Signature and spectral flow of $J$-unitary $S^1$-Fredholm operators

TL;DR

This paper studies $J$-unitary operators on Krein spaces, introducing spectral flow and signature invariants for $S^1$-Fredholm operators, and explores their topological properties and classifications.

Contribution

It introduces a homotopy invariant based on the signature of $J$ on eigenspaces, and extends spectral flow concepts to $J$-unitary $S^1$-Fredholm operators.

Findings

Defined an intersection index similar to Conley-Zehnder index.

Classified components of essentially $S^1$-gapped operators using signature invariants.

Provided examples illustrating the spectral and topological properties.

Abstract

Operators conserving the indefinite scalar product on a Krein space $(K,J)$ are called $J$-unitary. Such an operator $T$ is defined to be $S^1$-Fredholm if $T-z$ is Fredholm for all $z$ on the unit circle $S^1$, and essentially $S^1$-gapped if there is only discrete spectrum on $S^1$. For paths in the $S^1$-Fredholm operators an intersection index similar to the Conley-Zehnder index is introduced. The strict subclass of essentially $S^1$-gapped operators has a countable number of components which can be distinguished by a homotopy invariant given by the signature of $J$ restricted to the eigenspace of all eigenvalues on $S^1$. These concepts are illustrated by several examples.