### Abstract

Composition operators C_{φ} on the Hilbert Hardy space H^{2} over the unit disk are considered. We investigate when convergence of sequences {φ_{n}} of symbols, (i.e., of analytic selfmaps of the unit disk) towards a given symbol φ, implies the convergence of the induced composition operators, C_{φn} → C_{φ}. If the composition operators C_{φn} are Hilbert-Schmidt operators, we prove that convergence in the Hilbert-Schmidt norm, ∥C_{φn} - C_{φ}∥_{HS} → 0 takes place if and only if the following conditions are satisfied: ∥φ_{n} - φ∥_{2} → 0, ∫ 1/(1 - φ ^{2}) < ∞, and ∫ 1/(1 - φ_{n} ^{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 | |

Publication status | Published - May 15 2005 |

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### Keywords

- Composition operators
- Convergence

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics