  The linearized Boltzmann collision operator is fundamental in many studies of
the Boltzmann equation and its main properties are of substantial importance.
The decomposition into a sum of a positive multiplication operator, the
collision frequency, and an integral operator is trivial. Compactness of the
integral operator for monatomic single species is a classical result, while
corresponding results for monatomic mixtures and polyatomic single species are
more recently obtained. This work concerns the compactness of the operator for
a multicomponent mixture of polyatomic species, where the polyatomicity is
modeled by a discrete internal energy variable. With a probabilistic
formulation of the collision operator as a starting point, compactness is
obtained by proving that the integral operator is a sum of Hilbert-Schmidt
integral operators and operators, which are uniform limits of Hilbert-Schmidt
integral operators, under some assumptions on the collision kernel. The
assumptions are essentially generalizations of the Grad's assumptions for
monatomic single species. Self-adjointness of the linearized collision operator
follows. Moreover, bounds on - including coercivity of - the collision
frequency are obtained for a hard sphere like model. Then it follows that the
linearized collision operator is a Fredholm operator, and its domain is also
obtained.
