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Family index for Fredholm extensions of semi-Fredholm operators

Marina Prokhorova

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 $K^0$ index of a family of Fredholm realizations of semi-Fredholm operators in terms of the corresponding family of boundary conditions. Similarly, we find the $K^1$ 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.

Family index for Fredholm extensions of semi-Fredholm operators

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 index of a family of Fredholm realizations of semi-Fredholm operators in terms of the corresponding family of boundary conditions. Similarly, we find the 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.
Paper Structure (14 sections, 27 theorems, 137 equations)

This paper contains 14 sections, 27 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Grassmanian
  3. Regular operators and Fredholm pairs of subspaces
  4. $K^0$ index
  5. Extension pairs of operators
  6. Pairs $\Gamma\subset\Gamma'$
  7. Push-forward for transversal subspaces
  8. Index formula for Fredholm pairs
  9. Index formula for Fredholm realizations
  10. The case of compact base space
  11. Self-adjoint operators and Lagrangian Grassmanian
  12. $K^1$ index
  13. Index formula for Fredholm Lagrangian pairs
  14. Index formula for Fredholm self-adjoint extensions

Key Result

Theorem 1

Let $(L_x)$ be a continuous family of Fredholm boundary conditions for a Fredholm extension pair $\mathcal{A} = (A,A')$. Suppose that $A$ is injective and $A'$ is surjective. Then

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5: Neu, Lemma 1.5
  • ...and 21 more