Year: 2010
Type: preprint
Abstract: For any analytic self-map $ψ$ of $\{z: |z| < 1\}$, J. H. Shapiro has established that the square of the essential norm of the composition operator $C_ψ$ on the Hardy Space $H^2$ is precisely $\limsup_{|w|\rightarrow 1^-}N_ψ(w)/(1-|w|)$; where $N_ψ$ is the Nevanlinna counting function for $ψ$. In this paper we show that this quantity is equal to $\limsup_{|a|\rightarrow 1^-}(1 - |a|^2)||1/(1 - \bar{a}ψ)||_{H^2}^2.$ This alternative expression provides a link between the one given by Shapiro and earlier measure-theoretic notions. Applications are given.
Source: arXiv (Cornell University)
Author: Akeroyd, John
Institution: University of Arkansas at Fayetteville
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2032696397