Convergent sequences of composition operators

Composition operators Cφ on the Hilbert Hardy space H 2 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.  2004 Elsevier Inc. All rights reserved.