Self-adjoint local boundary problems on compact surfaces. II. Family index

Prokhorova, Marina L.

doi:10.48550/arxiv.1809.04353

Cited by 1 publication

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“…We exclude this result from the current version of the paper to make the exposition more clear. The generalization of that result can be found in the next paper, see [19,Theorem 11.5].…”

Section: Introductionmentioning

confidence: 63%

“…Generalization to arbitrary families. In the next 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 spectral flow then is replaced by the analytical index and the integer-valued invariant Ψ is replaced by the topological index.…”

Section: Continuity Criteriamentioning

confidence: 99%

“…We will need only properties (1-3) in this paper. Properties (4-5) will be used in the next paper [19].…”

Section: Universality Of ψmentioning

confidence: 99%

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Self-Adjoint Local Boundary Problems on Compact Surfaces. I. Spectral Flow

Prokhorova

2019

J Geom Anal

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The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems. We address here the simplest nontrivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space X. We compute the K 1 (X)valued index in terms of the topological data over the boundary. The second result is universality of the index: we show that the index is a universal additive homotopy invariant for such families, if the vanishing on families of invertible operators is required. Contents1. ind a (γ 1 ) = ind a (γ 2 ). Family index for self-adjoint unbounded operatorsIn order to deal with unbounded self-adjoint operators (in particular, with self-adjoint differential operators) directly, one needs an analogue of the Atiyah-Singer theory [2]. Cf.[5], [7], [9]. This section is devoted to such an analogue adapted to our framework. The functor K 1 . Let H be a Hilbert space. Denote by B(H) the space of bounded linear operators H → H with the norm topology.

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“…We exclude this result from the current version of the paper to make the exposition more clear. The generalization of that result can be found in the next paper, see [19,Theorem 11.5].…”

Section: Introductionmentioning

confidence: 63%

Section: Continuity Criteriamentioning

confidence: 99%