Hypercyclic and mixing operator semigroups

Abstract

We describe a class of topological vector spaces admitting a mixing uniformly continuous operator group$\smash{\{T_t\}_{t\in\mathbb{C}^n}}$with holomorphic dependence on the parametert. This result builds on those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups$\smash{\{T_t\}_{t\geq0}}$.

A review of some recent work on hypercyclicity

“…This result implies the result of Bayart and Matheron [15] that for every hypercyclic operator T on the countable product of copies of K = C or R, we have also that T ⊕ T is hypercyclic. Further, Shkarin [105] described a class of topological vector spaces admitting a mixing uniformly continuous operator group {T t } t∈C n with holomorphic dependence on the parameter t, and a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups {T t } t≥0 .…”

Strongly continuous semigroups on some Fréchet spaces

“…This question arose in the context of hypercyclicity: It is shown in [Con07,Theorem 2.7] that no such semigroup on ω can be hypercyclic. Although Shkarin [Shk11] proved that there are not even supercyclic strongly continuous semigroups on ω, the question of Conejero remained open. Only a partial answer is contained in [ABR10].…”

On normed semigroups

“…Here is yet another definition of a normed semigroup (see, for example, [11]): "Let A be an abelian monoid. A norm on A is a function | • | : A → [0, ∞) satisfying the conditions |na| = n|a| and |a + b| ≤ |a| + |b| for any positive integer n and a, b ∈ A.…”