On the spectrum of frequently hypercyclic operators

TL;DR

This paper investigates the spectral properties of frequently hypercyclic operators on Banach spaces, proving they cannot have isolated spectrum points, which implies such operators do not exist on certain complex and hereditarily indecomposable spaces.

Contribution

It establishes that frequently hypercyclic operators have spectra without isolated points, answering a long-standing open question negatively.

Findings

Spectrum of frequently hypercyclic operators has no isolated points

No such operators exist on certain hereditarily indecomposable Banach spaces

Provides a negative answer to the existence question on all separable infinite-dimensional spaces

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.