# 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 $F:X\to Y$ is a Fredholm operator between two vector spaces^{} $X$ and $Y$,
then there exists a bounded operator $G:Y\to X$ such that

$$GF-{1\mathrm{I}}_{X}\in \mathbb{K}(X),FG-{1\mathrm{I}}_{Y}\in \mathbb{K}(Y),$$ | (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 $\{F:X\to X\}$ is an open subset of the Banach algebra^{} of bounded operators $\{T:X\to X\}$.

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 |