Mixing convolution operators on spaces of entire functions

TL;DR

This paper investigates the dynamical properties of convolution operators on spaces of entire functions, establishing conditions under which these operators are mixing or hypercyclic, and providing counterexamples.

Contribution

It proves that convolution operators on certain nuclear spaces are mixing, and identifies cases where translation operators are not hypercyclic, expanding understanding of operator dynamics.

Findings

Convolution operators on $(DFN)$-spaces are mixing.

On $ ext{Hol}(E)$, nontrivial convolution operators are hypercyclic.

Translation operators on $ ext{Hol}( extbf{C}^ extbf{N})$ are not hypercyclic.

Abstract

We show that if $E$ is an arbitrary $(DFN)$-space, then every nontrivial convolution operator on the Fr\'echet 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 Fr\'echet 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.