About: Hilbert–Schmidt integral operator

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In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k : Ω × Ω → C with (that is, the L2(Ω×Ω; C) norm of k is finite), and the associated Hilbert–Schmidt integral operator is the operator K : L2(Ω; C) → L2(Ω; C) given by Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators). is compact. If

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rdfs:label	Hilbert–Schmidt integral operator (en)
rdfs:comment	In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k : Ω × Ω → C with (that is, the L2(Ω×Ω; C) norm of k is finite), and the associated Hilbert–Schmidt integral operator is the operator K : L2(Ω; C) → L2(Ω; C) given by Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators). is compact. If (en)
sameAs	Hilbert–Schmidt integral operator
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Functional_analysis
Subject	Operator theory
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Hilbert–Schmidt_integral_operator?oldid=986771357&ns=0
Wikipage page ID	12366193 (xsd:integer)
page length (characters) of wiki page	2916 (xsd:nonNegativeInteger)
Wikipage revision ID	986771357 (xsd:integer)
Link from a Wikipage to another Wikipage	Locally compact space Borel measure Self-adjoint operator Separable space Operator theory Dimension Hilbert space Hilbert–Schmidt operator Compact operator Connected space Continuous function Open set Spectral theorem Hausdorff space Integral transform Euclidean space Mathematics
has abstract	In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k : Ω × Ω → C with (that is, the L2(Ω×Ω; C) norm of k is finite), and the associated Hilbert–Schmidt integral operator is the operator K : L2(Ω; C) → L2(Ω; C) given by Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators). The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let X be a locally compact Hausdorff space equipped with a positive Borel measure. Suppose further that L2(X) is a separable Hilbert space. The above condition on the kernel k on Rn can be interpreted as demanding k belong to L2(X × X). Then the operator is compact. If then K is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces. See Chapter 2 of the book by Bump in the references for examples. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Hilbert–Schmidt_integral_operator
is Wikipage redirect of	Hilbert-Schmidt integral operator Hilbert-schmidt integral operator Hilbert-Schmidt kernel
is Link from a Wikipage to another Wikipage of	List of things named after David Hilbert List of transforms Fredholm alternative Hilbert-Schmidt integral operator Hilbert-schmidt integral operator Hilbert operator Hilbert–Schmidt Hilbert–Schmidt operator Compact operator Erhard Schmidt Kernel embedding of distributions Hilbert-Schmidt kernel Mercer's theorem
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Hilbert–Schmidt_integral_operator

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