# Fredholm operator

A Fredholm operator is a bounded operator between Banach spaces that has a finite dimensional kernel and cokernel (and closed range). Equivalently, it is invertible modulo compact operators. That is, if $F\colon X\to Y$ is a Fredholm operator between two vector spaces $X$ and $Y$, then there exists a bounded operator $G\colon Y\to X$ such that

 $GF-\mathord{\mathrm{1\!\!\!\>I}}_{X}\in\mathbb{K}(X),\quad FG-\mathord{\mathrm% {1\!\!\!\>I}}_{Y}\in\mathbb{K}(Y),$ (1)

where $\mathbb{K}(X)$ denotes the space of compact operators on $X$. (Another way to say this is that $F$ is invertible in the Calkin algebra). The set of Fredholm operators $\{F\colon X\to X\}$ is an open subset of the Banach algebra of bounded operators $\{T\colon X\to X\}$.

If $F$ is Fredholm then so is its adjoint, $F^{*}$. If $T\in\mathbb{K}(X,Y)$ is a compact operator then $F+T$ is also Fredholm.

Title Fredholm operator FredholmOperator 2013-03-22 12:58:52 2013-03-22 12:58:52 mhale (572) mhale (572) 15 mhale (572) Definition msc 47A53 FredholmIndex