Mappings preserving spectra of products of matrices
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- by Jor-Ting Chan, Chi-Kwong Li and Nung-Sing Sze
- Proc. Amer. Math. Soc. 135 (2007), 977-986
- DOI: https://doi.org/10.1090/S0002-9939-06-08568-6
- Published electronically: October 4, 2006
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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 \[ X \mapsto \xi S^{-1}XS \quad \mathrm { or } \quad A \mapsto \xi S^{-1}X^tS\] 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
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Bibliographic Information
- Jor-Ting Chan
- Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
- Email: jtchan@hku.hk
- Chi-Kwong Li
- Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795
- MR Author ID: 214513
- Email: ckli@math.wm.edu
- 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
- Email: NungSingSze@graduate.hku.hk
- 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. - Communicated by: Joseph A. Ball
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 977-986
- MSC (2000): Primary 15A04, 15A18
- DOI: https://doi.org/10.1090/S0002-9939-06-08568-6
- MathSciNet review: 2262897
Dedicated: Dedicated to Professor Ahmed Sourour on the occasion of his sixtieth birthday.