For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection wrt $g$ by $\nabla$, and the corresponding connection Laplacian by $\nabla^*\nabla$.

If I am not mistaken, then the Lichnerowicz formula says that $$ D^2 = \nabla^{\ast} \nabla + O, $$ where $O$ is a zero order operator.

I have two questions:

(i) Does there exist a global version of this proof? The two versions I've looked at are in Andre Moroianu's notes, and Thomas Friedrich's book. Both are given in local terms, but both seem realtively algebraic, causing me to suspect that there may exist a global version. I am right here?

(ii) I know that the Lichnerowicz formula can be used to prove that $D$ has compact resolvent, wrt the standard Hilbert space completion of the exterior algebra. However, I'm finding it difficult to see the wood for the trees in proofs I have to hand. Could someone explain why a relation between the Dirac-Laplacian and the connection Laplacian tells me some about $(1+D)^{-1}$? Is it some clear that $\nabla^{\ast} \nabla$ has nice properties as Hilbert space operator?