Fredholm property of the linearized Boltzmann operator for a polyatomic single gas model

In the following work, we consider Boltzmann equation that models a polyatomic gas by representing microscopic internal energy continuous variable I. Under some convenient assumptions on transition function $ \mathcal{B} $, prove linearized operator \mathcal{L} of this model is Fredholm operator. For this, write as perturbation collision frequency multiplication operator, and \mathcal{K} compact. The result established after inspecting kernel form proving it to be L^2 integrable over its domain using elementary arguments.