Generalized complex symmetric composition operators with applications

We characterize the weighted composition-differentiation operators $D_{\mfn,\psi,\varphi}$ acting on $\mathcal{H}_\gamma(\mathbb{D}^d)$ over the polydisk $\mathbb{D}^d$ which are complex symmetric with respect to the conjugation $\mathcal{J}$. We obtain necessary and sufficient conditions for $D_{\mfn,\psi,\varphi}$ to be self-adjoint. We also investigate complex symmetry of generalized weighted composition differentiation operators $M_{n, \psi, \varphi}=\displaystyle\sum_{j=1}^{n}a_jD_{j,\psi_j, \varphi},$ (where $a_j\in \mathbb{C}$ for $j=1, 2, \dots, n$) on the reproducing kernel Hilbert space $\mathcal{H}_\gamma(\mathbb{D})$ of analytic functions on the unit disk $\mathbb{D}$ with respect to a weighted composition conjugation $C_{\mu, \xi}$. Further, we discuss the structure of self-adjoint linear composition differentiation operators. Finally, the convexity of the Berezin range of composition operator on $\mathcal{H}_\gamma(\mathbb{D})$ are investigated. Additionally, geometrical interpretations have also been employed.