Editing Hilbert–Schmidt operator

{{Short description|Topic in mathematics}}

In [[mathematics]], a '''Hilbert–Schmidt operator''', named after [[David Hilbert]] and [[Erhard Schmidt]], is a [[bounded operator]] <math> A \colon H \to H </math> that acts on a [[Hilbert space]] <math> H </math> and has finite '''Hilbert–Schmidt norm'''

<math display="block">\|A\|^2_{\operatorname{HS}} \ \stackrel{\text{def}}{=}\ \sum_{i \in I} \|Ae_i\|^2_H,</math>

where <math>\{e_i: i \in I\}</math> is an [[orthonormal basis]].<ref name="MathWorld">{{cite web |last=Moslehian |first=M. S. |title=Hilbert–Schmidt Operator (From MathWorld) |url=http://mathworld.wolfram.com/Hilbert-SchmidtOperator.html}}</ref><ref name="EOM">{{eom |first=M. I. |last=Voitsekhovskii |title=Hilbert-Schmidt operator}}</ref> The index set <math> I </math> need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning.{{sfn | Schaefer | 1999 | p=177}} This definition is independent of the choice of the orthonormal basis. 
In finite-dimensional [[Euclidean space]], the Hilbert–Schmidt norm <math>\|\cdot\|_\text{HS}</math> is identical to the [[matrix norm#Frobenius norm|Frobenius norm]].

==‖·‖{{sub|HS}} is well defined==
The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if <math>\{e_i\}_{i\in I}</math> and <math>\{f_j\}_{j\in I}</math> are such bases, then
<math display="block">
\sum_i \|Ae_i\|^2 = \sum_{i,j} \left| \langle Ae_i, f_j\rangle \right|^2 = \sum_{i,j} \left| \langle e_i, A^*f_j\rangle \right|^2 = \sum_j\|A^* f_j\|^2.
</math>
If <math>e_i = f_i, </math> then <math display="inline"> \sum_i \|Ae_i\|^2 = \sum_i\|A^* e_i\|^2. </math> As for any bounded operator, <math> A = A^{**}. </math> Replacing <math> A </math> with <math> A^* </math> in the first formula, obtain <math display="inline"> \sum_i \|A^* e_i\|^2 = \sum_j\|A f_j\|^2. </math> The independence follows.

== Examples ==

An important class of examples is provided by [[Hilbert–Schmidt integral operator]]s. 
Every bounded operator with a finite-dimensional range (these are called operators of finite rank) is a Hilbert–Schmidt operator. 
The [[identity map|identity operator]] on a Hilbert space is a Hilbert–Schmidt operator if and only if the Hilbert space is finite-dimensional. 
Given any <math>x</math> and <math>y</math> in <math>H</math>, define <math>x \otimes y : H \to H</math> by <math>(x \otimes y)(z) = \langle z, y \rangle x</math>, which is a continuous linear operator of rank 1 and thus a Hilbert–Schmidt operator; 
moreover, for any bounded linear operator ''<math>A</math>'' on <math>H</math> (and into <math>H</math>), <math>\operatorname{Tr}\left( A\left( x \otimes y \right) \right) = \left\langle A x, y \right\rangle</math>.{{sfn | Conway | 1990 | p=268}} 

If <math>T: H \to H</math> is a bounded compact operator with eigenvalues <math>\ell_1, \ell_2, \dots</math> of <math>|T| := \sqrt{T^*T}</math>, where each eigenvalue is repeated as often as its multiplicity, then <math>T</math> is Hilbert–Schmidt if and only if <math display="inline">\sum_{i=1}^{\infty} \ell_i^2 < \infty</math>, in which case the Hilbert–Schmidt norm of <math>T</math> is <math display="inline">\left\| T \right\|_{\operatorname{HS}} = \sqrt{\sum_{i=1}^{\infty} \ell_i^2}</math>.{{sfn | Conway | 1990 | p=267}} 

If <math>k \in L^2\left( \mu \times \mu \right)</math>, where <math>\left( X, \Omega, \mu \right)</math> is a measure space, then the integral operator <math>K : L^2\left( \mu \right) \to L^2\left( \mu \right)</math> with kernel <math>k</math> is a Hilbert–Schmidt operator and <math>\left\| K \right\|_{\operatorname{HS}} = \left\| k \right\|_2</math>.{{sfn | Conway | 1990 | p=267}}

== Space of Hilbert–Schmidt operators ==

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 display="block">\langle A, B \rangle_\text{HS} = \operatorname{Tr}(B^* A) = \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 {{math|''H''}}. 
They also form a Hilbert space, denoted by {{math|''B''<sub>HS</sub>(''H'')}} or {{math|''B''<sub>2</sub>(''H'')}}, which can be shown to be [[Natural transformation|naturally]] isometrically isomorphic to the [[tensor product of Hilbert spaces]]

<math display="block">H^* \otimes H,</math>

where {{math|''H''<sup>∗</sup>}} is the [[dual space]] of {{math|''H''}}. 
The norm induced by this inner product is the Hilbert–Schmidt norm under which the space of Hilbert–Schmidt operators is complete (thus making it into a Hilbert space).{{sfn | Conway | 1990 | p=268}} 
The space of all bounded linear operators of finite rank (i.e. that have a finite-dimensional range) is a dense subset of the space of Hilbert–Schmidt operators (with the Hilbert–Schmidt norm).{{sfn | Conway | 1990 | p=268}} 

The set of Hilbert–Schmidt operators is closed in the [[norm topology]] if, and only if, {{math|''H''}} is finite-dimensional.

== Properties ==

* Every Hilbert–Schmidt operator {{math|''T'' : ''H'' → ''H''}} is a [[compact operator]].{{sfn | Conway | 1990 | p=267}} 
* A bounded linear operator {{math|''T'' : ''H'' → ''H''}} is Hilbert–Schmidt if and only if the same is true of the operator <math display="inline">\left| T \right| := \sqrt{T^* T}</math>, in which case the Hilbert–Schmidt norms of ''T'' and |''T''| are equal.{{sfn | Conway | 1990 | p=267}} 
* Hilbert–Schmidt operators are [[nuclear operator]]s of order 2, and are therefore [[compact operator]]s.{{sfn | Conway | 1990 | p=267}} 
* If <math>S : H_1 \to H_2</math> and <math>T : H_2 \to H_3</math> are Hilbert–Schmidt operators between Hilbert spaces then the composition <math>T \circ S : H_1 \to H_3</math> is a [[nuclear operator]].{{sfn | Schaefer | 1999 | p=177}}
* If {{math|''T'' : ''H'' → ''H''}} is a bounded linear operator then we have <math>\left\| T \right\| \leq \left\| T \right\|_{\operatorname{HS}}</math>.{{sfn | Conway | 1990 | p=267}} 
* {{math|''T''}} is a Hilbert–Schmidt operator if and only if the [[trace class|trace]] <math>\operatorname{Tr}</math> of the nonnegative self-adjoint operator <math>T^{*} T</math> is finite, in which case <math>\|T\|^2_\text{HS} = \operatorname{Tr}(T^* T)</math>.<ref name="MathWorld"/><ref name="EOM"/>
* If {{math|''T'' : ''H'' → ''H''}} is a bounded linear operator on {{math|''H''}} and {{math|''S'' : ''H'' → ''H''}} is a Hilbert–Schmidt operator on {{math|''H''}} then <math>\left\| S^* \right\|_{\operatorname{HS}} = \left\| S \right\|_{\operatorname{HS}}</math>, <math>\left\| T S \right\|_{\operatorname{HS}} \leq \left\| T \right\| \left\| S \right\|_{\operatorname{HS}}</math>, and <math>\left\| S T \right\|_{\operatorname{HS}} \leq \left\| S \right\|_{\operatorname{HS}} \left\| T \right\|</math>.{{sfn | Conway | 1990 | p=267}} In particular, the composition of two Hilbert–Schmidt operators is again Hilbert–Schmidt (and even a [[trace class operator]]).{{sfn | Conway | 1990 | p=267}} 
* The space of Hilbert–Schmidt operators on {{math|''H''}} is an [[Ideal (ring theory)|ideal]] of the space of bounded operators <math>B\left( H \right)</math> that contains the operators of finite-rank.{{sfn | Conway | 1990 | p=267}}
* If {{math|''A''}} is a Hilbert–Schmidt operator on {{math|''H''}} then <math display="block">\|A\|^2_\text{HS} = \sum_{i,j} |\langle e_i, Ae_j \rangle|^2 = \|A\|^2_2</math> where <math>\{e_i: i \in I\}</math> is an [[orthonormal basis]] of ''H'', and <math>\|A\|_2</math> is the [[Schatten norm]] of <math>A</math> for {{math|1=''p'' = 2}}.  In [[Euclidean space]], <math>\|\cdot\|_\text{HS}</math> is also called the [[matrix norm#Frobenius norm|Frobenius norm]].

==See also==

* {{annotated link|Frobenius inner product}}
* {{annotated link|Sazonov's theorem}}
* {{annotated link|Trace class}}

==References==

{{Reflist}}
* {{cite book | last=Conway | first=John B.|authorlink = John B. Conway| title=A course in functional analysis | publisher=Springer-Verlag | publication-place=New York | year=1990 | isbn=978-0-387-97245-9 | oclc=21195908}} <!--- {{sfn | Conway | 1990 | p=}} ---> 
* {{cite book | last=Schaefer | first=Helmut H.|authorlink = Helmut H. Schaefer| title=Topological Vector Spaces | publisher=Springer New York Imprint Springer | series=[[Graduate Texts in Mathematics|GTM]] | volume=3 | location=New York, NY | year=1999 | isbn=978-1-4612-7155-0 | oclc=840278135 }} <!-- {{sfn | Schaefer | 1999 | p=}} -->

{{Hilbert space}}
{{Topological tensor products and nuclear spaces}}
{{Functional analysis}}

{{DEFAULTSORT:Hilbert-Schmidt Operator}}

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