  The non-cutoff Kac operator is a kinetic model for the non-cutoff radially
symmetric Boltzmann operator. For Maxwellian molecules, the linearization of
the non-cutoff Kac operator around a Maxwellian distribution is shown to be a
function of the harmonic oscillator, to be diagonal in the Hermite basis and to
be essentially a fractional power of the harmonic oscillator. This linearized
operator is a pseudodifferential operator, and we provide a complete asymptotic
expansion for its symbol in a class enjoying a nice symbolic calculus. Related
results for the linearized non-cutoff radially symmetric Boltzmann operator are
also proven.
