When do finite Blaschke products commute?
type: Item
instance of: scholarly article
title: When do finite Blaschke products commute?
zbMATH Open document ID: 1082.30024
DOI: 10.1017/S0004972700039861
author: Raymond Mortini
author: Isabelle Chalendar
published in: Bulletin of the Australian Mathematical Society
publication date: 2001-01-01T00:00:00Z
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 1679517
zbMATH Keywords: Blaschke products
zbMATH Keywords: composition operators
MaRDI profile type: Publication
cites work: Commuting Analytic Functions
cites work: The group of the invariants of a finite blaschke product
cites work: Commuting Analytic Functions Without Fixed Points
cites work: Commuting finite Blaschke products
cites work: Composition operators and classical function theory
cites work: A decomposition of finite blaschke products
cites work: Commuting endomorphisms of the circle
cites work: Absolutely continuous conjugacies of Blaschke products
cites work: Absolutely Continuous Conjugacies of Blaschke Products II
cites work: Absolutely continuous conjugacies of Blaschke products. III
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review text: In this paper, commuting pairs of finite Blaschke products are studied: if \(u\) is an automorphism of the unit disk \(\mathbb{D}\), and \(T_{u}: f \mapsto f\circ u\) is the composition operator associated to \(u\), then it is possible to describe completely which finite Blaschke products are eigenvectors of \(T_{u}\) (Proposition \(2.4\)). Then pairs of finite Blaschke products \((B,C)\) such that \(B\circ C=C\circ B\) are studied, which leads to counterexamples to several conjectures stated in [\textit{C. Cowen}, ``Commuting analytic functions'', Trans. Am. Math. Soc. 283, 685--695 (1984; Zbl 0542.30030)].
reviewed by: Sophie Grivaux
description: scientific article; zbMATH DE number 1679517