compact operators and Fredholm determinant

Stream: Is there code for X?

[William Coram]

Is anyone working (or have worked) on compact operators over Banach spaces?
I will eventually need Fredholm determinants and specifically prop 7 in Serre - Endomorphismes completment continus des espaces de Banach p-adiques.

[William Coram]

Screenshot 2026-02-09 at 17.54.46.png
For reference (in French)

[Kevin Buzzard]

We have talked about this in the past and I am wondering whether one should leave the classical ($\R$ or $\mathbb{C}$) analysts to do their thing, and just develop the nonarchimedean theory in a nonarchimedean namespace. It's hard to imagine much of a "base" theory which is useful in both the classical and nonarchimedean world.

Coleman developed the theory for Banach modules over an affinoid algebra and I generalised it further in my eigenvarieties paper (and fixed up a couple of issues with Coleman's exposition); one should perhaps revisit those sources and try and figure out the correct generality for all of this is in the nonarchimedean world.

[Kevin Buzzard]

Previous discussion here #Is there code for X? > Finite rank continuous linear map is compact @ 💬 . I think the conclusion is that even if there are people working on the classical theory, the nonarchimedean people should develop their own nonarchimedean theory. As I know you know well, this is what we have done for adeles (developing the theory of finite and infinite adeles separately rather than taking a restricted product over all completions at once; we immediately separate off the archimedean and nonarchimedean completions despite it being technically possible to begin the story describing them both as completions of a number field at a place; it is there that the similarities in the arch and nonarch world end).

[Bhavik Mehta]

@Thomas Browning and I have been doing a little for compact operators over complex Banach spaces recently, we have the Fredholm alternative and spectral theorem in this context. But we don't plan to get much further. I've started making PRs for the things we have already.