Recurrence of multiples of composition operators on weighted Dirichlet spaces

TL;DR

This paper characterizes when scalar multiples of composition operators induced by linear fractional maps are recurrent on weighted Dirichlet spaces, completing previous work on Hardy spaces and identifying specific conditions for recurrence.

Contribution

It provides a complete characterization of recurrence for scalar multiples of composition operators on weighted Dirichlet spaces, extending prior results on Hardy spaces.

Findings

Characterization of recurrence for scalar multiples of composition operators

Identification of conditions on parameters for non-recurrence

Extension of previous Hardy space results to 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.