On hypercyclicity of weighted composition operators on Stein manifolds

Abstract

In this manuscript, we study the hypercyclicity of weighted composition operators defined on the set of holomorphic complex functions on a connected Stein $n$-manifold $\M$. We show that a weighted composition operator $\K_{\psi, \omega}$ (associated to a holomorphic self-map $\psi$ and a holomorphic function $\omega$ on $\M$) is hypercyclic with respect to an increasing sequence $(n_{l})_{l}$ of natural numbers if and only if at every $p \in \M$ we have $\omega(p) \neq 0$ and the self-map $\psi$ is injective without any fixed points in $\M$, $\psi(\M)$ is a Runge domain and for every $\M$-convex compact subset $C \subset \M$ there is a positive integer number $k$ such that the sets $C$ and $\psi^{[n_{k}]}(C)$ are separable in $\M$.