Composition Operators on Dirichlet Spaces over the Half-plane

TL;DR

This paper extends the study of composition operators on Dirichlet spaces from the unit disk to the unbounded half-plane, providing rational approximation results and characterizations of operators with dense range, and exploring connections with Hardy spaces.

Contribution

It introduces the analysis of composition operators on Dirichlet spaces over the half-plane, including rational approximation and operator characterization, advancing the understanding beyond bounded domains.

Findings

Established rational approximation in Dirichlet spaces over the half-plane

Characterized composition operators with dense range on these spaces

Explored the relationship between Dirichlet and Hardy spaces on 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.