Composition operators on Hardy spaces of a half plane

TL;DR

This paper characterizes when composition operators are bounded on Hardy spaces of the right half-plane, linking boundedness to the behavior of the inducing map at infinity and providing explicit formulas for operator norms and spectral radius.

Contribution

It establishes a precise criterion for boundedness of composition operators on Hardy spaces of a half-plane based on the inducing map's behavior at infinity.

Findings

Boundedness of composition operators is equivalent to the inducing map fixing infinity non-tangentially.

The norm, essential norm, and spectral radius of such operators are all equal to the square root of the angular derivative at infinity.

Provides explicit formulas connecting the operator's spectral properties to the inducing map's behavior.

Abstract

We prove that a composition operator is bounded on the Hardy space $H^2$ of the right half-plane if and only if the inducing map fixes the point at infinity non-tangentially, and has a finite angular derivative $\lambda$ there. In this case the norm, essential norm, and spectral radius of the operator are all equal to $\sqrt{\lambda}$.