Generalized complex symmetric composition operators with applications

We characterize the weighted composition-differentiation operators D, , acting on H_ (Dᵈ) over the polydisk Dᵈ which are complex symmetric with respect to the conjugation J. We obtain necessary and sufficient conditions for D, , to be self-adjoint. We also investigate complex symmetry of generalized weighted composition differentiation operators M₍, , =₉=₁^naⱼD₉, 䲛, , (where aⱼ C for j=1, 2, , n) on the reproducing kernel Hilbert space H_ (D) of analytic functions on the unit disk D with respect to a weighted composition conjugation C,. Further, we discuss the structure of self-adjoint linear composition differentiation operators. Finally, the convexity of the Berezin range of composition operator on H_ (D) are investigated. Additionally, geometrical interpretations have also been employed.