The main result is that the ellipticity and the Fredholm property of a $\Psi $ DO acting on Sobolev spaces in the Weyl-Hörmander calculus are equivalent when the Hörmander metric is geodesically temperate and its associated Planck function vanishes at infinity. The proof is essentially related to the following result that we prove for geodesically temperate Hörmander metrics: If $\lambda \mapsto a_{\lambda }\in S(1,g)$ is a $\mathcal {C}^N$ , $0\leq N\leq \infty $ , map such that each $a_{\lambda }^w$ is invertible on $L^2$ , then the mapping $\lambda \mapsto b_{\lambda }\in S(1,g)$ , where $b_{\lambda }^w$ is the inverse of $a_{\lambda }^w$ , is again of class $\mathcal {C}^N$ . Additionally, assuming also the strong uncertainty principle for the metric, we obtain a Fedosov-Hörmander formula for the index of an elliptic operator. At the very end, we give an example to illustrate our main result.

主要结果是， 当 Hörmander 度量是测地温带且其相关的普朗克函数在无穷远处消失时，作用于 Weyl-Hörmander 演算中 Sobolev 空间的 $\Psi $ DO 的椭圆率和 Fredholm 属性是等价的。该证明本质上与我们为测地温带 Hörmander 度量证明的以下结果相关：如果 $\lambda \mapsto a_{\lambda }\in S(1,g)$ 是 $\mathcal {C}^N$ ， $0\leq N\leq \infty $ ，映射使得每个 $a_{\lambda }^w$ 在 $L^2$ 上可逆 ，那么映射 $\lambda \mapsto b_{\lambda }\in S(1 ,g)$ ，其中 $b_{\lambda }^w$ 是 $a_{\lambda }^w$ 的倒数，又是 $\mathcal {C}^N$ 类。此外，还假设度量的强不确定性原理，我们获得了椭圆算子索引的 Fedosov-Hörmander 公式。最后，我们举一个例子来说明我们的主要结果。