In Salamon's notes on Floer homology, 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.