On the Closed Range Weighted Composition–Differentiation Operator

We provide a number of conditions which characterize the closed range weighted composition–differentiation operator $$D_{(u,\psi ,n)}$$ D ( u , ψ , n ) on Fock spaces. We, in particular, prove $$D_{(u,\psi ,n)}$$ D ( u , ψ , n ) has a closed range if and only if it is either trivial or surjective, and the latter happens if and only if $$\psi (z)= az+b$$ ψ ( z ) = a z + b and a belongs to the unit circle.