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
Canonical name	FredholmOperator
Date of creation	2013-03-22 12:58:52
Last modified on	2013-03-22 12:58:52
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	15
Author	mhale (572)
Entry type	Definition
Classification	msc 47A53
Related topic	FredholmIndex