View source for Hilbert–Schmidt integral operator

In [[mathematics]], a '''Hilbert–Schmidt integral operator''' is a type of [[integral transform]]. Specifically, given a [[Domain_(mathematical_analysis)|domain]] {{math|&Omega;}} in {{math|''n''}}-[[dimension]]al [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}}, then the [[square-integrable function]] {{math|''k''&nbsp;:&nbsp;&Omega;&nbsp;&times;&nbsp;&Omega;&nbsp;&rarr;&nbsp;'''C'''}} belonging to {{math|''L''<sup>2</sup>(&Omega;×&Omega;)}} such that
:<math>\int_{\Omega} \int_{\Omega} | k(x, y) |^{2} \,dx \, dy < \infty ,</math>
is called a '''Hilbert–Schmidt kernel''' and the associated [[Integral_transform|integral operator]] {{math|''T''&nbsp;:&nbsp;''L''<sup>2</sup>(&Omega;)&nbsp;&rarr;&nbsp;''L''<sup>2</sup>(&Omega;)}} given by
:<math>(Tf) (x) = \int_{\Omega} k(x, y) f(y) \, dy, \quad f \in L^2(\Omega),</math>
is called a  '''Hilbert–Schmidt integral operator'''.{{sfn | Simon | 1978|p=14}}{{sfn | Bump | 1998 | pp=168}}
Then {{math|''T''}} is a [[Hilbert–Schmidt operator]] with Hilbert–Schmidt norm

:<math>\Vert T \Vert_\mathrm{HS} = \Vert k \Vert_{L^2}.</math>

Hilbert–Schmidt integral operators are both [[continuous function|continuous]] and [[compact operator|compact]].{{sfn | Renardy | Rogers | 2004 | pp=260,262}}

The concept of a Hilbert–Schmidt operator may be extended to any [[locally compact space|locally compact]] [[Hausdorff space]]s. Specifically, let {{math|''L''<sup>2</sup>(''X'')}} be a [[separable space|separable]] [[Hilbert space]] and {{math|''X''}} a locally compact Hausdorff space equipped with a positive [[Borel measure]]. The initial condition on the kernel {{math|''k''}} on {{math|&Omega; ⊆ '''R'''<sup>n</sup>}} can be reinterpreted as demanding {{math|''k''}} belong to {{math|''L''<sup>2</sup>(''X × X'')}}. Then the operator
:<math>(Tf)(x) = \int_{X} k(x,y)f(y)\,dy,</math>
is [[compact operator|compact]]. If 
:<math>k(x,y) = \overline{k(y,x)},</math>
then {{math|''T''}} is also [[self-adjoint operator|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.{{sfn | Bump | 1998 | pp=168-185}}

== See also ==
* [[Hilbert–Schmidt operator]]

== Notes ==
{{reflist}}
==References==

* {{cite book | last=Renardy | first=Michael | last2=Rogers | first2=Robert C. | title=An Introduction to Partial Differential Equations | publisher=Springer Science & Business Media | publication-place=New York Berlin Heidelberg | date=2004-01-08 | isbn=0-387-00444-0}}

* {{cite book | last=Bump | first=Daniel | title=Automorphic Forms and Representations | publisher=Cambridge University Press | date=1998 | isbn=0-521-65818-7}}
* {{cite web | last=Simon | first=B. | title=An Overview of Rigorous Scattering Theory | date=1978 | url=https://api.semanticscholar.org/CorpusID:16913591}}

{{Spectral theory}}
{{Functional analysis}}

{{DEFAULTSORT:Hilbert-Schmidt integral operator}}

[[Category:Linear operators]]
[[Category:Operator theory]]