Equivalence of Ellipticity and Fredholmness in the Weyl-H\"ormander calculus

The main result is that the Fredholm property of a $\mathrm{\Psi }$DO acting on Sobolev spaces in the Weyl-H"ormander calculus and the ellipticity are equivalent for geodesically temperate H"ormanders metrics whose associated Planck’s functions vanish at infinity. Additionally, we prove that when the H"ormander metric is geodesically temperate, and consequently the calculus is spectrally invariant, the inverse $\lambda \mapsto b_{\lambda }\in S(1,g)$ of every ${\mathcal{C}}^N$, $0\le N\le \mathrm{\infty }$, map $\lambda \mapsto a_{\lambda }\in S(1,g)$ comprised of invertible elements on $L^2$ is again of class ${\mathcal{C}}^N$.