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Angular Derivatives and Compact Composition Operators on the Hardy and Bergman Spaces

Published online by Cambridge University Press:  20 November 2018

Barbara D. MacCluer  and
Joel H. Shapiro
Barbara D. MacCluer
Affiliation:
University of Virginia, Charlottesville, Virginia
Joel H. Shapiro
Affiliation:
Michigan State University, East Lansing, Michigan
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Let U denote the open unit disc of the complex plane, and φ a holomorphic function taking U into itself. In this paper we study the linear composition operator Cφ defined by Cφf = f º φ for f holomorphic on U. Our goal is to determine, in terms of geometric properties of φ, when Cφ is a compact operator on the Hardy and Bergman spaces of φ. For Bergman spaces we solve the problem completely in terms of the angular derivative of φ, and for a slightly restricted class of φ (which includes the univalent ones) we obtain the same solution for the Hardy spaces Hp (0 < p < ∞). We are able to use these results to provide interesting new examples and to give unified explanations of some previously discovered phenomena.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 4 , 01 August 1986 , pp. 878 - 906
Copyright
Copyright © Canadian Mathematical Society 1986

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