Reference Request: Elliptic differential operators in the Fréchet setting

Question

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) which exposes the theory completely in the setting of smooth sections, that is using Fréchet space techniques.

In particular, I'm interested in how the Fredholm theory translates into Fréchet spaces. The basic results should transfer without big modifications, however some results are not so straightforward (like "Fredholm operators are open in the space of all continuous linear maps" since the theory of dual spaces is more complicated).

Answer 1

The book Locally Convex Spaces by Osborne has a statement on p152 of the Fredholm alternative for Hausdorff locally convex spaces. Also consider the exercises from 26 onwards of the same chapter.

Answer 2

A few bits and pieces of Fredholm theory in the locally convex (and, in particular, Fréchet) setting are discussed in the literature. The most comprehensive treatment I could find were two old articles by Schaefer:

• Über singuläre Integralgleichungen und eine Klasse von Homomorphismen in lokalkonvexen Räumen, Math. Z., Springer Nature, 1956, 66, 147-163
• On the Fredholm alternative in locally convex linear spaces, Studia Math., 1959, 18, 229-245

There he discusses how the basic elements of Fredholm theory like invertibility modulo compact operators, existence of a parametrix and the Fredholm alternative generalize from Banach to locally convex spaces.

However, most applications of Fredholm theory use families of operators and things like the invariance of the index under small deformations etc. These results do not directly generalize to locally spaces since the set of invertible operators fails to be open, leading to many pathological (?) counterexamples. In order to discuss families of Fredholm operators and extend the important results about them to the locally convex setting, in my Phd Thesis, I introduced the notation of a "uniformly regular" family of operators. Roughly speaking, these are families of operators that posses a family of (generalized) inverses. This allows to extend most classical results to this setting. For example, if you have a uniformly regular family of operators and one of them happens to be Fredholm, then all of them are Fredholm with the same index. Finally, a classical result about elliptic operators (Theorem II.3.3.3 in Hamilton, R. S., The inverse function theorem of Nash and Moser) can be used to show that families of elliptic operators are uniformly regular so that these results apply in particular to elliptic operators. A streamlined (but perhaps less pedagogical) exposition of uniformly regularity and Fredholm theory can be found in the article

• Diez, T. & Rudolph, G., Normal form of equivariant maps in infinite dimensions, Ann. Glob. Anal. Geom., 2021