Estimates of solutions of linear Boltzmann equation at large time and spectral singularities

The spectral analysis of the dissipative linear transport (Boltzmann) operator with polynomial collision integral by the Szokefalvi-Nagy - Foias functional model is given. An exact estimate for the reminder in the asymptotic of the corresponding evolution semigroup is proved in the isotropic case. In the general case, it is shown that the operator has finitely many eigenvalues and spectral singularities and an absolutely continuous essential spectrum, and an upper estimate for the reminder is established.