Hypercyclic and mixing operator semigroups

TL;DR

This paper characterizes classes of topological vector spaces that admit mixing operator groups with holomorphic dependence, and also identifies spaces that do not support supercyclic strongly continuous semigroups, advancing understanding of operator dynamics.

Contribution

It introduces new classes of topological vector spaces with specific operator semigroup properties and clarifies conditions preventing supercyclic semigroups, extending existing literature.

Findings

Existence of mixing operator groups with holomorphic dependence in certain spaces

Non-existence of supercyclic strongly continuous semigroups in other spaces

Generalization of previous results on operator semigroup dynamics

Abstract

We describe a class of topological vector spaces admitting a mixing uniformly continuous operator group ${T_t}_{t\in\C^n}$ with holomorphic dependence on the parameter $t$. This result covers those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups ${T_t}_{t\geq 0}$.