Remarks on common hypercyclic vectors

TL;DR

This paper investigates the existence of common hypercyclic vectors for families of linear operators, providing negative results for certain parameter sets and positive results for others, including scalar multiples and translation operators.

Contribution

It establishes conditions under which families of operators lack or possess common hypercyclic vectors, answering open questions and extending previous results in the field.

Findings

Families with parameters of zero three-dimensional Lebesgue measure have no common hypercyclic vectors.

Certain scalar multiple families of operators do have common hypercyclic vectors.

Translation families on entire functions space possess common hypercyclic vectors.

Abstract

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.