Components of linear-fractional composition operators.
type: Item
instance of: scholarly article
title: Components of linear-fractional composition operators.
zbMATH Open document ID: 1043.47021
DOI: 10.1016/S0022-247X(03)00004-0
author: Paul S. Bourdon
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: Let \(\phi\) denote a holomorphic function on the open unit disc \(U\) in the complex plane with \(\phi(U) \subset U\). The composition operator \(C_{\phi}(f):=f \circ \phi\) is bounded in the Hardy space \(H^2\). Denote by \(\text{comp} (H^2)\) the collection of all composition operators on \(H^2\) endowed with the metric induced by the operator norm. \textit{J. H. Shapiro} and \textit{C. Sundberg} [Pac. J. Math. 145, 117--121 (1990; Zbl 0732.30027)], improving work by Berkson, studied isolated points in \(\text{comp} (H^2)\) and showed, e.g., that only extreme points of \(H^{\infty}(U)\) can induce isolated composition operators. They also conjectured that if two composition operators \(C_{\phi}\) and \(C_{\psi}\) belong to the same component of \(\text{comp} (H^2)\), then \(C_{\phi} - C_{\psi}\) is compact. Two analytic self-maps \(\phi\) and \(\psi\) are said to have the same first order boundary data if, whenever \(\phi\) or \(\psi\) has a finite angular derivative at \(\zeta\) in the boundary of \(U\), then \(\phi\) and \(\psi\) have the same radial limit and the same angular derivative at \(\zeta\). \textit{B. D. MacCluer} [Integral Equations Oper. Theory 12, 725--738 (1989; Zbl 0685.47027)] showed that if \(C_{\phi}\) and \(C_{\psi}\) belong to the same component of \(\text{comp} (H^2)\), then \(\phi\) and \(\psi\) have the same first order boundary data. The author proves the converse, if both \(\phi\) and \(\psi\) are linear fractional, and even that \(C_{\phi}\) and \(C_{\psi}\) can be joined by a continuous path in \(\text{comp} (H^2)\). In the final section, the author proves that, for analytic self-maps \(\phi\) and \(\psi\) which extend to be \(C^2\) on the closed unit disc, a necessary condition for \(C_{\phi}- C_{\psi}\) to be compact is that \(\phi\) and \(\psi\) have the same second order boundary data, thus obtaining an example disproving the conjecture of Shapiro and Sundberg. He also shows that the component of a non-automorphic, linear fractio…
reviewed by: José Bonet
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 1915745
zbMATH Keywords: composition operator
zbMATH Keywords: Hardy space
zbMATH Keywords: linear fractional
zbMATH Keywords: first order boundary data
MaRDI profile type: Publication
cites work: Composition Operators Isolated in the Uniform Operator Topology
cites work: Cyclic phenomena for composition operators
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cites work: Isolation amongst the composition operators
cites work: Composition operators and classical function theory
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description: scientific article; zbMATH DE number 1915745