Surjective and closed range differentiation operator

We identify Fock-type spaces \(\mathcal {F}_{(m,p)}\) ${\mathcal{F}}_{(m,p)}$ 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 \(m=1\) $m=1$ . Moreover, since the operator is unbounded on the classical Fock spaces, we consider the modified or the weighted composition–differentiation operator, \(D_{(u,\psi ,n)} f= u\cdot \big ( f^{(n)}\circ \psi \big )\) $D_{(u,\psi ,n)}f=u·(f^{(n)}\circ \psi )$ , on these spaces and describe conditions under which the operator admits closed range, surjective, and order bounded structures.