### 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.

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.

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