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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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Mbekhta’s subspaces and a spectral theory of compact operators
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by Weibang Gong and Libin Wang PDF
Proc. Amer. Math. Soc. 131 (2003), 587-592 Request permission

Abstract:

Let $A$ be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta’s subspaces $H_{0}(A)$ and $K(A)$, we give a spectral theory of compact operators. The main results are: Let $A$ be compact. $1$. The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of $A;$ (2) $K(A)$ is closed; (3) $K(A)$ is of finite dimension; (4) $K(A^{\ast })$ is closed; (5) $K(A^{\ast })$ is of finite dimension; $2$. sufficient conditions for $0$ to be an isolated point in $\sigma (A)$; $3$. sufficient and necessary conditions for $0$ to be a pole of the resolvent of $A$.
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Additional Information
  • Weibang Gong
  • Affiliation: Department of Mathematics, Qufu Normal University, Qufu, Shandong, 273165, People’s Republic of China
  • Email: gongwb@ji-public.sd.cninfo.net
  • Libin Wang
  • Affiliation: Department of Mathematics, Qufu Normal University, Qufu, Shandong, 273165, People’s Republic of China
  • Received by editor(s): April 19, 2001
  • Received by editor(s) in revised form: October 2, 2001
  • Published electronically: July 17, 2002
  • Additional Notes: This paper is project 19871048 supported by the NSFC
  • Communicated by: Joseph A. Ball
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 587-592
  • MSC (2000): Primary 47A10, 47A11
  • DOI: https://doi.org/10.1090/S0002-9939-02-06639-X
  • MathSciNet review: 1933350