Cyclicity of bicyclic operators and completeness of translates

Abstract

We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if ${S: (a_{n})_{n\in \mathbb Z}\longmapsto (a_{n-1})_{n\in \mathbb Z}}$ is the shift operator acting on the weighted space of sequences ${\ell_{\omega }^{2}(\mathbb{Z})}$, if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if ${{\rm lim}_{n\rightarrow +\infty}{\rm log}\omega(-n)/\sqrt{n}=0}$. On the other hand one can see that S is not cyclic if the series ${\sum_{n\geq 1} {\rm log}{\omega (-n)}/n^{2}}$ diverges. We show that the question of Herrero whether either S or S* is cyclic on ${\ell_{\omega }^{2}(\mathbb Z)}$ admits a positive answer when the series ${\sum_{n\in\mathbb Z} {\rm log} ||S^{n}||/(n^{2}+1)}$ is convergent. We also prove completeness results for translates in certain Banach spaces of functions on ${\mathbb R}$.