Convergent sequences of composition operators

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.