Convergent sequences of composition operators |
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Authors: | Valentin Matache |
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Affiliation: | Department of Mathematics, University of Nebraska, Omaha, NE 68182-0243, USA |
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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/(1−2|φ|)<∞, and ∫1/(1−2|φn|)→∫1/(1−2|φ|). The convergence of the sequence of powers of a composition operator is studied. |
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Keywords: | Composition operators Convergence |
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