Continuity of the norm of a composition operator
type: Item
instance of: scholarly article
title: Continuity of the norm of a composition operator
zbMATH Open document ID: 1053.47019
author: David B. Pokorny
author: Jonathan E. Shapiro
published in: Integral Equations and Operator Theory
publication date: 2003-12-08T00:00:00Z
review text: Let \(ASM({\mathbb{D}})^1\) (respectively, \(ASM({\mathbb{D}})^\infty\)) denote the set of all analytic self-maps of the unit disc, considered as a subset of \(H^1\) (respectively, \(H^\infty \)). Let \(N_i: ASM({\mathbb{D}})^i \to {\mathbb{R}}\) (\(i=1, \infty \)) be given by \(N_i(\varphi )= \| C_\varphi \|\), where \(C_\varphi \) is the composition operator. The authors study the continuity of these maps by showing that \(N_1\) is continuous at any \(\varphi\) which either is an inner map or \(\varphi (0)=0\), and that \(N_\infty \) is continuous at any \(\varphi\) which satisfies \(\varphi ({\mathbb{D}}) \subset r{\mathbb{D}}\) for some \(r<1\). In addition, the authors study similar questions with the operator norm replaced with either the essential norm \(\| \cdot \|_e\) or the Hilbert-Schmidt norm \(\| \cdot \|_{HS}\). They show that if the map \(\varphi \mapsto \| C_\varphi \|_e\) on \(ASM({\mathbb{D}})^\infty\) is continuous at \(\varphi \), then \(C_\varphi\) is compact; and the continuity of \(N_{HS} \) at any \(\varphi\) with \(\| \varphi \|_{\infty } <1\). Finally, the authors characterize when the norm of \(C_\varphi \) is minimal: when \(\varphi (0) \neq 0\), the lower bound \(\sqrt{\frac{1}{1-| \varphi (0)| ^2}}\) for \(\| C_\varphi \|\) is attained only when \(\varphi\) is constant.
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47A30
zbMATH DE Number: 1925526
zbMATH Keywords: composition operator
zbMATH Keywords: norm continuity
zbMATH Keywords: compactness
zbMATH Keywords: minimality
reviewed by: María del Carmen Calderón-Moreno
MaRDI profile type: Publication
full work available at URL: s000200300010
OpenAlex ID: W2074119690
DOI: 10.1007/S000200300010
description: scientific article; zbMATH DE number 1925526