Generalized weighted composition-differentiation operators on weighted Bergman spaces

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 \(L_{n, \psi, \phi}\).