The numerical ranges of automorphic composition operators
type: Item
instance of: scholarly article
title: The numerical ranges of automorphic composition operators
zbMATH Open document ID: 1048.47014
author: Paul S. Bourdon
author: Joel H. Shapiro
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-10-11T00:00:00Z
review text: Let \(H^2\) be the Hardy space of functions holomorphic on the unit open disc \(\mathbb{U}\). Let \(\varphi\) be a holomorphic self-map of \(\mathbb{U}\). Let \(C_\varphi\) be the bounded composition operator induced by \(\varphi\), the numerical range of \(C_\varphi\) is denoted by \(W(C_\varphi)\). Some important results established in this paper include the following: Result 1: Suppose \(\varphi\) is a conformal automorphism of \(\mathbb{V}\) that is either parabolic or hyperbolic. Then \(W(C_\varphi)\) is a disc centered at the origin. Result 2: If \(\varphi\) is a canonical hyperbolic automorphism of \(\mathbb{U}\) with dilation parameter \(0< r< 1\), then \(W(C_\varphi)\) is the open disc of radius \(1/\sqrt{r}\) centered at the origin. Result 3: Suppose that \(\varphi\) is an elliptic automorphism of \(\mathbb{U}\) with rotation parameter \(\omega\). If \(\omega\) is not a root of unity, then \(\overline W(C_\varphi)\) is a disc centered at the origin.
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47A12
zbMATH DE Number: 1556947
zbMATH Keywords: composition operators
zbMATH Keywords: numerical range
zbMATH Keywords: parabolic, elliptic and hyperbolic automorphisms
reviewed by: K. Chandrasekhara Rao
MaRDI profile type: Publication
OpenAlex ID: W2031187999
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DOI: 10.1006/JMAA.2000.7072
description: scientific article; zbMATH DE number 1556947