  We establish an instantaneous smoothing property for decaying solutions on
the half-line $(0,+\infty)$ of certain degenerate Hilbert space-valued
evolution equations arising in kinetic theory, including in particular the
steady Boltzmann equation. Our results answer the two main open problems posed
by Pogan and Zumbrun in their treatment of $H^1$ stable manifolds of such
equations, showing that $L^2_{loc}$ solutions that remain sufficiently small in
$L^\infty$ (i) decay exponentially, and (ii) are $C^\infty$ for $t>0$, hence
lie eventually in the $H^1$ stable manifold constructed by Pogan and Zumbrun
