Surjective and closed range differentiation operator

TL;DR

This paper characterizes when the differentiation operator has closed range on Fock-type spaces, showing it is surjective only when a specific parameter equals one, and explores properties of a modified weighted operator.

Contribution

It identifies conditions for the differentiation operator to have closed range and be surjective on Fock-type spaces, and analyzes a modified weighted differentiation operator.

Findings

D has closed range only if it is surjective, which occurs when m=1.

The differentiation operator is unbounded on classical Fock spaces.

Conditions are described for the weighted composition-differentiation operator to have closed range and be surjective.

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.