While learning more about the analytic background for the Atiyah-Singer Index Theorem, I was curious about the following question (although not needed for the ASID): what are some general conditions which imply that weakly elliptic operators are strongly elliptic?
Context: $M$: closed smooth manifold; $E,F\rightarrow M$ smooth, $\mathbb{C}$-vector bundles over $M$; $P:\Gamma(E)\rightarrow \Gamma(F)$.
Def. 1: $P$ is called weakly elliptic if, for each $m\in M$ and $\xi\neq 0\in T^{\ast}_mM$, its symbol $\sigma( P)(m,\xi)$ is invertible.
Def. 2: $P$ is called strongly elliptic if there exists a constant $C>0$ such that
$$ (\sigma( P)(m,\xi)v,v)\geq C\|v \|^2 $$
for all $v\in E$ and $\|\xi\|=1$.
So, again, I'm wondering:
$$ \underline{\textbf{Question:}} \text{ Under what conditions do the above notions coincide?} $$
Or, at least some references?