Namespaces
Variants
Actions

Difference between revisions of "Hilbert-Schmidt norm"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX done)
 
Line 1: Line 1:
The norm of a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473401.png" /> acting from a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473402.png" /> into a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473403.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473404.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473405.png" /> is an orthonormal basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473406.png" />. The Hilbert–Schmidt norm satisfies all the axioms of a norm and is independent of the choice of the basis. Its properties are: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h0473409.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h04734010.png" /> is the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h04734011.png" /> in the Hilbert space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h04734012.png" />, then
+
The norm of a linear operator $T$ acting from a Hilbert space $H$ into a Hilbert space $H_1$, given by
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047340/h04734013.png" /></td> </tr></table>
+
|T| = \left({\sum_{\alpha\in A} \Vert Te_\alpha \Vert^2}\right)^{1/2} \,,
 +
$$
 +
where $\{e_\alpha : \alpha \in A \}$ is an orthonormal basis in $H$. The Hilbert–Schmidt norm satisfies all the axioms of a norm and is independent of the choice of the basis. Its properties are: $\Vert T \Vert \le |T|$, $|T| = |T^*|$, $|T_1T_2| \le \Vert T_1\Vert \cdot |T_2|$, where $\Vert T\Vert$ is the [[operator norm]] of $T$ in the Hilbert space. If $H_1 = H$, then
 +
$$
 +
|T|^2 = \sum_{\alpha,\beta\in A} (Te_\alpha,e_\beta)^2 \ .
 +
$$
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
 +
A [[Hilbert–Schmidt operator]], or operator of Hilbert–Schmidt class, is one for which the Hilbert–Schmidt norm is well-defined: it is necessarily a [[compact operator]].
  
 +
====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR>
 +
</table>
  
====References====
+
{{TEX|done}}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR></table>
 

Latest revision as of 17:07, 29 October 2017

The norm of a linear operator $T$ acting from a Hilbert space $H$ into a Hilbert space $H_1$, given by $$ |T| = \left({\sum_{\alpha\in A} \Vert Te_\alpha \Vert^2}\right)^{1/2} \,, $$ where $\{e_\alpha : \alpha \in A \}$ is an orthonormal basis in $H$. The Hilbert–Schmidt norm satisfies all the axioms of a norm and is independent of the choice of the basis. Its properties are: $\Vert T \Vert \le |T|$, $|T| = |T^*|$, $|T_1T_2| \le \Vert T_1\Vert \cdot |T_2|$, where $\Vert T\Vert$ is the operator norm of $T$ in the Hilbert space. If $H_1 = H$, then $$ |T|^2 = \sum_{\alpha,\beta\in A} (Te_\alpha,e_\beta)^2 \ . $$

References

[1] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
[2] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)


Comments

A Hilbert–Schmidt operator, or operator of Hilbert–Schmidt class, is one for which the Hilbert–Schmidt norm is well-defined: it is necessarily a compact operator.

References

[a1] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
How to Cite This Entry:
Hilbert-Schmidt norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Schmidt_norm&oldid=22573
This article was adapted from an original article by V.B. Korotkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article