Editing Hilbert–Schmidt operator (section)

== 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]].