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 ${\mathrm{limsup}}_{|w|\to 1^{-}}{N}_{\psi }(w)/(1-|w|)$; where $N_{\psi }$ is the Nevanlinna counting function for $\psi$. In this paper we show that this quantity is equal to ${\mathrm{limsup}}_{|a|\to 1^{-}}(1-|a{|}^2)||1/(1-\bar{a}\psi )|{|}_{H^2}^2.$ This alternative expression provides a link between the one given by Shapiro and earlier measure-theoretic notions. Applications are given.

Cited by in corpus (1)

• Decay rates for approximation numbers of composition operators