Generalized Weighted Composition Operators on Vector-Valued Weighted Bergman Space

In this research article the necessary and sufficient conditions for the norm of composition operator $C_{\Phi}$ on $\mathcal{A}_{\alpha}^2(H)$ to be one are obtained. Moreover, $C_{\Phi}$ is unitary on $\mathcal{A}_{\alpha}^2(H)$ if and only if it is co-isometry. The necessary and sufficient condition for Hermitian and normal composition operators on $\mathcal{A}_{\alpha}^2(H)$ are also explored. Also, the characterization for boundeness of generalized weighted composition operator is obtained under some condition on $\Phi.$