Hermann Shulz-Baldes: Signature and spectral flow for J-unitary S¹ Fredholm operators

Operators conserving the indefinite scalar product on a Krein space (K,J) are called J-unitary. Such an operator T is defined to be S¹-Fredholm if T-z is Fredholm for all z on the unit circle S¹, and essentially S¹-gapped if there is only discrete spectrum on S¹. For paths in the S¹-Fredholm operators an intersection index similar to the Conley-Zehnder index is introduced. The strict subclass of essentially S¹-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 S¹. These concepts are illustrated by several examples.