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On local irreducibility of the spectrum

Authors:	Constantin Costara Thomas Ransford

Institution:	Faculty of Mathematics and Informatics, Ovidius University of Constanta, Mamaia Boul. No. 124, 900527, Romania ; Département de mathématiques et de statistique, Université Laval, Québec (QC), Canada G1K 7P4

Abstract:	Let $\mathcal M_n$ be the algebra of $n\times n$ complex matrices, and for $x\in\mathcal M_n$ denote by $\sigma(x)$ and $\rho(x)$ the spectrum and spectral radius of respectively. Let be a domain in $\mathcal M_n$ containing 0, and let $F:D\to\mathcal M_n$ be a holomorphic map. We prove: (1) if $\sigma(F(x))\cap\sigma(x)\ne\emptyset$ for $x\in D$ , then $\sigma(F(x))=\sigma(x)$ for $x\in D$ ; (2) if $\rho(F(x))=\rho(x)$ for $x\in D$ , then again $\sigma(F(x))=\sigma(x)$ for $x\in D$ . Both results are special cases of theorems expressing the irreducibility of the spectrum $\sigma(x)$ near .

Keywords:	Spectrum algebroid multifunction irreducible spectral radius preserver

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