Composition operators with closed range for smooth injective symbols R→Rd

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: F ∈ E (Rd )} for a smooth injective symbol ψ : R→Rd 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). This yields a characterization of A(ψ) =A(ψ). © 2011 Elsevier Inc. All rights reserved.