Weighted holomorphic spaces with trivial closed range multiplication operators

Abstract

We deal with the space H∞ v consisting of those analytic functions f on the unit disc D such that ||f|| v := sup z ∈ D v(z)|f(z)| < ∞, with v(z) = v(|z|). We determine the critical rate of decay of v such that the pointwise multiplication operator M φ , M φ (f)(z) = φ(z)f(z) and φ analytic, has closed range in H∞ v only in the trivial case that φ is the product of an invertible function in H∞ and a finite Blaschke product.