Positivity of an operator on a compact subset of a manifold Let $E$ and $F$ be two vector bundles over manifold $X$. Let $P:\Gamma(E)\to \Gamma(F)$ be a self-adjoint differential operator over $X$. Define inner product on the spaces $\Gamma(E)$ of smooth sections over $X$ by the following pointwise inner product:
$$(s_1,s_2):=\int_X <s_1(x),s_2(x)>,\quad s_1,s_2\in\Gamma(E)$$
My question is how to prove that there exists a constant $k>0$ such that
$$P+k\ Id\geq 0$$  holds on a compact subset $K\subset X$. That is,
$$((P+k\ Id) s,s)\geq 0, \quad s\in\Gamma(E).$$ holds on a compact subset $K\subset X$.
I'm a little confused to prove the inequality. Could you please give me some help with the details? Thanks!
 A: I don’t think there’s any completely general method for doing so. In the case of the generalised Dirac operator $D$ induced by a Clifford connection $\nabla$ on a Clifford module bundle $E$ on a complete Riemannian manifold $X$ (e.g., the spin Dirac operator on a complete Riemannian spin manifold), you have a Lichnerowicz formula $$D^2 = \nabla^\ast \nabla + \frac{1}{4}\operatorname{Sc} \operatorname{Id} + \mathcal{F}^{E/S},$$ where $\operatorname{Sc}$ is the scalar curvature of $X$ and $\mathcal{F}^{E/S}$ is a pointwise symmetric smooth bundle endomorphism of $E$ encoding the so-called twisting curvature of the Clifford module bundle $\mathcal{E}$ (e.g., $\mathcal{F}^{E/S} = 0$ when $D$ is a spin Dirac operator). Then
$$
 \forall \sigma \in \Gamma_c(E), \quad (D^2s,s) = \lVert \nabla s \rVert^2_2 +\left((\tfrac{1}{4}\operatorname{Sc}\operatorname{Id}+\mathcal{F}^{E/S})s,s\right) \geq \left((\tfrac{1}{4}\operatorname{Sc}\operatorname{Id}+\mathcal{F}^{E/S})s,s\right).
$$
But now, since $\tfrac{1}{4}\operatorname{Sc}\operatorname{Id}+\mathcal{F}^{E/S}$ is a smooth (hence continuous) bundle endomorphism, the usual extreme value theorem together with a tiny bit of care imply that for every compact $K \subset X$, there exists $\kappa_1[K] > 0$, such that $-\left(\tfrac{1}{4}\operatorname{Sc}\operatorname{Id}+\mathcal{F}^{E/S}\right) \leq \kappa_1[K]\operatorname{Id}$ pointwise on $K$, in which case
$$
 \forall \sigma \in \Gamma(E), \quad ((D^2+\kappa_1[K]\operatorname{id})s,s)_K \geq \left((\tfrac{1}{4}\operatorname{Sc}\operatorname{Id}+\mathcal{F}^{E/S})s,s\right)_K + \left(\kappa_1[K]\operatorname{id}s,s\right)_K \geq 0.
$$
The existence of $\kappa_1[K]$ now follows from the following straightforward lemma:

Let $\mathcal{F}$ be a pointwise symmetric smooth bundle endomorphism of a Hermitian vector bundle $E$ on a smooth manifold $X$. For every compact $K \subset X$, there exists constant $C[K] > 0$, such that $\mathcal{F} \leq C[K]\operatorname{id}$ pointwise on $K$, i.e.,
$$
 \forall x \in K, \, \forall \sigma_x \in E_x, \quad \langle\mathcal{F}\sigma_x,\sigma_x\rangle_x \leq C[K]\langle \sigma_x,\sigma_x \rangle_x.
$$

Proof. Since $X$ is locally Euclidean and $E$ is locally trivial, without loss of generality, $X = \mathbb{R}^n$ and $E$ is trivial with fibre $\mathbb{C}^r$. By applying Gram–Schmidt to the standard ordered basis of $\mathbb{C}^r$, we therefore obtain an orthonormal frame $(e_1,\dotsc,e_r)$ for $E$ consisting of smooth global sections. Thus, since $K$ is compact, let
$$
 C[K] := 1+r^2 \max_{1\leq i,j \leq r} \max_{x \in K} \lvert \langle \mathcal{F}_x (e_i)_x,(e_j)_x \rangle _x \rvert.
$$
Then, for all $x \in X$ and $\sigma_x \in E_x$,
$$\begin{align}
\langle \mathcal{F}_x \sigma_x,\sigma_x \rangle_x &= \sum_{i,j=1}^r \langle \mathcal{F}_x (e_i)_x,(e_j)_x\rangle_x \langle \sigma_x, (e_i)_x \rangle_x \overline{\langle \sigma_x, (e_j)_x \rangle_x}\\
&\leq \sum_{i,j=1}^r \lvert\langle \mathcal{F}_x (e_i)_x,(e_j)_x\rangle_x \rvert \cdot \lvert\langle \sigma_x, (e_i)_x \rangle_x \rvert \cdot \lvert \overline{\langle \sigma_x, (e_j)_x \rangle_x}\rvert\\
&\leq \sum_{i,j=1}^r \lvert\langle \mathcal{F}_x (e_i)_x,(e_j)_x\rangle_x \rvert \cdot \sqrt{\langle \sigma_x,\sigma_x \rangle_x} \cdot \sqrt{\langle \sigma_x,\sigma_x \rangle_x}\\
&\leq \sum_{i,j=1}^r \left(\max_{y \in K} \lvert\langle \mathcal{F}_y (e_i)_y,(e_j)_y\rangle_x \rvert\right) \langle \sigma_x,\sigma_x \rangle_x\\
&\leq C[K] \langle \sigma_x,\sigma_x \rangle_x. 
\end{align}$$
There are obviously slicker ways to get sharper estimates, but this is good enough for this particular purpose.
