Generalized complex symmetric composition operators with applications

Abstract

We characterize the weighted composition-differentiation operators $D_{\mfn,ψ,φ}$ acting on ${\mathcal{H}}_{\hat{I}}^3({\mathbb{D}}^d)$ over the polydisk ${\mathbb{D}}^d$ which are complex symmetric with respect to the conjugation $\mathcal{J}$. We obtain necessary and sufficient conditions for $D_{\mfn,ψ,φ}$ to be self-adjoint. We also investigate complex symmetry of generalized weighted composition differentiation operators $M_{n,\ddot{I}ˆ,\ddot{I}†}=\sum _{j=1}^n{a}_j{D}_{j,\ddot{I}{ˆ}_j,\ddot{I}†},$ (where $a_j\in \mathbb{C}$ for $j=1,2,\dots ,n$) on the reproducing kernel Hilbert space ${\mathcal{H}}_{\hat{I}}^3(\mathbb{D})$ of analytic functions on the unit disk $\mathbb{D}$ with respect to a weighted composition conjugation $C_{\hat{I}¼,\hat{I}¾}$. Further, we discuss the structure of self-adjoint linear composition differentiation operators. Finally, the convexity of the Berezin range of composition operator on ${\mathcal{H}}_{\hat{I}}^3(\mathbb{D})$ are investigated. Additionally, geometrical interpretations have also been employed.

38 pages and 25 figures