Surjective and closed range differentiation operator

Abstract

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, $D_{(u,\psi,n)} f= u\cdot\big( f^{(n)}\circ \psi\big)$, on these spaces and describe conditions under which the operator admits closed range, surjective, and order bounded structures.