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 $C_{\mathrm{\Phi }}$ on ${\mathcal{A}}_{\alpha }^2(H)$ to be one are obtained. Moreover, $C_{\mathrm{\Phi }}$ 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 $\mathrm{\Phi }\mathrm{.}$