# Banach Lecture 19

This lecture ( Banach19 ) was given on 10/9/09

### Excerpts

Thm II.3.1 For $T \in \mathscr{L}(X,Y)$ TFAE

1. $T$ is Fredholm.
2. $\exists S \in \mathscr{L}(Y,X)$ such that $ST - I$ and $TS - I$ have finite rank.
3. $\exists S \in \mathscr{L}(Y,X)$ such that $ST - I$ and $TS - I$ are compact.

Thm II.3.2 if $T \in \mathscr{L}(X,Y) and$latex S \in \mathscr{L}(Y,Z)\$ are Fredholm then $ST$ is Fredholm of index

$ind(ST) = ind(S) + ind(T)$.

Thm II.3.3 (Stability thm) Let $T \in \mathscr{L}(X,Y)$ be Fredholm then $\exists \epsilon > 0$ s.t. if $U \in \mathscr{L}(X,Y)$ with $||T - U|| < \epsilon$ then $U$ is Fredholm and $ind(U) = ind(T)$.

Cor. If $T \in \mathscr{L}(X,Y)$ if Fredholm, $K \in L(X,Y)$ is compact then $T + K$ is Fredholm and $ind(T +K) = ind (T)$.