In this paper, we discuss the hypercyclicity, supercyclicity and cyclicity of the adjoint of a weighted composition operator on a Hilbert space of analytic functions.
Let $E$ be a complex Banach space, with the unit ball $B_E$. We study the spectrum of a bounded weighted composition operator $uC_{\varphi }$ on $H^\infty (B_E)$ determined by an analytic symbol $\varphi $ with a fixed point in $B_E$ such that $\varphi (B_E)$ is a relatively compact subset of $E$, where $u$ is an analytic function on $B_E$.