Dissipation form of the linearized Boltzmann operator ($-⟨h,\mathcal{L}h{⟩}_M=\frac{1}{4}\int BM{M}_{\ast }(\Delta h{)}^{2}dvd{v}_{\ast }dn$)

Write $f=M(1+h)$ around a fixed Maxwellian and expand the nonlinear Boltzmann collision operator to first order. The weak Boltzmann collision identity gives the nonnegative dissipation form in $L^2(Mdv)$, and polarization proves symmetry on an appropriate common domain. Its zero modes are the collision invariants when the scattering kernel is nondegenerate. A spectral gap, and self-adjointness of an unbounded realization, require additional domain and kernel hypotheses.