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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A condition for spectral continuity of positive elements
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by S. Mouton PDF
Proc. Amer. Math. Soc. 137 (2009), 1777-1782 Request permission

Abstract:

Let $a$ be an element of a Banach algebra $A$. We introduce a compact subset $T(a)$ of the complex plane, show that the function which maps $a$ onto $T(a)$ is upper semicontinuous and use this fact to provide a condition on $a$ which ensures that if $(a_n)$ is a sequence of positive elements converging to $a$, then the sequence of the spectral radii of the terms $a_n$ converges to the spectral radius of $a$ in the case that $A$ is partially ordered by a closed and normal algebra cone and $a$ is a positive element.
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Additional Information
  • S. Mouton
  • Affiliation: Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa
  • Email: smo@sun.ac.za
  • Received by editor(s): June 29, 2007
  • Received by editor(s) in revised form: April 22, 2008, and July 21, 2008
  • Published electronically: November 4, 2008
  • Additional Notes: The author thanks the referee for making useful suggestions.
  • Communicated by: N. Tomczak-Jaegermann
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 1777-1782
  • MSC (2000): Primary 46H05, 47A10
  • DOI: https://doi.org/10.1090/S0002-9939-08-09715-3
  • MathSciNet review: 2470837