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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?

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    $\begingroup$ The passage from (1) to (2) is the passage from pointwise positivity of $(\sigma( P)(m,\xi)v,v)$ to having a uniform positive lower bound for it. It is possibly under the combination of continuity and compactness. I don't think there is anything else going on. $\endgroup$
    – user147263
    Commented Jun 14, 2014 at 1:50
  • $\begingroup$ @wordsthatendinGRY Ah, yeah. That sounds reasonable to me. Thanks! $\endgroup$ Commented Jun 14, 2014 at 2:48

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