In the paper "Index theory for skew-adjoint Fredholm operators" by Atiyah and Singer, the corollary at page 3 states that the space $\mathcal{F}_{s}(H)$ has two contractible components, characterized by the sign of the essential spectrum. Is there a way, to extend this statement in terms of the normal spectrum? Particularly one needs to check, that the space $$\mathcal{F}_s^+(H):=\{ T\in \mathcal{F}_s(H):\sigma(T)>0 \}$$ is path-connected (or not). But since I am not aware of an immediate homomorphism - which for the essential spectrum exists by definition of the Calkin algebra - I am having trouble showing this.
2026-03-26 09:16:40.1774516600
Extending Atiyah-Singer for selfadjoint Fredholm operators
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