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The linearized Boltzmann collision operator has a central role in manyimportant applications of the Boltzmann equation. Recently some importantclassical properties of the linearized collision operator for monatomic singlespecies were extended to multicomponent monatomic gases and polyatomic singlespecies. For multicomponent polyatomic gases, the case where the polyatomicityis modelled by a discrete internal energy variable was considered lately. Herewe considers the corresponding case for a continuous internal energy variable.Compactness results, saying that the linearized operator can be decomposed intoa sum of a positive multiplication operator, the collision frequency, and acompact operator, bringing e.g., self-adjointness, is extended from theclassical result for monatomic single species, under reasonable assumptions onthe collision kernel. With a probabilistic formulation of the collisionoperator as a starting point, the compactness property is shown by adecomposition, such that the terms are, or at least are uniform limits of,Hilbert-Schmidt integral operators and therefore are compact operators.Moreover, bounds on - including coercivity of - the collision frequency areobtained for a hard sphere like, as well as hard potentials with cutoff like,models, from which Fredholmness of the linearized collision operator follows,as well as its domain.

Compactness property of the linearized Boltzmann collision operator for a mixture of monatomic and polyatomic species