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.
DOI: http://dx.doi.org/10.7900/jot.2017aug03.2194
Keywords: hypercyclic operator, common hypercyclic functions,
translation\break operator
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