On the Closed Range Weighted Composition–Differentiation Operator

We provide a number of conditions which characterize the closed range weighted composition–differentiation operator $$D_{(u,\psi ,n)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <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:math> on Fock spaces. We, in particular, prove $$D_{(u,\psi ,n)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <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:math> has a closed range if and only if it is either trivial or surjective, and the latter happens if and only if $$\psi (z)= az+b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ψ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mi>z</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> and a belongs to the unit circle.