Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a psuedodifferential operator $P:\Gamma(E)\rightarrow \Gamma(F)$ of order $n$ determines a Fredholm operator $H_s(E)\rightarrow H_{s-n}(F)$ between $L^2$ Sobolev spaces of sections for any $s\in\mathbb{R}$ if and only if the operator $P$ is elliptic; see for instance section 19.5 of volume III of Hörmander's book.
What can be said about the invertibility of the symbol of $P$ if this operator is (merely) assumed to be Fredholm as an operator $\Gamma(E)\rightarrow\Gamma(F)$ on $C^{\infty}$ sections?
Since I am interested in differential operators primarily, let me ask the following.
- Suppose that a differential operator $P:\Gamma(E)\rightarrow\Gamma(F)$ is Fredholm as a continuous map among Fréchet spaces of $C^{\infty}$ sections, is it then true that $P$ is elliptic?
I know that parametrices for certain classes of non-elliptic hypoelliptic and subelliptic operators can be constructed, but as far as I am aware, for such operators to determine Freholm operators, one needs to have them act on weighted Sobolev-Hilbert scales inequivalent to the usual one $\{H_s(E)\}_{s\in \mathbb{R}}$, so that taking the limit of the scale may not produce a Fredholm map $\Gamma(E)\rightarrow \Gamma(F)$.
