When seeing a proof of Fredholm's alternative I don't get the following:
Let $T$ be compact from a Banach space $X$ to itself, and $\lambda \neq 0$. Define $S=I-T$, $S^k$ its $k$-th power and $N_k=$ker$(S^k)$.
Is it true that $S(N_{k-1})=N_{k}$? One of the inclusions is clear, but the other one isn't, because $S$ is not necessarily onto (is it?). And in the proof it seems to me that we need that to happen.
Thank you.
Even in the finite-dimensional case and for $k=2$ it is not necessarily true that $S(\ker S^k) = \ker S^{k-1}$: $\text{ran}(S) \cap \ker S^K$ might be $\{0\}$, e.g. for $$ S = \pmatrix{1 & 0\cr 0 & 0\cr}$$