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

MacCluer, Barbara D. ; Shapiro, Joel H.

Canadian journal of mathematics, Tome 38 (1986) no. 4, pp. 878-906

Résumé

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.

Détail

DOI : 10.4153/CJM-1986-043-4

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