Surjective and closed range differentiation operator

We identify Fock-type spaces $$\mathcal {F}_{(m,p)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> </mml:math> 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 $$m=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Moreover, since the operator is unbounded on the classical Fock spaces, we consider the modified or the weighted composition–differentiation operator, $$D_{(u,\psi ,n)} f= u\cdot \big ( f^{(n)}\circ \psi \big )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ψ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msub> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>u</mml:mi> <mml:mo>·</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>∘</mml:mo> <mml:mi>ψ</mml:mi> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , on these spaces and describe conditions under which the operator admits closed range, surjective, and order bounded structures.