  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 $\Phi.$
