The Calderón Projection: New Definition and Applications

We consider an arbitrary linear elliptic first–order differential operator A with smooth coefficients acting on sections of a complex vector bundle E over a compact smooth manifold M with smooth boundary Σ. We describe the analytic and topological properties of A in a collar neighborhood U of Σ 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 A 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.