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 $$T^n (U)\cap U\ne \emptyset$$ . 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 $$\mathcal {S}_\nu$$ ; in particular on the Bergman space, the Hardy space, and the Dirichlet space. Consequently, we complete previous work of Costakis, Manoussos, and Parissis on the recurrence of linear fractional composition operators on Hardy space. In this manner, we determine the triples $$(\lambda ,\nu ,\phi )\in {\mathbb {C}}\times \mathbb {R}\times \mathrm{LFM}(\mathbb {D})$$ for which the scalar multiple of composition operator $$\lambda C_\phi$$ acting on $$\mathcal {S}_\nu$$ fails to be recurrent.