfying $ \operatorname{Sp}(A_1*\cdots*A_k) = \operatorname{Sp}(\phi(A_1)* \cdots*\phi(A_k))$ for all $ A_1, \dots, A_k \in M_n$ has the form

$\displaystyle X \mapsto \xi S^{-1}XS \quad \hbox{ or } \quad A \mapsto \xi S^{-1}X^tS$

for some invertible $ S \in M_n$ and scalar $ \xi$. The result covers the special cases of the usual product $ A_1* \cdots * A_k = A_1 \cdots A_k$, the Jordan triple product $ A_1*A_2 = A_1*A_2*A_1$, and the Jordan product $ A_1*A_2 = (A_1A_2+A_2A_1)/2$. Similar results are obtained for Hermitian matrices.

References [Enhancements On Off] (What's this?)

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Additional Information

Jor-Ting Chan
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

Chi-Kwong Li
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795

Nung-Sing Sze
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong.
Address at time of publication: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Keywords: Eigenvalue, spectrum, preserve
Received by editor(s): April 7, 2005
Received by editor(s) in revised form: November 10, 2005
Published electronically: October 4, 2006
Additional Notes: This research was partially supported by Hong Kong RCG CERG grant HKU 7007/03P. The second author was also supported by a USA NSF grant.
The second author is also an honorary professor of the Heilongjiang University, and an honorary professor of the University of Hong Kong.
Dedicated: Dedicated to Professor Ahmed Sourour on the occasion of his sixtieth birthday.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society

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