Closed range multiplication operators on weighted Bergman spaces

Let ψ be a bounded analytic function on a simply connected domain Ω⊆C. For a large family of weights we characterize when a pointwise multiplication operator Mψ, Mψ(f)(z)=ψ(z)f(z), defined on a weighted Bergman space Awp(Ω) on Ω has closed range. In particular, the result holds for weights w(z)=ξ(d(z,∂S)), ξ:R+→R+,ξ⩾0, defined on a strip S or weights w(z)=(Rez)α,α>-1p, defined on a right half plane.