Generalized weighted composition-differentiation operators on weighted Bergman spaces

TL;DR

This paper investigates the complex symmetry, spectral properties, and conditions for Hermitian and normal behavior of generalized weighted composition-differentiation operators on weighted Bergman spaces, providing necessary and sufficient criteria.

Contribution

It introduces a comprehensive analysis of complex symmetry and spectral characteristics of these operators, deriving explicit conditions for symmetry, Hermitian, and normal properties in weighted Bergman spaces.

Findings

Derived necessary and sufficient conditions for $ C_{,} $-symmetry.

Established criteria for Hermitian and normal operators.

Analyzed spectral properties and kernel of the adjoint operator.

Abstract

Let $ \mathcal{H}(\mathbb{D}) $ be the class of all holomorphic functions in the unit disk $ \mathbb{D} $. We aim to explore the complex symmetry exhibited by generalized weighted composition-differentiation operators, denoted as $L_{n, \psi, \phi}$ and is defined by \begin{align*} L_{n, \psi, \phi}:=\sum_{k=1}^{n}c_kD_{k, \psi_k, \phi},\; \mbox{where }\; c_k\in\mathbb{C}\; \mbox{for}\; k=1, 2, \ldots, n, \end{align*} where $ D_{k, \psi, \phi}f(z):=\psi(z)f^{(k)}(\phi(z)),\; f\in \mathcal{A}^2_{\alpha}(\mathbb{D}), $ in the reproducing kernel Hilbert space, labeled as $\mathcal{A}^2_{\alpha}(\mathbb{D})$, which encompasses analytic functions defined on the unit disk $\mathbb{D}$. By deriving a condition that is both necessary and sufficient, we provide insights into the $ C_{\mu, \eta} $-symmetry exhibited by $L_{n, \psi, \phi}$. The explicit conditions for which the operator T is Hermitian and normal are obtained through our investigation. Additionally, we conduct an in-depth analysis of the spectral properties of $ L_{n, \psi, \phi} $ under the assumption of $ C_{\mu, \eta} $-symmetry and thoroughly examine the kernel of the adjoint operator of $L_{n, \psi, \phi}$.