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On the spectrum of frequently hypercyclic operators

Author: Stanislav Shkarin
Journal: Proc. Amer. Math. Soc. 137 (2009), 123-134
MSC (2000): Primary 47A16, 37A25
Published electronically: August 28, 2008
MathSciNet review: 2439433
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Abstract: A bounded linear operator $ T$ on a Banach space $ X$ is called frequently hypercyclic if there exists $ x\in X$ such that the lower density of the set $ \{n\in\mathbb{N}:T^nx\in U\}$ is positive for any non-empty open subset $ U$ of $ X$. Bayart and Grivaux have raised a question whether there is a frequently hypercyclic operator on any separable infinite dimensional Banach space. We prove that the spectrum of a frequently hypercyclic operator has no isolated points. It follows that there are no frequently hypercyclic operators on all complex and on some real hereditarily indecomposable Banach spaces, which provides a negative answer to the above question.

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Stanislav Shkarin
Affiliation: Department of Pure Mathematics, Queens’s University Belfast, University Road, Belfast, BT7 1NN, United Kingdom

Keywords: Frequently hypercyclic operators, hereditarily indecomposable Banach spaces, quasinilpotent operators
Received by editor(s): July 26, 2007
Published electronically: August 28, 2008
Additional Notes: Partially supported by Plan Nacional I+D+I grant No. MTM2006-09060 and Junta de Andalucía FQM-260.
Communicated by: Marius Junge
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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