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In [[mathematics]], a '''Hilbert–Schmidt operator''', named for [[David Hilbert]] and [[Erhard Schmidt]], is a [[bounded operator]] ''A'' on a [[Hilbert space]] ''H'' with finite '''Hilbert–Schmidt norm''' | |||
:<math>\|A\|^2_{HS}={\rm Tr} |(A^{{}^*}A)|:= \sum_{i \in I} \|Ae_i\|^2 </math> | |||
where <math>\|\ \|</math> is the norm of ''H'' and <math>\{e_i : i\in I\} </math> an [[orthonormal basis]] of ''H'' for an index set <math>I</math>.<ref>{{cite web|last=Moslehian|first=M.S.|title=Hilbert–Schmidt Operator (From MathWorld)|url=http://mathworld.wolfram.com/Hilbert-SchmidtOperator.html}}</ref><ref>{{eom|first=M.I.|last=Voitsekhovskii|id=H/h047350}}</ref> Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore | |||
:<math>\|A\|^2_{HS}=\sum_{i,j} |A_{i,j}|^2 = \|A\|^2_2</math> | |||
for <math>A_{i,j}=\langle e_i, Ae_j \rangle</math> and <math>\|A\|_2</math> the [[Schatten norm]] of <math>A</math>. In [[Euclidean space]] <math>\|\ \|_{HS}</math> is also called [[Matrix norm#Frobenius norm|Frobenius norm]], named for [[Ferdinand Georg Frobenius]]. | |||
The product of two Hilbert–Schmidt operators has finite [[trace class| trace class norm]]; therefore, if ''A'' and ''B'' are two Hilbert–Schmidt operators, the '''Hilbert–Schmidt inner product''' can be defined as | |||
:<math>\langle A,B \rangle_\mathrm{HS} = \operatorname{tr} (A^*B) | |||
= \sum_{i} \langle Ae_i, Be_i \rangle.</math> | |||
The Hilbert–Schmidt operators form a two-sided [[ideal (ring theory)|*-ideal]] in the [[Banach algebra]] of bounded operators on ''H''. They also form a Hilbert space, which can be shown to be [[Natural transformation|natural]]ly isometrically isomorphic to the [[tensor product of Hilbert spaces]] | |||
:<math>H^* \otimes H, \, </math> | |||
where ''H*'' is the [[dual space]] of ''H''. | |||
The set of Hilbert–Schmidt operators is closed in the [[norm topology]] if, and only if, ''H'' is finite dimensional. | |||
An important class of examples is provided by [[Hilbert–Schmidt integral operator]]s. | |||
Hilbert–Schmidt operators are [[nuclear operator]]s of order 2, and are therefore compact. | |||
==Functionals== | |||
A mapping <math>\phi:H_1\times H_2\to C</math> is a '''Hilbert-Schmidt functional''' if it is a bounded bilinear functional.<ref>{{Citation | last1=Kadison | first1=Richard V. | last2=Ringrose | first2=John R. | title=Fundamentals of the theory of operator algebras. Vol. I | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-0819-1 | id={{MathSciNet | id = 1468229}} | year=1997 | volume=15}} (''see p. 127'')</ref> | |||
A bounded linear mapping <math>L:H_1\times H_2\to K</math> is '''weakly Hilbert-Schmidt''' if for all <math>v\in K</math> the mapping | |||
:<math>\phi_v = (u_1,u_2)\mapsto\langle L(u_1,u_2), v\rangle</math> | |||
is a Hilbert-Schmidt functional and <math>\|\phi_v\|\leq M\|v\|</math> for some real number <math>M\geq 0</math>.<ref>Kadison and Ringrose, (''see p. 131'')</ref> | |||
==References== | |||
{{Reflist}} | |||
{{Functional Analysis}} | |||
{{DEFAULTSORT:Hilbert-Schmidt Operator}} | |||
[[Category:Operator theory]] | |||
Latest revision as of 19:30, 26 January 2014
In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm
where is the norm of H and an orthonormal basis of H for an index set .[1][2] Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore
for and the Schatten norm of . In Euclidean space is also called Frobenius norm, named for Ferdinand Georg Frobenius.
The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if A and B are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as
The Hilbert–Schmidt operators form a two-sided *-ideal in the Banach algebra of bounded operators on H. They also form a Hilbert space, which can be shown to be naturally isometrically isomorphic to the tensor product of Hilbert spaces
where H* is the dual space of H.
The set of Hilbert–Schmidt operators is closed in the norm topology if, and only if, H is finite dimensional.
An important class of examples is provided by Hilbert–Schmidt integral operators.
Hilbert–Schmidt operators are nuclear operators of order 2, and are therefore compact.
Functionals
A mapping is a Hilbert-Schmidt functional if it is a bounded bilinear functional.[3]
A bounded linear mapping is weakly Hilbert-Schmidt if for all the mapping
is a Hilbert-Schmidt functional and for some real number .[4]
References
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- ↑ Template:Cite web
- ↑ Template:Eom
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