# compact operator

Let $X$ and $Y$ be two Banach spaces^{}.
A compact operator^{} (completely continuous operator) is a linear operator $T:X\to Y$
that maps the unit ball in $X$ to a set in $Y$ with compact closure. It can be shown that a compact operator is necessarily a bounded operator^{}.

The set of all compact operators on $X$, commonly denoted by $\mathbb{K}(X)$, is a closed two-sided ideal of the set of all bounded operators on $X$, $\mathbb{B}(X)$.

Any bounded operator which is the norm limit of a sequence of finite rank operators is compact^{}.
In the case of Hilbert spaces^{}, the converse is also true.
That is, any compact operator on a Hilbert space is a norm limit of finite rank operators.

###### Example 1 (Integral operators)

Let $k\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}$, with $x\mathrm{,}y\mathrm{\in}\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{]}$, be a continuous function^{}.
The operator defined by

$$(T\psi )(x)={\int}_{0}^{1}k(x,y)\psi (y)dy,\psi \in C([0,1])$$ |

is compact.

Title | compact operator |
---|---|

Canonical name | CompactOperator |

Date of creation | 2013-03-22 14:26:59 |

Last modified on | 2013-03-22 14:26:59 |

Owner | mhale (572) |

Last modified by | mhale (572) |

Numerical id | 8 |

Author | mhale (572) |

Entry type | Definition |

Classification | msc 46B99 |

Synonym | completely continuous |