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

In the following work, we consider the Boltzmann equation that models a polyatomic gas by representing the microscopic internal energy by a continuous variable I. Under some convenient assumptions on the transition function B, we prove that the linearized Boltzmann operator L of this model is a Fredholm operator. For this, we write L as a perturbation of the collision frequency multiplication operator, and we prove that the perturbation operator K is compact. The result is established after inspecting the kernel form of K and proving it to be L² integrable over its domain using elementary arguments.