Linear dynamics of operators on non-metrizable topological vector spaces

In this talk we will explore the following notions of chaos for operators on non-metrizable topological vector spaces: mixing, hypercyclicity, cyclicity, n-supercyclicity, Li-Yorke chaos and Devaney chaos. We will discuss the main differences between these notions for operators on Fréchet and on non-metrizable topological vector spaces.

To illustrate some of these differences, we will explore results about the linear dynamics of convolution operators on spaces of entire functions of finitely and infinitely many complex variables. A classical result due to Godefroy and Shapiro states that every nontrivial convolution operator on the Fréchet space $\mathcal{H}(\mathbb{C}^n)$ of all entire functions of $n$ complex variables is hypercyclic. In sharp contrast to this result, in a joint work with J. Mujica it was showed that no translation operator on the space $\mathcal{H}(\mathbb{C}^\mathbb{N})$ (which is a complete non-metrizable locally convex space) of entire functions of infinitely many complex variables is hypercyclic. In 2020, in a joint work with B. Caraballo, it was showed that no convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is cyclic or $n$-supercyclic for any positive integer $n$. In the opposite direction, it was proved that every nontrivial convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is mixing. Exploring the concept of Li-Yorke chaos on non-metrizable topological vector spaces, it was also proved that nontrivial convolution operators on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ are Li-Yorke chaotic. Recently, we also proved that every nontrivial convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ is (Devaney) chaotic.