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The linearized Boltzmann collision operator is fundamental in many studies ofthe Boltzmann equation and its main properties are of substantial importance.The decomposition into a sum of a positive multiplication operator, thecollision frequency, and an integral operator is trivial. Compactness of theintegral operator for monatomic single species is a classical result, whilecorresponding results for monatomic mixtures and polyatomic single species aremore recently obtained. This work concerns the compactness of the operator fora multicomponent mixture of polyatomic species, where the polyatomicity ismodeled by a discrete internal energy variable. With a probabilisticformulation of the collision operator as a starting point, compactness isobtained by proving that the integral operator is a sum of Hilbert-Schmidtintegral operators and operators, which are uniform limits of Hilbert-Schmidtintegral operators, under some assumptions on the collision kernel. Theassumptions are essentially generalizations of the Grad's assumptions formonatomic single species. Self-adjointness of the linearized collision operatorfollows. Moreover, bounds on - including coercivity of - the collisionfrequency are obtained for a hard sphere like model. Then it follows that thelinearized collision operator is a Fredholm operator, and its domain is alsoobtained.

Compactness property of the linearized Boltzmann collision operator for a multicomponent polyatomic gas