  The linearized collision operator of the Boltzmann equation can in a natural
way be written as a sum of a positive multiplication operator, the collision
frequency, and an integral operator. Compactness of the integral operator for
monatomic single species is a classical result, while corresponding result for
mixtures is more recently obtained. In this work the compactness of the
operator for polyatomic single species, where the polyatomicity is modeled by a
discrete internal energy variable, is studied. 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 approximately Hilbert-Schmidt integral operators, under some
assumptions on the collision kernel. Self-adjointness of the linearized
collision operator follows. Moreover, bounds on - including coercivity of - the
collision frequency are obtained for a hard sphere model. Then it follows that
the linearized collision operator is a Fredholm operator.
  The results can be extended to mixtures. For brevity, only the case of
mixtures for monatomic species is accounted for.
