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On local irreducibility of the spectrum

Authors: Constantin Costara and Thomas Ransford
Journal: Proc. Amer. Math. Soc. 135 (2007), 2779-2784
MSC (2000): Primary 32Hxx; Secondary 15A18, 32A12, 47B49
Published electronically: February 6, 2007
MathSciNet review: 2317952
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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 $ x$ respectively. Let $ D$ 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 $ x=0$.

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Additional Information

Constantin Costara
Affiliation: Faculty of Mathematics and Informatics, Ovidius University of Constanţa, Mamaia Boul. No. 124, 900527, Romania

Thomas Ransford
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec (QC), Canada G1K 7P4

Keywords: Spectrum, algebroid multifunction, irreducible, spectral radius, preserver
Received by editor(s): April 6, 2006
Received by editor(s) in revised form: May 5, 2006
Published electronically: February 6, 2007
Additional Notes: The second author was supported by grants from NSERC and the Canada Research Chairs program
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society

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