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Spectrum nonincreasing maps on matrices. (English) Zbl 1267.15008

A function from the set \(M_{n}(\mathbb{C})\) of all complex \(n\times n\) matrices into itself is said to have standard form if there exists an invertible matrix \(S\) such that \(A\longmapsto SAS^{-1}\) for all \(A\) or \(A\longmapsto SA^{t}S^{-1}\) for all \(A\). Let \(\Phi\) be a function from \(M_{n}(\mathbb{C})\) into itself such that \(\Phi(0)=0\) and the spectrum of \(\Phi(A)-\Phi(B)\) is contained in the spectrum of \(A-B\) for all \(A,B\in M_{n}(\mathbb{C})\). It has previously been proved that \(\Phi\) must be of standard form if we assume either (i) \(\Phi\) is continuous (see [C. Costara, Linear Algebra Appl. 435, No. 11, 2674–2680 (2011; Zbl 1231.15003)]), or (ii) \(\Phi\) is surjective (see [M. Bendaoud, M. Douimi and M. Sarih, “Maps on matrices preserving local spectra”, Linear Multilinear Algebra 61, No. 7, 871–880 (2013; {doi:10\.1080/03081087.2012.716429})]).
In the present paper, the authors prove that \(\Phi\) is of standard form without making any additional assumption on \(\Phi\). A critical step in the proof is that the hypothesis on \(\Phi\) implies that there exists a dense subset \(\mathcal{K}\) of matrices in \(M_{n}(\mathbb{C})\) such that the trace identity \(\operatorname{Tr}(AB)=\operatorname{Tr}(\Phi(A)\Phi(B))\) holds whenever \(A-B\in\mathcal{K}\) (\(\mathcal{K}\) consists of the matrices with \(n\) distinct eigenvalues which are linearly independent over \(\mathbb{Z}\)).

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A86 Linear preserver problems

Citations:

Zbl 1231.15003
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References:

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