Signatures for ‐hermitians and ‐unitaries on Krein spaces with Real structures

For J‐hermitian operators on a Krein space satisfying an adequate Fredholm property, a global Krein signature is shown to be a homotopy invariant. It is argued that this global signature is a generalization of the Noether index. When the Krein space has a supplementary Real structure, the sets of J‐hermitian Fredholm operators with Real symmetry can be retracted to certain of the classifying spaces of Atiyah and Singer. Secondary ‐invariants are introduced to label their connected components. Related invariants are also analyzed for J‐unitary operators.