Mixing convolution operators on spaces of entire functions

Abstract

We show that if $E$ is an arbitrary $(DFN)$-space, then every nontrivial convolution operator on the Fréchet nuclear space $\mathcal{H}(E)$ is mixing, in particular hypercyclic. More generally we obtain the same conclusion when $E=F_c^{\prime },$ where $F$ is a separable Fréchet space with the approximation property. On the opposite direction we show that a translation operator on the space $\mathcal{H}({\mathbb{C}}^{\mathbb{N}})$ is never hypercyclic.

17 pages