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Abstract. We study the gap (= "projection norm" = "graph distance") topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators.

Abstract. We consider an arbitrary linear elliptic first-order differential operator A with smooth coefficients acting between sections of complex vector bundles E, F 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 selfadjoint Fredholm extensions under continuous variation of the data.

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