Generalized complex symmetric composition operators with applications

We characterize the weighted composition-differentiation operators D,ψ,φ acting on H_γ(Dᵈ) over the polydisk Dᵈ which are complex symmetric with respect to the conjugation J. We obtain necessary and sufficient conditions for D,ψ,φ to be self-adjoint. We also investigate complex symmetry of generalized weighted composition differentiation operators Mn, ψ, φ=∑ⱼ₌₁ⁿaⱼDj,ψⱼ, φ, (where aⱼ∈ C for j=1, 2, , n) on the reproducing kernel Hilbert space H_γ(D) of analytic functions on the unit disk D with respect to a weighted composition conjugation Cμ, ξ. Further, we discuss the structure of self-adjoint linear composition differentiation operators. Finally, the convexity of the Berezin range of composition operator on H_γ(D) are investigated. Additionally, geometrical interpretations have also been employed.