Fredholm operator

A Fredholm operator is a bounded operatorMathworldPlanetmathPlanetmath between Banach spacesMathworldPlanetmath that has a finite dimensional kernel and cokernel (and closed range). Equivalently, it is invertiblePlanetmathPlanetmathPlanetmath modulo compact operatorsMathworldPlanetmath. That is, if F:XY is a Fredholm operator between two vector spacesMathworldPlanetmath X and Y, then there exists a bounded operator G:YX such that

GF-1IX𝕂(X),FG-1IY𝕂(Y), (1)

where 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:XX} is an open subset of the Banach algebraMathworldPlanetmath of bounded operators {T:XX}.

If F is Fredholm then so is its adjointPlanetmathPlanetmathPlanetmath, F*. If T𝕂(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