The Calderón projection: New definition and applications

We consider an arbitrary linear elliptic first-order differential operator AA with smooth coefficients acting between sections of complex vector bundles E,FE,F over a compact smooth manifold MM with smooth boundary ΣΣ. We describe the analytic and topological properties of AA in a collar neighborhood UU of ΣΣ and analyze various ways of writing A↾U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of AA by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderón projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderón projection and of well-posed self-adjoint Fredholm extensions under continuous variation of the data.