Mixing convolution operators on spaces of entire functions

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 $E=F^{\prime}_c,$ 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.

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