compact operator

Let X and Y be two Banach spacesMathworldPlanetmath. A compact operatorMathworldPlanetmath (completely continuous operator) is a linear operator T:XY 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 operatorMathworldPlanetmathPlanetmath.

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

Any bounded operator which is the norm limit of a sequence of finite rank operators is compactPlanetmathPlanetmath. In the case of Hilbert spacesMathworldPlanetmath, 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(x,y), with x,y[0,1], be a continuous functionMathworldPlanetmath. The operator defined by


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