Convergent sequences of composition operators

Abstract. Composition operators Cϕ on the Hilbert Hardy spaceH2 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. 1