Remarks on common hypercyclic vectors

Stanislav Shkarin

Published 2012-09-06Version 1

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator $T$ on a complex Fr\'echet space $X$ and a set $\Lambda\subseteq \R_+\times\C$ which is not of zero three-dimensional Lebesgue measure, the family $\{aT+bI:(a,b)\in\Lambda\}$ has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fr\'echet space to have a common hypercyclic vector. It allows to show that if $\D=\{z\in\C:|z|<1\}$ and $\phi\in \H^\infty(\D)$ is non-constant, then the family $\{zM_\phi^\star:b^{-1}<|z|<a^{-1}\}$ has a common hypercyclic vector, where $M_\phi:\H^2(\D)\to \H^2(\D)$, $M_\phi f=\phi f$, $a=\inf\{|\phi(z)|:z\in\D\}$ and $b=\sup\{|\phi(z)|:|z|\in\D\}$, providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family $\{aT_b:a,b\in\C\setminus\{0\}\}$ has a common hypercyclic vector, where $T_bf(z)=f(z-b)$ acts on the Fr\'echet space $\H(\C)$ of entire functions on one complex variable.