Compactness property of the linearized Boltzmann operator for a diatomic single gas model

<p style='text-indent:20px;'>In the following work, we consider Boltzmann equation that models a diatomic gas by representing microscopic internal energy continuous variable I. Under some convenient assumptions on collision cross-section <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, prove linearized operator id="M2">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> of this model is Fredholm operator. For this, write id="M3">\begin{document}$ as perturbation frequency multiplication operator, and id="M4">\begin{document}$ \mathcal{K} compact. The result established after inspecting kernel form id="M5">\begin{document}$ proving it to be id="M6">\begin{document}$ L^2 integrable over its domain using elementary arguments.This implies id="M7">\begin{document}$ Hilbert-Schmidt operator.</p>