In Voisin's beautiful book on Hodge theory she gives a proof that every cohomology class can be represented by a harmonic form by referring to the the following theorem on elliptic operators:

Let $P : E → F \$ be an elliptic differential operator on a compact manifold. Assume that $E$ and $F$ are of the same rank, and are equipped with metrics. Then $\operatorname{ker} P \subset C^{\infty}(E)\$ is ﬁnite-dimensional, $P( C^{\infty}(E)) \subset C^{\infty}(F) \$ is closed and of ﬁnite codimension, and we have a decomposition as an orthogonal direct sum (for the $L^2$ metric) $$C^{\infty}(E) = \operatorname{ker} P \oplus P^{\ast}( C^{\infty}(F)),$$ which she then applies to the Laplace operator.

Unfortunately, her reference for this (Demailly - Théorie de Hodge L2 et Théor`emes d'annulation) is in French, which I don't understand.

I am primarily hoping for a reference on elliptic operators which proves this theorem; however, at the same time I also feel like I should know more about elliptic operators in general, and 'how they can be used in geometry' (can they, actually? I'm thinking about Atiyah-Singer here, which I don't understand at all, so I'm not sure.), so if you can give me a reference about that, it would be much appreciated, too.

One more suggestion: Wells Differential Geometry on Complex Manifolds has a complete proof, and focuses on the sort of examples which will interest someone who is reading Voisin's book.

The book Topology and Analysis: Atiyah-Singer Index Formula and Gauge-theoretic Physics by Booss and Bleecker (Springer , Universitxt) seems to be exactly what you are loooking for.
It is an introduction to the Atiyah-Singer index formula, with all prerequisites carefully developed :Fredholm operators, (pseudo-)differential operators on manifolds, Sobolev spaces, vector bundles and much more. The style is quite friendly, with many examples,remarks, exercises, interspersed throughout the text.
To sum up, I find the blend of analysis, manifold theory and topology in this book quite remarkable and unusual.

I am only really familiar with the German original version, written by Booss alone. The English translation I refer to is enriched by about 100 pages due to Bleecker and devoted to applications in mathematical physics: they seem very interesting but I haven't looked at them seriously .

• I took a look on part II, and I'm very impressed. Pictures, explanations, motivation, and interesting comments are everywhere. Thank you! – Jonas M. Sep 11 '11 at 15:35

Unfortunately, I am not an exert in this area, but I have collected for my purposes some links about that, so I would like to share.

First of all, I would take a look to "Lectures on pseudodifferential operators" of Richard Melrose, Chapter 6 is dedicated to ellipticity, where you can try to find some proofs that you are after.

Next, I would refer to the classical source of this subject, the book of R.Melrose "The Atiyah-Patodi-Singer Index Theorem", a copy is generously provided by the author here.

Another very interesting online resource is Michael E. Taylor's "Pseudodifferential operators. Four Lectures at MSRI, September 2008"

And if you want a quick yet brilliant introduction to the topic don't miss out Rafe Mazzeo's presentation "The Atiyah–Singer Index Theorem: What it is and why you should care"

Some additional googling revealed a plenitude of various texts on the topic. For instance, this one: Nigel Higson and John Roe, Lectures on Operator K-Theory and the Atiyah-Singer Index Theorem

"Notes on the Atiyah-Singer Index Theorem" by Liviu I. Nicolaescu also deserves a very good mention! (just google it!)

I hope that this will be more than enough to satisfy your curiosity.

EDIT. And I forgot to mention Victor Guillemin's homepage where you can find notes on elliptic operators!

• Thank you very much! All these notes look very good. I especially like those on elliptic operators by Professor Guillemin, as they seem to be completely devoted to proving the theorem in my question. – Jonas M. Sep 11 '11 at 15:37