On the spectrum of frequently hypercyclic operators

A bounded linear operator T on a Banach space X  is called frequently hypercyclic if there exists x ∈ X  such that the lower density of the set { n ∈ N : Tⁿx ∈ 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.