This is an announcement for the paper "On the spectrum of frequently hypercyclic operators" by Stanislav Shkarin.

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 ${n\in\N:T^nx\in U}$ 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.

Archive classification: math.FA math.DS

Mathematics Subject Classification: 47A16, 37A25

Citation: Proc. AMS 137 (2009), 123-134

Submitted from: s.shkarin@qub.ac.uk

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1209.1221

or

http://arXiv.org/abs/1209.1221