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

## TL;DR

This paper proves that in the Weyl-H"ormander calculus, ellipticity and the Fredholm property are equivalent for certain metrics, and establishes spectral invariance and smoothness of inverses in this setting.

## Contribution

It demonstrates the equivalence of ellipticity and Fredholmness in the Weyl-H"ormander calculus under specific conditions and proves spectral invariance and smoothness of inverses for invertible elements.

## Key findings

- Ellipticity is equivalent to Fredholm property in the Weyl-H"ormander calculus.
- Spectral invariance holds for the calculus with geodesically temperate metrics.
- Inverses of invertible elements are smooth functions of the parameter.

## Abstract

The main result is that the Fredholm property of a $\Psi$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 $\mathcal{C}^N$, $0\leq N\leq \infty$, map $\lambda\mapsto a_\lambda\in S(1,g)$ comprised of invertible elements on $L^2$ is again of class $\mathcal{C}^N$.