Syndetically Hypercyclic Operators

Abstract.

Given a continuous linear operator T ∈ L(x) defined on a separable $\mathcal{F}$ -space X, we will show that T satisfies the Hypercyclicity Criterion if and only if for any strictly increasing sequence of positive integers $\{ n_k \} _k$ such that $\sup _k \{ n_{k + 1} - n_k \} < \infty ,$ the sequence $\{ T^{n_k } \} _k$ is hypercyclic. In contrast we will also prove that, for any hypercyclic vector x ∈ X of T, there exists a strictly increasing sequence $\{ n_k \} _k$ such that $\sup _k \{ n_{k + 1} - n_k \} = 2$ and $\{ T^{n_k } x\} _k$ is somewhere dense, but not dense in X. That is, T and $\{ T^{n_k } \} _k$ do not share the same hypercyclic vectors.