J-class sequences of linear operators

In this paper we first introduce the extended limit set $J_{\{T^n\}}(x)$ for a sequence of bounded linear operators $\{T_n{\}}_{n=1}^{\mathrm{\infty }}$ on a separable Banach space $X$ . Then we study the dynamics of sequence of linear operators by using the extended limit set. It is shown that the extended limit set is strongly related to the topologically transitive of a sequence of linear operators. Finally we show that a sequence of operators $\{T_n{\}}_{n=1}^{\mathrm{\infty }}\subseteq \mathcal{B}(X)$ is hypercyclic if and only if there exists a cyclic vector $x\in X$ such that $J_{\{T^n\}}(x)=X$.