  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 collision cross-section
$\mathcal{B}$, we prove that the linearized Boltzmann operator $\mathcal{L}$ of
this model is a Fredholm operator. For this, we write $\mathcal{L}$ as a
perturbation of the collision frequency multiplication operator, and we prove
that the perturbation operator $\mathcal{K}$ is compact. The result is
established after inspecting the kernel form of $\mathcal{K}$ and proving it to
be $L^2$ integrable over its domain using elementary arguments.
