Schatten class membership of products of composition operators and their adjoints

In this paper, we investigate the Schatten class membership of the product of a composition operator and the adjoint of another composition operator on the Hardy and Bergman spaces. We provide several sufficient conditions and necessary conditions for these operators to belong to the Schatten classes. Moreover, for some specific symbols including linear fractional maps, finite Blaschke products, and inner functions, we obtain necessary and sufficient conditions.