On Linear Dynamics of Sets of Operators

TL;DR

This paper extends the concept of hypercyclicity to subsets of operators on complex topological vector spaces, providing new criteria and applications for regularized groups of operators.

Contribution

It introduces the hypercyclic criterion for subsets of operators and generalizes known results from single operators and semigroups to these subsets.

Findings

Extended hypercyclicity concepts to subsets of operators

Developed hypercyclic criterion for operator subsets

Applied results to C-regularized groups of operators

Abstract

Let $X$ be a complex topological vector space with $dim(X)>1$ and $\mathcal{B}(X)$ the set of all continuous linear operators on $X$. The concept of hypercyclicity for a subset of $\mathcal{B}(X)$, was introduced in \cite{AKH}. In this work, we introduce the notion of hypercyclic criterion for a subset of $\mathcal{B}(X)$. We extend some results known for a single operator and $C_0$-semigroup to a subset of $\mathcal{B}(X)$ and we give applications for $C$-regularized groups of operators.