Convergent sequences of composition operators

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 φ n → C φ . If the composition operators C φ n are Hilbert–Schmidt operators, we prove that convergence in the Hilbert–Schmidt norm, ‖ C φ n − 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.