Remarks on common hypercyclic vectors

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 $\{aT+bI:(a,b)\in \mathrm{\Lambda }\}$ 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 $\text{D}=\{z\in \text{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ and $\phi\in \H^\infty(\D)$isnon-constant,thenthefamily${zM_\phi^\star:b^{-1}<|z|<a^{-1}}$hasacommonhypercyclicvector,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 $\{a{T}_b:a,b\in \text{C}\setminus \{0\}\}$ has a common hypercyclic vector, where $T_{b}f(z)=f(z-b)$ acts on the Fr'echet space $\stackrel{˝}{(}\text{C})$ of entire functions on one complex variable.