Composition Operators on Dirichlet Spaces over the Half-plane

Abstract

As continuation of the study of polynomial approximation and composition operators on Dirichlet spaces of unit disk, which has settled a problem posed by Cima in 1976, the present paper aims to consider the case of the unbounded domains, such as the half-plane. Specifically, we may obtain the rational approximations in the Dirichlet spaces and characterize the composition operators which has dense range on the Dirichlet spaces over the half-plane. Moreover, this paper also considers the relationship between the Dirichlet spaces and Hardy spaces on half-plane.Problem 1.1. Characterize those ϕ in the unit ball of H ∞ for which C ϕ has dense range.Cima indicated that if C ϕ has dense range, then for ψ(z) = rϕ(z), 0 < r < 1, C ψ has dense range. He said that no conjecture on this question. This is an old 2010 Mathematics Subject Classification. 30H25.