Some versions of supercyclicity for a set of operators

Abstract

Let $X$ be a complex topological vector space and $\mathcal{B}(X)$ the algebra of all bounded linear operators on $X.$
An operator $T\in  \mathcal{B}(X)$ is supercyclic if there is $x\in X$ such that;
$\mathbb{C}Orb(T,x)=\{\alpha T^{n}x\mbox{ : }\alpha\in\mathbb{C}\mbox{, } n\geq0\},$
is dense in $X.$
In this paper, we extend this notion from a single operator $T\in  \mathcal{B}(X)$ to a subset of operators
$\Gamma \subseteq \mathcal{B}(X).$
We prove that most of related proprieties to supercyclicity in the case of a single operator $T$
remains true for subset of operators $\Gamma.$
This leads us to obtain some results for
$C$-regularized groups of operators.