Fredholm theory of non elliptic operators

Question

In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in the head of the following post:The subject of the conference is "Fredholm Theory of Non Elliptic Operators". I read the abstract of talks in the conference but I did not underestand that what is a precise non elliptic differential operator whose index is finite. In the above MO post it is discussed that ellipticity play a crucial role for Fredholm ness but there are some sence of index theory for non eliptic operators. I would appreaciate if you give answer, 1 item per answer, which present either a precise example of non eliptic operator of finite index or contains an extended explaination of one of the abstract listed in the above conference on "Fredholm Theory for non Elliptic operators".

I ask moderators to consider this question as a wiki question.

Answer 1

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 An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary by Christian Baer and Alexander Strohmaier 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.