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).

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