On the spectrum of frequently hypercyclic operators

Shkarin, Stanislav

doi:10.1090/S0002-9939-08-09655-X

On the spectrum of frequently hypercyclic operators
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Proc. Amer. Math. Soc. 137 (2009), 123-134 Request permission

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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