Generalized weighted composition operators on Hardy space $H^2({\mathbb{D}}^n)$

Abstract

In this paper, we explore the complex symmetrical characteristics of weighted composition operators $W_{u,v}$ and weighted composition-differentiation operators $W_{u,v,k_1,k_2,\dots ,k_n}$ on the Hardy space $H^2({\mathbb{D}}^n)$ over the Polydisk ${\mathbb{D}}^n$, with respect to the standard conjugation $\mathcal{J}$. We specify explicit conditions that confirm the Hermitian characteristics of the operator $W_{u,v,k_1,k_2,\dots ,k_n}$ and describe the conditions necessary for it to exhibit normal behavior. Additionally, we identify the kernels of the generalized weighted composition-differentiation operators and their corresponding adjoint operators.