Hilbert–Schmidt composition–differentiation operators on the unit ball

We use the notion of radial differential operator of order t>0 to introduce the weighted composition–differentiation operator Eψ,φt(f)=ψ·(Rtf)∘φ on the Hardy and Bergman spaces of the unit ball and the polydisk in Cn. We obtain necessary and sufficient conditions on the functions φ and ψ to ensure that the operator Eψ,φt is Hilbert–Schmidt.