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 - Cφ∥HS → 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 language | English (US) |
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Pages (from-to) | 659-668 |
Number of pages | 10 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 305 |
Issue number | 2 |
DOIs | |
State | Published - May 15 2005 |
Keywords
- Composition operators
- Convergence
ASJC Scopus subject areas
- Analysis
- Applied Mathematics