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An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473501.png" /> acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473502.png" /> such that for any orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473503.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473504.png" /> the following condition is met:
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<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/h047350/h0473505.png" /></td> </tr></table>
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 +
 
 +
An operator  $  A $
 +
acting on a Hilbert space  $  H $
 +
such that for any orthonormal basis  $  \{ x _ {i} \} $
 +
in  $  H $
 +
the following condition is met:
 +
 
 +
$$
 +
\| A \|  ^ {2}  = \
 +
\sum _ { i } \| Ax _ {i} \|  ^ {2}  < \infty
 +
$$
  
 
(however, this need be true for some basis only). A Hilbert–Schmidt operator is a [[Compact operator|compact operator]] for which the condition
 
(however, this need be true for some basis only). A Hilbert–Schmidt operator is a [[Compact operator|compact operator]] for which the condition
  
<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/h047350/h0473506.png" /></td> </tr></table>
+
$$
 +
\sum _ { i } | \lambda _ {i} ( A) |  ^ {2}
 +
\leq  \sum _ { i } s _ {i}  ^ {2} ( A)  = \
 +
\| A \|  ^ {2}  =   \mathop{\rm Tr} ( A  ^ {*} A);
 +
$$
  
applies to its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473507.png" />-numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473508.png" /> and its eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h0473509.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735010.png" /> is a trace-class operator (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735011.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735013.png" /> is the trace of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735014.png" />). The set of all Hilbert–Schmidt operators on a fixed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735015.png" /> forms a Hilbert space with scalar product
+
applies to its $  s $-
 +
numbers $  s _ {i} ( A) $
 +
and its eigen values $  \lambda _ {i} ( A) $;  
 +
here $  A  ^ {*} A $
 +
is a trace-class operator ( $  A  ^ {*} $
 +
is the adjoint of $  A $
 +
and $  \mathop{\rm Tr}  C $
 +
is the trace of an operator $  C $).  
 +
The set of all Hilbert–Schmidt operators on a fixed space $  A $
 +
forms a Hilbert space with scalar product
  
<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/h047350/h04735016.png" /></td> </tr></table>
+
$$
 +
\langle  A, B \rangle  = \
 +
\mathop{\rm Tr} ( AB  ^ {*} ).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735017.png" /> is the resolvent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735018.png" /> and
+
If $  R _  \lambda  ( A) = ( A - \lambda E )  ^ {-} 1 $
 +
is the resolvent of $  A $
 +
and
  
<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/h047350/h04735019.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm det} _ {2} ( E - zA)  = \
 +
\prod _ { i } ( 1 - z \lambda _ {i} ( A)) e ^ {z \lambda _ {i} ( A) }
 +
$$
  
 
is its regularized characteristic determinant, then the Carleman inequality
 
is its regularized characteristic determinant, then the Carleman inequality
  
<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/h047350/h04735020.png" /></td> </tr></table>
+
$$
 +
\left \|  \mathop{\rm det} _ {2} \left ( E - {
 +
\frac{1} \lambda
 +
} A \right )
 +
R _  \lambda  ( A) \right \|
 +
\leq  | \lambda |  \mathop{\rm exp} \left [
 +
{
 +
\frac{1}{2}
 +
} \left ( 1 +
 +
\frac{\| A \|  ^ {2} }{| \lambda |  ^ {2} }
 +
 
 +
\right )  \right ]
 +
$$
  
 
holds.
 
holds.
  
 
A typical representative of a Hilbert–Schmidt operator is a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt integral operator]] (which explains the origin of the name).
 
A typical representative of a Hilbert–Schmidt operator is a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt integral operator]] (which explains the origin of the name).
 
 
  
 
====Comments====
 
====Comments====
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735022.png" />-numbers or [[singular value]]s of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735023.png" /> are the (positive) eigen values of the self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735024.png" />. Instead of Hilbert–Schmidt operator one also says  "operator of Hilbert–Schmidt class" . A bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735025.png" /> on a Hilbert space is said to be of trace class if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735026.png" /> for arbitrary complete orthonormal systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735028.png" />. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735029.png" /> is of trace class if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735030.png" />. The trace of such an operator is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735032.png" /> is any orthonormal basis. The product of two Hilbert–Schmidt operators is of trace class and the converse is also true.
+
The $  s $-
 +
numbers or [[singular value]]s of $  A $
 +
are the (positive) eigen values of the self-adjoint operator $  A  ^ {*} A $.  
 +
Instead of Hilbert–Schmidt operator one also says  "operator of Hilbert–Schmidt class" . A bounded operator $  T $
 +
on a Hilbert space is said to be of trace class if $  \sum \langle  T \phi _ {j,\ } \psi _ {j} \rangle < \infty $
 +
for arbitrary complete orthonormal systems $  \{ \phi _ {j} \} $,  
 +
$  \{ \psi _ {j} \} $.  
 +
Equivalently, $  T $
 +
is of trace class if $  \sum s _ {i} ( T) < \infty $.  
 +
The trace of such an operator is defined as $  \sum \langle  T \phi _ {j} , \phi _ {j} \rangle $,  
 +
where $  \phi _ {j} $
 +
is any orthonormal basis. The product of two Hilbert–Schmidt operators is of trace class and the converse is also true.
  
The norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735033.png" /> in the above article is not the usual operator norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047350/h04735034.png" /> but its [[Hilbert–Schmidt norm|Hilbert–Schmidt norm]].
+
The norm $  \| A \| $
 +
in the above article is not the usual operator norm of $  A $
 +
but its [[Hilbert–Schmidt norm|Hilbert–Schmidt norm]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a3]</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>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a3]</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 22:10, 5 June 2020


An operator $ A $ acting on a Hilbert space $ H $ such that for any orthonormal basis $ \{ x _ {i} \} $ in $ H $ the following condition is met:

$$ \| A \| ^ {2} = \ \sum _ { i } \| Ax _ {i} \| ^ {2} < \infty $$

(however, this need be true for some basis only). A Hilbert–Schmidt operator is a compact operator for which the condition

$$ \sum _ { i } | \lambda _ {i} ( A) | ^ {2} \leq \sum _ { i } s _ {i} ^ {2} ( A) = \ \| A \| ^ {2} = \mathop{\rm Tr} ( A ^ {*} A); $$

applies to its $ s $- numbers $ s _ {i} ( A) $ and its eigen values $ \lambda _ {i} ( A) $; here $ A ^ {*} A $ is a trace-class operator ( $ A ^ {*} $ is the adjoint of $ A $ and $ \mathop{\rm Tr} C $ is the trace of an operator $ C $). The set of all Hilbert–Schmidt operators on a fixed space $ A $ forms a Hilbert space with scalar product

$$ \langle A, B \rangle = \ \mathop{\rm Tr} ( AB ^ {*} ). $$

If $ R _ \lambda ( A) = ( A - \lambda E ) ^ {-} 1 $ is the resolvent of $ A $ and

$$ \mathop{\rm det} _ {2} ( E - zA) = \ \prod _ { i } ( 1 - z \lambda _ {i} ( A)) e ^ {z \lambda _ {i} ( A) } $$

is its regularized characteristic determinant, then the Carleman inequality

$$ \left \| \mathop{\rm det} _ {2} \left ( E - { \frac{1} \lambda } A \right ) R _ \lambda ( A) \right \| \leq | \lambda | \mathop{\rm exp} \left [ { \frac{1}{2} } \left ( 1 + \frac{\| A \| ^ {2} }{| \lambda | ^ {2} } \right ) \right ] $$

holds.

A typical representative of a Hilbert–Schmidt operator is a Hilbert–Schmidt integral operator (which explains the origin of the name).

Comments

The $ s $- numbers or singular values of $ A $ are the (positive) eigen values of the self-adjoint operator $ A ^ {*} A $. Instead of Hilbert–Schmidt operator one also says "operator of Hilbert–Schmidt class" . A bounded operator $ T $ on a Hilbert space is said to be of trace class if $ \sum \langle T \phi _ {j,\ } \psi _ {j} \rangle < \infty $ for arbitrary complete orthonormal systems $ \{ \phi _ {j} \} $, $ \{ \psi _ {j} \} $. Equivalently, $ T $ is of trace class if $ \sum s _ {i} ( T) < \infty $. The trace of such an operator is defined as $ \sum \langle T \phi _ {j} , \phi _ {j} \rangle $, where $ \phi _ {j} $ is any orthonormal basis. The product of two Hilbert–Schmidt operators is of trace class and the converse is also true.

The norm $ \| A \| $ in the above article is not the usual operator norm of $ A $ but its Hilbert–Schmidt norm.

References

[a1] M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972)
[a2] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977)
[a3] 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 operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Schmidt_operator&oldid=47227
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article