Surjective and closed range differentiation operator

We identify Fock-type spaces ${\mathcal{F}}_{(m,p)}$ 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, $D_{(u,\psi ,n)}f=u\cdot (f^{(n)}\circ \psi )$, on these spaces and describe conditions under which the operator admits closed range, surjective, and order bounded structures.