  Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on
$m$-columns of half-densities on a closed manifold $M$, whose principal symbol
is assumed to have simple eigenvalues. We show that the spectrum of $A$
decomposes, up to an error with superpolynomial decay, into $m$ distinct
series, each associated with one of the eigenvalues of the principal symbol of
$A$. These spectral results are then applied to the study of propagation of
singularities in hyperbolic systems. The key technical ingredient is the use of
the carefully devised pseudodifferential projections introduced in the first
part of this work, which decompose $L^2(M)$ into almost-orthogonal
almost-invariant subspaces under the action of both $A$ and the hyperbolic
evolution.
