  Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold
$M$, let $P \in \psi^m(M; E_0, E_1)$ be a $G$--invariant, classical
pseudodifferential operator acting between sections of two vector bundles $E_i
\to M$, $i = 0,1$, and let $\alpha$ be an irreducible representation of the
group $G$. Then $P$ induces a map $\pi_\alpha(P) : H^s(M; E_0)_\alpha \to
H^{s-m}(M; E_1)_\alpha$ between the $\alpha$-isotypical components. We prove
that the map $\pi_\alpha(P)$ is Fredholm if, and only if, $P$ is {\em
transversally $\alpha$-elliptic}, a condition defined in terms of the principal
symbol of $P$ and the action of $G$ on the vector bundles $E_i$.
