Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-07-08779-5
Published electronically: February 6, 2007
MathSciNet review: 2317952
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.


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

  • 1. A. Akbari and M. Aryapoor, On linear transformations preserving at least one eigenvalue, Proc. Amer. Math. Soc. 132 (2004), 1621-1625. MR 2051122 (2005b:15002)
  • 2. L. Baribeau and T. Ransford, Non-linear spectrum preserving maps, Bull. London Math. Soc. 32 (2000), 8-14. MR 1718765 (2000j:15008)
  • 3. M. Brešar and P. Šemrl, Linear maps preserving the spectral radius. J. Funct. Anal. 142 (1996), 360-368. MR 1423038 (97i:47070)
  • 4. C. Costara, A Cartan type theorem for finite-dimensional algebras, preprint.
  • 5. A. Edigarian and W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. (Basel) 84 (2005), 364-374. MR 2135687 (2006b:32020)
  • 6. S. Krantz, Function theory of several complex variables (second edition), American Mathematical Society, Providence RI, 2001. MR 1162310 (93c:32001)
  • 7. M. Marcus and R. Purves, Linear transformations on algebra of matrices II: the invariance of the elementary symmetric functions, Canad. J. Math., 11 (1959), 383-396. MR 0105425 (21:4167)
  • 8. T. Ransford, A Cartan theorem for Banach algebras, Proc. Amer. Math. Soc. 142 (1996), 243-247. MR 1307559 (96d:46063)
  • 9. T. Ransford and M. White, Holomorphic self-maps of the spectral unit ball, Bull. London Math. Soc. 23 (1991), 256-262. MR 1123334 (92g:32049)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32Hxx, 15A18, 32A12, 47B49

Retrieve articles in all journals with MSC (2000): 32Hxx, 15A18, 32A12, 47B49


Additional Information

Constantin Costara
Affiliation: Faculty of Mathematics and Informatics, Ovidius University of Constanţa, Mamaia Boul. No. 124, 900527, Romania
Email: cdcostara@univ-ovidius.ro

Thomas Ransford
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec (QC), Canada G1K 7P4
Email: ransford@mat.ulaval.ca

DOI: https://doi.org/10.1090/S0002-9939-07-08779-5
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

American Mathematical Society