Composition operators with closed range for smooth injective symbols R{double-strcuk}→R{double-strcuk}~d

摘要

In 1998, Allan, Kakiko, O'Farrell, and Watson proved a description of the closure (with respect to the uniform convergence of all derivatives on compact sets) of A(Ψ)={F°Ψ:Fψ({double-strcuk}d)} for a smooth injective symbol Ψ:R→R~d in terms of formal Taylor series. In that article it was conjectured that A(Ψ) is closed if Ψ is proper and has only critical points of finite order. In the present paper we first give a simple counterexample and then rectify the conjecture by adding a geometrical property for the curve Ψ(R{double-strcuk}). This yields a characterization of A(Ψ)-=A(Ψ).