Recurrence of multiples of composition operators on weighted Dirichlet spaces

Abstract

A bounded linear operator $T$ acting on a Hilbert space $\mathcal{H}$ is said to be recurrent if for every non-empty open subset $U\subset \mathcal{H}$ there is an integer $n$ such that $T^n (U)\cap U\neq\emptyset$. In this paper, we completely characterize the recurrence of scalar multiples of composition operators, induced by linear fractional self maps of the unit disk, acting on weighted Dirichlet spaces $S_\nu$; in particular on the Bergman space, the Hardy space, and the Dirichlet space. Consequently, we complete a previous work of Costakis et al. \cite{costakis} on recurrence of linear fractional composition operators on Hardy space. In this manner, we determine the triples $(\lambda,\nu,\phi)\in \mathbb{C}\times \mathbb{R}\times LFM(\mathbb{D})$ for which the scalar multiple of composition operator $\lambda C_\phi$ acting on $S_\nu$ fails to be recurrent.