Composition operators on weighted Bergman spaces of a half plane

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 $\lambda^{(2+\alpha)/2}$.