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I think I am in a good position to answer this. The Fredholm property of elliptic operators as maps between Sobolev spaces on compact manifolds rests on elliptic regularity properties. If an operators is not elliptic on a closed manifold this operator will not be Fredholm. This is however no longer true of you have a manifold with boundary and you impose boundary conditions, i.e. modify the domain. If the boundary conditions are carefully chosen then Hoermander's propagation of singularity theorem, replacing elliptic theory, can be used to show the Fredholm property even in non-elliptic situations. For example the Lorentzian Dirac operator is not elliptic, but it is Fredholm under some conditions (compact Cauchy surface, etc) if APS boundary conditions are imposed on spacelike hypersurfaces. Take a look at https://arxiv.org/abs/1506.00959 where an index theorem is proved for such operators. You can also look at the published version American Journal of Mathematics, 1421-1455 141.5.