مشخصات پژوهش

A bounded linear operator $T$ on a separable Hilbert space $\mathcal{H}$ is called \emph{hypercyclic} if there exists a vector $x \in \mathcal{H}$ whose orbit $\{T^n x : n \in \mathbb{N}\}$ is dense in $\mathcal{H}$. In this paper, we characterize the hypercyclicity of the weighted composition operators $C_{u, \varphi}$ on $\ell^2(\mathbb{Z})$ in terms of their weight functions and symbols. First, a necessary and sufficient condition is given for $C_{u, \varphi}$ to be hypercyclic. Then it is shown that the finite direct sums of the hypercyclic weighted composition operators are also hypercyclic. In particular, we conclude that the class of the hypercyclic weighted composition operators is weakly mixing. Finally, several examples are presented to illustrate the hypercyclicity of the weighted composition operators.