Signatures for $J$-hermitians and $J$-unitaries on Krein spaces with Real structures

For $J$-hermitian operators on a Krein space $(\mathcal{K},J)$ 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 $\mathbb{Z}_2$-invariants are introduced to label their connected components. Related invariants are also analyzed for $J$-unitary operators.