Closed range multiplication operators on weighted Bergman spaces

Abstract

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 M ψ , M ψ ( f ) ( z ) = ψ ( z ) f ( z ) , defined on a weighted Bergman space A w p ( 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 ) = ( Re z ) α , α > - 1 p , defined on a right half plane.