Isometries of the product of composition operators on weighted Bergman space

Abstract: In this paper the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on $\mathcal{H}^2(\mathbb{D})$ and $\mathcal{A}{\alpha}^2(\mathbb{D}),$ the composition operator on $\mathcal{S}^2(\mathbb{D})$ induced by an analytic self map on $\mathbb{D}$ with fixed origin need not be of norm one. We have generalized the Schwartz's well known result on $\mathcal{A}{\alpha}^2(\mathbb{D})$ which characterizes the almost multiplicative operator on $\mathcal{H}^2(\mathbb{D}).$