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compact operator
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(Definition)
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Let $X$ and $Y$ be two Banach spaces. A compact operator (completely continuous operator) is a linear operator $T \colon X \to Y$ 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 $\Kset(X)$ , is a closed two-sided ideal of the set of all bounded operators on $X$ , $\Bset(X)$ .
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 $k(x,y)$ , with $x,y \in [0,1]$ , be a continuous function. The operator defined by $$(T\psi)(x) = \int_0^1 k(x,y) \psi(y) \,\d y, \qquad \psi \in C([0,1])$$ is compact.
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"compact operator" is owned by mhale. [ full author list (2) ]
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completely continuous |
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Cross-references: continuous function, converse, Hilbert spaces, finite rank, sequence, limit, norm, two-sided ideal, closed, bounded operator, closure, compact, unit ball, maps, linear operator, operator, Banach spaces
There are 11 references to this entry.
This is version 5 of compact operator, born on 2004-06-26, modified 2008-06-29.
Object id is 5966, canonical name is CompactOperator.
Accessed 7021 times total.
Classification:
| AMS MSC: | 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous) |
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Pending Errata and Addenda
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