Family index for Fredholm extensions of semi-Fredholm operators

2024/01/29 by Prokhorova, Marina
#Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2401.16060

Abstract

This paper is devoted to Fredholm realizations of semi-Fredholm operators in a Hilbert space. Such a realization is determined by an abstract boundary condition, which is a subspace of the space of abstract boundary values. We find the K⁰ index of a family of Fredholm realizations of semi-Fredholm operators in terms of the corresponding family of boundary conditions. Similarly, we find the K¹ index of a family of self-adjoint Fredholm extensions of symmetric semi-Fredholm operators. Our approach is based on passing from a Fredholm operator to its graph. The graph forms a Fredholm pair with the horizontal subspace, and we prove the index formula by deforming the horizontal subspace instead of the operator.

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