Abstract
The paper deals with first order self-adjoint elliptic differential operators on a smooth compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. The first result is the computation of the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required. In the next paper of the series we generalize these results to families of such operators parametrized by points of an arbitrary compact space instead of an interval. The integer-valued spectral flow is replaced then by the family index taking values in the $K^1$-group of the base space.
Highlights
A closed linear operator A on a Hilbert space H is a linear operator acting from a linear subspace dom(A) ⊂ H to H such that its graph is closed in H ⊕ H
The natural topology on the space of closed operators on H is the so-called graph topology induced by the metric δ (A1, A2) = P1 − P2, where Pi denotes the orthogonal projection of H ⊕ H onto the graph of Ai
The space FRsa(H) of regular Fredholm selfadjoint operators on H equipped with the graph topology is path-connected, and its fundamental group is isomorphic to Z
Summary
In the paper we consider the simplest non-trivial case, namely the case of a two-dimensional manifold M For such M, we compute the spectral flow in terms of the topological data extracted from the corresponding one-parameter family of operators and boundary conditions. To prove Theorem A, we use the homotopy invariance of the spectral flow, its additivity with respect to direct sums, and vanishing of the spectral flow on paths of invertible operators. In the Appendix we give a general criterion of being graph continuous for families of closed operators in Hilbert and Banach spaces; see Proposition A.8 We apply this criterion to elliptic boundary value problems. In the paper [19] we generalize Theorems A, B, and C to families of self-adjoint elliptic local boundary value problems on a compact surface parametrized by points of an arbitrary compact space X. The idea of proofs remains essentially the same, but constant loops are replaced by “locally constant” families of boundary value problems, that is, fixed boundary value problems twisted by vector bundles over X
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