Pseudo-differential projections and the topology of certain spaces of elliptic boundary value problems

We calculate the homotopy groups of the space of elliptic boundary value problems for an elliptic differential operator A of a first order and of the space of elliptic self-adjoint boundary value problems when A is a formally self-adjoint. In particular we show that the spectral flow of an S1 family of self-adjoint elliptic boundary value problems is well defined. This provides some information on spectral properties along the lines of the Vafa-Witten approach to spectral inequalities.