I'm studying an application of inverse function theorem for Banach spaces.
The theorem I would like to prove is:
$X$ is a compact manifold.
And $p:(a,b) \rightarrow (c,d)$ is an increasing function. ($p'>0$)
For any function $g \in C^{ \infty }(X)$ with $c < g < d$ there is a unique $f \in C^{\infty}(X)$ with $a < f < b$ solving the equation:
$p(f) - \Delta f = g$.
The function that should be defined is:
$P: C^{ 2 + \alpha} \rightarrow C^{\alpha} $
$~~~~~~ f \mapsto p(f) - \Delta f$
$ DP(f)h = p'(f)h - \Delta h$
I would like to prove that $DP(f)$ is invertible. The Hamilton's article about Nash Moser theorem, page 111 (where I've found this example) asserts that since laplaian is selfadjoint $DP(f)$ has index is zero.
I don't understand why $DP(f)$ has index zero. I just can see it is a Fredholm operator.
Another possible problem is that laplacian is selfadjoint in $L^2(X)$ but maybe not in these other spaces.
