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On the Uniqueness of the $G$-Equivariant Spectral Flow

Marek Izydorek, Joanna Janczewska, Maciej Starostka, Nils Waterstraat

Abstract

The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators. Lesch as well as Pejsachowicz, Fitzpatrick and Ciriza independently showed that it is uniquely characterised by its elementary properties. The authors recently introduced a $G$-equivariant spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the action of a compact Lie group $G$. The purpose of this paper is to show that the $G$-equivariant spectral flow is uniquely characterised by the same elementary properties when appropriately restated. As an application, we introduce an alternative definition of the $G$-equivariant spectral flow via a $G$-equivariant Maslov index.

On the Uniqueness of the $G$-Equivariant Spectral Flow

Abstract

The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators. Lesch as well as Pejsachowicz, Fitzpatrick and Ciriza independently showed that it is uniquely characterised by its elementary properties. The authors recently introduced a -equivariant spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the action of a compact Lie group . The purpose of this paper is to show that the -equivariant spectral flow is uniquely characterised by the same elementary properties when appropriately restated. As an application, we introduce an alternative definition of the -equivariant spectral flow via a -equivariant Maslov index.

Paper Structure

This paper contains 11 sections, 10 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The G-equivariant spectral flow and its uniqueness
  3. Proof of Theorem \ref{['thm-uniqueness']}
  4. The G-equivariant cogredient parametrix
  5. G-equivariant approximation schemes
  6. Invariance under conjugation
  7. Finite-dimensional approximation and end of proof
  8. The $\boldsymbol{G}$-equivariant spectral flow as $\boldsymbol{G}$-equivariant Maslov index
  9. The G-equivariant Maslov index
  10. The G-equivariant spectral flow as G-equivariant Maslov index
  11. An example

Key Result

Theorem 2.1

Let $G$ be a compact Lie group and a map that is defined for any separable Hilbert space $H$. If $\mu$ satisfies $(\mathcal{Z}_G)$, $(\mathcal{A}_G)$, $(\mathcal{H}_G)$ and $(\mathcal{M}_G)$, then

Theorems & Definitions (12)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Remark 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Definition 4.1
  • ...and 2 more