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The Calderón projection: New definition and applications. (English) Zbl 1221.58016
Let $$A$$ be an arbitrary linear elliptic first-order differential operator with smooth coefficients acting between sections of complex vector bundles $$E,F$$ over a compact smooth manifold $$M$$ with smooth boundary $$\Sigma.$$ The authors describe the analytic and topological properties of $$A$$ in a collar neighborhood $$U$$ of $$\Sigma$$ and analyze various ways of writing $$A \upharpoonright U$$ in product form. They 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. The construction is applied 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.

##### MSC:
 58J32 Boundary value problems on manifolds 35J67 Boundary values of solutions to elliptic equations and elliptic systems 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 57Q20 Cobordism in PL-topology
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