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.  2004 Elsevier Inc. All rights reserved