Hello!

I'm going through the proof of the fact, that given a compact operator $\displaystyle K$ on a Banach space $\displaystyle X$ its spectrum $\displaystyle \sigma(K)$ is at most countable and 0 as the only possible accumulation point (which may or may not belong to the spectrum).

The reading has been fairly easy so far, but at the moment I'm somehow stuck at one step, namely:

The overall objective is to show that all points $\displaystyle \lambda\in\sigma(K)\setminus\{0\}$ are isolated.

Let $\displaystyle N:=N_\lambda^m(K)=\text{Ker}(\lambda I-K)^m$ and $\displaystyle K|_N$ be the restriction of $\displaystyle K$ to $\displaystyle N$

There is a moment when it is evident that we need finiteness of $\displaystyle \sigma(K|_N)$.

So they argue that this follows from the fact, that $\displaystyle N$ is finite-dimensional.

I can't see at the moment how exactly it follows.

Does anybody have an idea?

P.S.:

I think I should mention that $\displaystyle m$ is chosen so that:

$\displaystyle X=R+N$ with $\displaystyle R\cap N=\{0\}$, where $\displaystyle R:=R_\lambda^m(K)=\text{Im}(\lambda I-K)^m$.

Thanks in advance,

HAL