Convergent sequences of composition operators

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Abstract

Composition operators Cφ on the Hilbert Hardy space H2 over the unit disk are considered. We investigate when convergence of sequences {φn} of symbols, (i.e., of analytic selfmaps of the unit disk) towards a given symbol φ, implies the convergence of the induced composition operators, Cφn → Cφ. If the composition operators Cφn are Hilbert-Schmidt operators, we prove that convergence in the Hilbert-Schmidt norm, ∥Cφn - CφHS → 0 takes place if and only if the following conditions are satisfied: ∥φn - φ∥2 → 0, ∫ 1/(1 - φ 2) < ∞, and ∫ 1/(1 - φn 2) → ∫ 1/(1 - φ 2). The convergence of the sequence of powers of a composition operator is studied.

Original languageEnglish (US)
Pages (from-to)659-668
Number of pages10
JournalJournal of Mathematical Analysis and Applications
Volume305
Issue number2
DOIs
Publication statusPublished - May 15 2005

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Keywords

  • Composition operators
  • Convergence

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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