In [Salamon's notes on Floer homology](https://people.math.ethz.ch/%7Esalamon/PREPRINTS/floer.pdf), it's claimed that under some non-degenerancy assumptions the operator $$D:= \partial\_s+J\_0\partial\_t+S(s,t): W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\rightarrow L^p(\mathbb{R}\times S^1,\mathbb{R}^{2n})$$ is Fredholm for $1<p<\infty$, and moreover its Fredholm index is given by index $$D=\mu\_{CZ}(\Psi^{+})-\mu\_{CZ}(\Psi^{-}).$$


Now I was trying to see why this index formula holds up. In the lectures notes the author claims that this formula is true for $p=2$, where we are dealing with Hilbert spaces, by using the spectral flow of the family of operators $J\_0\partial\_t+S(s,t)$. However nothing else is said for the case where $p\neq 2$. It's clear that in the end this Fredholm index won't depend on the choice of $p$, but a priori we would need to prove that this is true so that we get the desired result just by proving the case $p=2$. Is this what is happening in this situation ? We can show that $D$ has the same Fredholm index for any $p$ without using the Conley-Zehnder index ?


Any insight is appreciated, thanks in advance.



Computing the Fredholm index in Floer theory
The idea is to show that the kernel & cokernel consist of smooth sections, and thus are independent of $p$. Since the Fredholm index is the difference between the dimensions of these, it doesn't depend on $p$.


This is proved, admittedly in the case without punctures, in an appendix of McDuff & Salamon's J-holomorphic curve book.


Another good reference, this time with punctures, is from Chris Wendl's lecture notes/book draft. The direct link to the PDF file is here:
[https://www.mathematik.hu-berlin.de/~wendl/Sommer2020/SFT/lecturenotes.pdf](https://www.mathematik.hu-berlin.de/%7Ewendl/Sommer2020/SFT/lecturenotes.pdf)


