Exercise :
Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$. Show that $\dim \ker ( \text{id} - A) < + \infty$.
Attempt/Thoughts :
The kernel of the operator $(\text{id}-A) : X \to X$ is : \begin{align*}\ker(\text{id}-A) &=\{x \in X : (\text{id}-A)(x) = 0 \} \\ &= \{x \in X :x-A(x) = 0 \} \\ &= \{x \in X : A(x) = x \}\end{align*} So essentialy, we are restricting $A$ to the kernel of $\text{id}-A$, as $A\bigg|_{\ker(\text{id}-A)} \equiv x \in \ker(\text{id}-A)$. But $A$ is compact, which means that if $\ker(\text{id}-A)$ is Banach, then it cannot be infinite dimensional, otherwise we would not be able to achieve compactness.
Am I missing something in my elaboration here or is it as straightforward ? Maybe, how would one imply that $\ker(\text{id}-A)$ is indeed Banach ?
Any tips, elaborations or corrections will be appreciated.