Signature and Spectral Flow of J-Unitary $${\mathbb{S}^1}$$ S 1 -Fredholm Operators

Abstract

Bijective operators conserving the indefinite scalar product on a Krein space \({(\mathcal{K}, J)}\) are called J-unitary. Such an operator T is defined to be \({\mathbb{S}^1}\) -Fredholm if T−z1 is Fredholm for all z on the unit circle \({\mathbb{S}^1}\) , and essentially \({\mathbb{S}^1}\) -gapped if there is only discrete spectrum on \({\mathbb{S}^1}\) . For paths in the \({\mathbb{S}^1}\) -Fredholm operators an intersection index similar to the Conley–Zehnder index is introduced. The strict subclass of essentially \({\mathbb{S}^1}\) -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 \({\mathbb{S}^1}\) . These concepts are illustrated by several examples.