# Banach Lecture 16,17

These lectures ( Banach16&17) were given on 10/2/09, 10/5/09

### Excerpts

In these lecture the following result was proved.

Thm: Let $X$ is an infinite dimensional Banach space, $K \in \mathscr{L}(X)$ is compact.  If $sp(K) - 0$ is nonempty then the set consists of eigenvalues of finite multiplicity (meaning $\dim \ker (K - \lambda I)$ is finite dimensional ) with no limit other than zero.  This will follow from the following properties.  With $M_j = (K-I)^jX$, $N_j = \ker (K-I)^j$, then

1. $\ker (K - I)$ is finite dimensional
2. $(K - I)X$ is closed and of finite codimension.
3. $\exists j,i$ such that $M_j = M_{j+1}$, $N_i = N_{i+1}$
4. $i = j$ and $X = M_i \oplus N_i$
5. $(K - I)|_{M_i}$ is invertible and $(K - I)|_{N_i}$ is nilpotent or order $i$