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Journal of Operator Theory

Volume 80, Issue 2,  Fall  2018  pp. 257-294.

Existence of common hypercyclic vectors for translation operators

Authors:  Nikos Tsirivas
Author institution: University College Dublin, School of Mathematical Sicences, Belfield, Dublin 4, Dublin, Ireland and Department of Mathematics, University of Ioannina, P.C. 45110, Panepistimiopolis Ioannina, Greece

Summary:  Let $\mathcal{H}(\mathbb{C})$ be the set of entire functions endowed with the topology $\mathcal{T}_\mathrm u$ of local uniform convergence. Fix a sequence of non-zero complex numbers $(\lambda_n)$ with $|\lambda_n|\!\!\to\!\! +\infty$ and $|\lambda_{n+1}|/|\lambda_n|\!\!\to\!\! 1$. We prove that there exists a residual set $G\!\!\subset\!\! \mathcal{H}(\mathbb{C})$ so that for every $f\!\!\in\!\! G$ and every non-zero complex number $a$ the set $\{ f(z\!\!+\!\!\lambda_na):n\!\!=\!\!1,2,\ldots \}$ is dense in $(\mathcal{H}(\mathbb{C}),\mathcal{T}_\mathrm u)$. This provides a very strong extension of a theorem by G.~Costakis and M.~Sambarino in \textit{Adv. Math.} \textbf{182}(2004), 278--306. Actually, in that article, the above result is proved only for the case $\lambda_n\!\!=\!\!n$. Our result is in a sense best possible, since there exist sequences $( \lambda_n )$, with $|\lambda_{n+1}|/|\lambda_n| \!\!\to\!\! l$ for certain $l\!\!>\!\!1$, for which the above result fails to hold, cf.\ F.~Bayart, \textit{Int. Math. Res. Notices} \textbf{21}(2016), 6512--6552.

Keywords:  hypercyclic operator, common hypercyclic functions, translation\break operator

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