Chaos in convolution operators on the space of entire functions of infinitely many complex variables

A classical result of Godefroy and Shapiro states that every nontrivial convolution operator on the space $\mathcal{H}({\mathbb{C}}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with this result F'avaro and Mujica show that no translation operator on the space $\mathcal{H}({\mathbb{C}}^{\mathbb{N}})$ of entire functions of infinitely many complex variables is hypercyclic. In this work we study the linear dynamics of convolution operators on $\mathcal{H}({\mathbb{C}}^{\mathbb{N}})$. First we show that no convolution operator on $\mathcal{H}({\mathbb{C}}^{\mathbb{N}})$ is neither cyclic nor $n$-supercyclic for any positive integer $n$. After we study the notion of Li–Yorke chaos in non-metrizable topological vector spaces and we show that every nontrivial convolution operator on $\mathcal{H}({\mathbb{C}}^{\mathbb{N}})$ is Li–Yorke chaotic.