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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant subspaces for Banach space operators with an annular spectral set


Author: Onur Yavuz
Journal: Trans. Amer. Math. Soc. 360 (2008), 2661-2680
MSC (2000): Primary 47A15; Secondary 47A60
Published electronically: December 11, 2007
MathSciNet review: 2373328
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Abstract: Consider an annulus $ \Omega=\{z\in\mathbb{C}:r_ {0}<\vert z\vert<1\}$ for some $ 0<r_{0}<1$, and let $ T$ be a bounded invertible linear operator on a Banach space $ X$ whose spectrum contains $ \partial\Omega$. Assume there exists a constant $ K>0$ such that $ \Vert p(T)\Vert~\leq~ K \sup\{\vert p(\lambda)\vert:\vert\lambda\vert\leq 1\}$ and $ \Vert p(r_0T^{-1})\Vert\leq K \sup\{\vert p(\lambda)\vert:\vert\lambda\vert\leq 1\}$ for all polynomials $ p$. Then there exists a nontrivial common invariant subspace for $ T^{*}$ and $ {T^{*}}^{-1}$.


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Additional Information

Onur Yavuz
Affiliation: Department of Mathematics, Indiana University, Rawles Hall, Bloomington, Indiana 47405
Address at time of publication: Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Email: oyavuz@indiana.edu, oyavuz@metu.edu.tr

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04324-3
PII: S 0002-9947(07)04324-3
Received by editor(s): May 2, 2005
Received by editor(s) in revised form: April 17, 2006
Published electronically: December 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.