Table of Contents
1. Fredholm Operator
Let \( H \) be a complex separable Hilbert space. Let \( \mathcal{B} \) be the Banach algebra of bounded linear operators on \( H \) with finite norm.
An operator \( T \in \mathcal{B} \) is called a Fredholm operator if it has finite dimensional kernel and finite dimensional cokernel.
1.1. Index
An index of Fredholm operator \( T \) is defined to be:
\begin{equation*} \mathrm{index}(T) = \mathrm{dim}(\mathrm{Ker}(T)) - \mathrm{dim}(\mathrm{Coker}(T)). \end{equation*}1.2. Properties
- For a Fredholm operator \( T\colon H\to H' \) where \( H, H' \) are finite dimensional vector spaces,
-For two Fredholm operator \( F\colon H\to H, G\colon H' \to H' \), the index of the direct sum operator \( F\oplus G\colon H\oplus H' \to H\oplus H' \) is additive
\begin{equation*} \mathrm{index}(F\oplus G) = \mathrm{index}(F) + \mathrm{index}(G). \end{equation*}- For two Fredholm operator \( F\colon H \to H', G\colon H' \to H'' \), the index of the composed operator \( G\circ F\colon H\to H'' \) is also additive: