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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Mbekhta's subspaces and a spectral theory of compact operators


Authors: Weibang Gong and Libin Wang
Journal: Proc. Amer. Math. Soc. 131 (2003), 587-592
MSC (2000): Primary 47A10, 47A11
Published electronically: July 17, 2002
MathSciNet review: 1933350
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06639-X
PII: S 0002-9939(02)06639-X
Keywords: Spectral theory of compact operators, isolated point of the spectrum, pole of the resolvent operator
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
Article copyright: © Copyright 2002 American Mathematical Society