  A new coercivity estimate on the spectral gap of the linearized Boltzmann
collision operator for multiple species is proved. The assumptions on the
collision kernels include hard and Maxwellian potentials under Grad's angular
cut-off condition. Two proofs are given: a non-constructive one, based on the
decomposition of the collision operator into a compact and a coercive part, and
a constructive one, which exploits the "cross-effects" coming from collisions
between different species and which yields explicit constants. Furthermore, the
essential spectra of the linearized collision operator and the linearized
Boltzmann operator are calculated. Based on the spectral-gap estimate, the
exponential convergence towards global equilibrium with explicit rate is shown
for solutions to the linearized multi-species Boltzmann system on the torus.
The convergence is achieved by the interplay between the dissipative collision
operator and the conservative transport operator and is proved by using the
hypocoercivity method of Mouhot and Neumann.
