IUMJ

Title: A new class of frequently hypercyclic operators

Authors: Sophie Grivaux

Issue: Volume 60 (2011), Issue 4, 1177-1202

Abstract:

We study in this paper a hypercyclicity property of linear dynamical systems: a bounded linear operator $T$ acting on a separable infinite-dimensional Banach space $X$ is said to be \emph{hypercyclic} if there exists a vector $x in X$ such that $\{T^nx; n geq 0\}$ is dense in $X$, and \emph{frequently hypercyclic} if there exists $x in X$ such that for any non-empty open subset $U$ of $X$, the set $\{n \geq 0; T^nx \in U\}$ has positive lower density. We prove in this paper that if $T \in \mathcal{B}(X)$ is an operator which has "sufficiently many" eigenvectors associated to eigenvalues of modulus $1$ in the sense that these eigenvectors are perfectly spanning, then $T$ is automatically frequently hypercyclic.