  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.
