Editing Hilbert–Schmidt operator (section)

== 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}}