Every year or so I make an attempt to "really" learn the AtiyahSinger index theorem. I always find that I give up because my analysis background is too weak  most of the sources spend a lot of time discussing the topology and algebra, but very little time on the analysis. Question : is there a "fun" source for reading about the appropriate parts of analysis?

I found Booss, Bleecker: "Topology and analysis, the AtiyahSinger index formula and gaugetheoretic physics" (review) very beautifull and had read it just for fun. It is a very nice piece of exposition, motivates everything and demands from the reader only very little preknowledge. 


I know it may seem rather "old", but the notes from the IAS "Seminar on the AtiyahSinger Index Theorem from back in 1965 (published by Princeton Univ. Press) may be just what you are looking for, since it covers all the analytic machinery in great detail. It was written to be easily accessible to a math graduate student who had a basic analysis course. 


You need to understand pseudodifferential operators if you want to understand the original statement of the full AtiyahSinger index theorem. However, in most applications to differential geometry, only the theorem for twisted Dirac operators is needed. (One of the main results of Atiyah and Singer is that the Bott periodicity theorem  or rather, its generalization to vector bundles, the Thom isomorphism theorem for Ktheory  reduces the general case to that of twisted Dirac operators.) If you want to learn the theory of pseudodifferential operators, I recommend the original papers of Kohn and Nirenberg and Hörmander. This theory is not needed to prove the AtiyahSinger index theorem: you can get away with the existence of an asymptotic solution of the heat equation. To see this in action, see the paper of McKean and Singer. One advantage of the heatkernel approach is that it is welladapted to study the generalizations of the theory, such as the theory of analytic torsion and the family index theorem. 


I first learnt about the AtiyahSinger index theorem from Shanahan's Springer notes (638). I liked it because while developing the main theory, it went through the standard examples (Dirac, Dolbeaut, de Rham, signature) in some detail. At the time I was primarily interested in using index theory so wasn't so bothered about the details of the proof, but it does at least sketch the proof (it may do more, I don't remember and don't have a copy on my shelves) but I do remember that the words "pseudodifferential operator" occur which suggests that even if sketchy, the main points are all there. He also covers the equivariant theory. Another place where it's put in context is Spin Geometry by Lawson and Michelsohn. That's quite nice because the whole theory of Dirac operators and Clifford algebras is developed from scratch so there are lots of "entry points" depending on whether or not you're more of an algebraist or geometer or other. MR numbers:



Chapter 4 of Wells's "Differential Analysis on Complex Manifolds" is titled "Elliptic Operator Theory" and is, I think, close to what you want. It certainly explains why elliptic operators have finitedimensional kernels and cokernels. 


I'm also rather weak in analysis background, but I found the original paper rather readable. 


There is also a "physicist's" proof of the index theorem. (No, really, keep reading!) Attached to every supersymmetric quantum field theory (or even quantum mechanics) there is something called the Witten index. There are quantum mechanical systems for which the Witten index coincides with the index of an elliptic operator (made out of the supercharges of the theory). It is not difficult to argue that the Witten index has a homotopy invariance property which allows one to compute it in different "temperature" regimes. At infinite temperature it simply counts the difference between the dimensions of the kernel and cokernel of the elliptic operator, whereas at zero temperature it can be evaluated to give an integral formula for the index in terms of characteristic classes. This can be made rigourous and the details can be found this paper of Getzler's. 


I agree with T. Booss and Bleecker is a very good book to learn about the index theorem from. It begins very gently indeed. (I'd have simply voted T's response up, but I don't have the permissions to do that yet.) 


Ålthough written from the Khomology point of view, the book "Analytic Khomology" by Higson and Roe should be quite useful (both for basics about elliptic differential operators and index theory; iirc they sketch the proof of the index theorem for Spin^c manifolds). 


I'm reading the original paper right now and finding it a little terse in some places, but a good guide. "Spin Geometry" by Michelson and Lawson has a chapter that covers in great detail all the analysis you need for index theory as well as complete proofs of the index theorem in all of its forms. It also has some decent chapters on Ktheory (What you really need is to understand the Thom Isomorphism in Ktheory. Segal's "Equivariant KTheory" also has a nice description.) Hope this helps. 


If you have access to Atiyah's collected works at your library, try taking a look at those. There are a few transcribed lectures and short expository papers where he explains the context and motivation of the theorems. (If I remember correctly, they are classified as miscellanea and appear in the first volume, but there might also be some in the second volume. I don't have it at hand to check.) He writes beautifully, and for myself, I didn't feel like I "got" the index theorem until I read these. EDIT: Oops, I didn't read your question carefully enough: you are mostly looking for the analysis part. In that case I would just second the recommendation of the relevant chapter in Warner or Wells. 


Antony Wasserman has some course notes including the index theorem itself, at least in special cases, but also with some very readable notes on background aspects of op alg and op theory. IMHO the style is pretty compressed, but the tools used are relatively accessible. 


Perhaps you will also like "Heat kernels and Dirac operators" by Nicole Berline,Ezra Getzler,Michèle Vergne
Perhaps not the "full" AtiyahSinger index theorem, but that for DiracOperators. So if you are more a differentialgeometer than an "analysisguy", I think this is an appropriate reference. In addition another book by Gilkey (compare the post of José FigueroaO'Farrill and the comment by Ryan Budney) should be very readable: 


Hi, If you're still interested in improving your background for understanding the ASIT, you can try with these lecture notes, from a course given at the Utrecht University by E.P van den Ban and M. Crainic. The course spent time in the analytic part of (this) proof, going through pseudodifferential operators and it's symbols, and showing the role of Fourier Analysis in it's construction. Greetings. 


Liviu Nicolaescu's "Lectures on the Geometry of Manifolds" has a long chapter on differential operators on manifolds including elliptic operators. 


I have also good experience with the book of Lawson and Michelsohn. The theory of pseudoelliptic differential operators is also well explained in the book of Wells, named "Analysis and complex geometry" (or something like that). 

