On Shapiro's Compactness Criterion for Composition Operators

Abstract

For any analytic self-map $\psi$ of $\{z: |z| < 1\}$, J. H. Shapiro has established that the square of the essential norm of the composition operator $C_\psi$ on the Hardy Space $H^2$ is precisely $\limsup_{|w|\rightarrow 1^-}N_\psi(w)/(1-|w|)$; where $N_\psi$ is the Nevanlinna counting function for $\psi$. In this paper we show that this quantity is equal to $\limsup_{|a|\rightarrow 1^-}(1 - |a|^2)||1/(1 - \bar{a}\psi)||_{H^2}^2.$ This alternative expression provides a link between the one given by Shapiro and earlier measure-theoretic notions. Applications are given.