, Volume 51, Issue 2, pp 275-281

Syndetically Hypercyclic Operators

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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.