Composition operators on weighted Bergman spaces of a half plane

TL;DR

This paper characterizes when composition operators on weighted Bergman spaces of the right half-plane are bounded, showing they are bounded if the inducing map fixes infinity non-tangentially with a finite angular derivative, and provides explicit formulas for their norms and spectral radius.

Contribution

It establishes necessary and sufficient conditions for boundedness of composition operators on weighted Bergman spaces of a half-plane, including explicit norm and spectral radius formulas.

Findings

Boundedness characterized by fixed point and finite angular derivative.

Norm, essential norm, and spectral radius are equal and given by a specific formula.

Provides a complete spectral analysis of these operators.

Abstract

We use induction and interpolation techniques to prove that a composition operator induced by a map $\phi$ is bounded on the weighted Bergman space $\A^2_\alpha(\mathbb{H})$ of the right half-plane if and only if $\phi$ fixes $\infty$ non-tangentially, and has a finite angular derivative $\lambda$ there. We further prove that in this case the norm, essential norm, and spectral radius of the operator are all equal, and given by $\lambda^{(2+\alpha)/2}$.