compact operator
Let and be two Banach spaces. A compact operator (completely continuous operator) is a linear operator that maps the unit ball in to a set in with compact closure. It can be shown that a compact operator is necessarily a bounded operator.
The set of all compact operators on , commonly denoted by , is a closed two-sided ideal of the set of all bounded operators on , .
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)
Title | compact operator |
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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 |