Surjective and closed range differentiation operator

Abstract

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