Surjective and closed range differentiation operator

Abstract

We identify Fock-type spaces F(m,p) on which the differentiation operator D has closed range. We prove that D has closed range only if it is surjective, and this happens if and only if m = 1. Moreover, since the operator is unbounded on the classical Fock spaces, we consider the modified or the weighted composition–differentiation operator, D(u,ψ,n) f = u · f (n) ◦ ψ , on these spaces and describe conditions under which the operator admits closed range, surjective, and order bounded structures.