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''of eigen values''
 
''of eigen values''
  
A special type of relationship connecting the eigen values of a completely-continuous [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639601.png" /> (cf. also [[Completely-continuous operator|Completely-continuous operator]]) with the maximum and minimum values of the associated quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639602.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639603.png" /> be a completely-continuous self-adjoint operator on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639604.png" />. The spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639605.png" /> consists of a finite or countable set of real eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639606.png" /> having unique limit point zero. The root subspaces corresponding to the non-zero eigen values consist of eigen vectors and are finite dimensional; the eigen subspaces associated with distinct eigen values are mutually orthogonal; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639607.png" /> has a complete system of eigen vectors. The spectral decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639608.png" /> (cf. [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]]) has the form: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m0639609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396010.png" /> are the distinct eigen values, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396011.png" /> are the projection operators onto the corresponding eigen spaces, and the series converges in the operator norm. The norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396012.png" /> coincides with the maximum modulus of the eigen values and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396013.png" />; the maximum is attained at the corresponding eigen vector.
+
A special type of relationship connecting the eigen values of a completely-continuous [[Self-adjoint operator|self-adjoint operator]] $  A $(
 +
cf. also [[Completely-continuous operator|Completely-continuous operator]]) with the maximum and minimum values of the associated quadratic form $  ( A x , x ) $.  
 +
Let $  A $
 +
be a completely-continuous self-adjoint operator on a Hilbert space $  H $.  
 +
The spectrum of $  A $
 +
consists of a finite or countable set of real eigen values $  \lambda _ {n} $
 +
having unique limit point zero. The root subspaces corresponding to the non-zero eigen values consist of eigen vectors and are finite dimensional; the eigen subspaces associated with distinct eigen values are mutually orthogonal; $  A $
 +
has a complete system of eigen vectors. The spectral decomposition of $  A $(
 +
cf. [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]]) has the form: $  A = \sum \lambda _ {i} P _ {i} $,  
 +
where $  \lambda _ {i} $
 +
are the distinct eigen values, $  P _ {i} $
 +
are the projection operators onto the corresponding eigen spaces, and the series converges in the operator norm. The norm of $  A $
 +
coincides with the maximum modulus of the eigen values and with $  \max  \{ {| ( A x , x ) | } : {x \in H,  | x | = 1 } \} $;  
 +
the maximum is attained at the corresponding eigen vector.
 +
 
 +
Let  $  \lambda _ {1}  ^ {+} \geq  \lambda _ {2}  ^ {+} \geq  \dots $
 +
be the positive eigen values of  $  A $,
 +
where each eigen value is repeated as often as its multiplicity. Then
 +
 
 +
$$ \tag{1 }
 +
\left . \begin{array}{c}
 +
 
 +
\lambda _ {1}  ^ {+}  = \
 +
\max _ {x \in H } 
 +
\frac{( A x , x ) }{| x |  ^ {2} }
 +
,
 +
\\
 +
 
 +
\lambda _ {n+} 1  ^ {+}  = \
 +
\min _ {y _ {1} \dots y _ {n} } \
 +
\max _ { {( x , y _ {i} ) = 0 }  {i = 1 \dots n } } \
 +
 
 +
\frac{( A x , x ) }{| x |  ^ {2} }
 +
,\ \
 +
n > 1 ,
 +
\end{array}
 +
 
 +
\right \}
 +
$$
 +
 
 +
where  $  x , y _ {1} \dots y _ {n} $
 +
are arbitrary non-zero vectors in  $  H $.
 +
Similar relations hold for the negative eigen values  $  \lambda _ {1}  ^ {-} \geq  \lambda _ {2}  ^ {-} \geq  \dots $:
 +
 
 +
$$ \tag{2 }
 +
\left . \begin{array}{c}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396014.png" /> be the positive eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396015.png" />, where each eigen value is repeated as often as its multiplicity. Then
+
\lambda _ {1}  ^ {-}  = \
 +
\min _ {x \in H } 
 +
\frac{( A x , x ) }{| x |  ^ {2} }
 +
,
 +
\\
  
<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/m/m063/m063960/m06396016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\lambda _ {n+} 1  ^ {-= \
 +
\max _ {y _ {1} \dots y _ {n} } \
 +
\min _ { {( x , y _ {i} ) = 0 }  {i = 1 \dots n } } \
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396017.png" /> are arbitrary non-zero vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396018.png" />. Similar relations hold for the negative eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396019.png" />:
+
\frac{( A x , x ) }{| x |  ^ {2} }
 +
,\ \
 +
n > 1 .  
 +
\end{array}
  
<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/m/m063/m063960/m06396020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\right \}
 +
$$
  
Relations (1) and (2) are applied for finding the eigen values of integral operators with a symmetric kernel. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396022.png" /> are completely-continuous self-adjoint operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396023.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396024.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396026.png" /> the sequences of their positive eigen values, listed in decreasing order, where each value is repeated as often as its multiplicity, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063960/m06396027.png" />.
+
Relations (1) and (2) are applied for finding the eigen values of integral operators with a symmetric kernel. If $  A $
 +
and $  B $
 +
are completely-continuous self-adjoint operators, $  A \leq  B $(
 +
that is, $  ( Ax , x) \leq  ( B x , x ) $),  
 +
$  \lambda _ {n} $
 +
and $  \mu _ {n} $
 +
the sequences of their positive eigen values, listed in decreasing order, where each value is repeated as often as its multiplicity, then $  \lambda _ {n} \leq  \mu _ {n} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Wiley  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Wiley  (1988)</TD></TR></table>

Latest revision as of 08:00, 6 June 2020


of eigen values

A special type of relationship connecting the eigen values of a completely-continuous self-adjoint operator $ A $( cf. also Completely-continuous operator) with the maximum and minimum values of the associated quadratic form $ ( A x , x ) $. Let $ A $ be a completely-continuous self-adjoint operator on a Hilbert space $ H $. The spectrum of $ A $ consists of a finite or countable set of real eigen values $ \lambda _ {n} $ having unique limit point zero. The root subspaces corresponding to the non-zero eigen values consist of eigen vectors and are finite dimensional; the eigen subspaces associated with distinct eigen values are mutually orthogonal; $ A $ has a complete system of eigen vectors. The spectral decomposition of $ A $( cf. Spectral decomposition of a linear operator) has the form: $ A = \sum \lambda _ {i} P _ {i} $, where $ \lambda _ {i} $ are the distinct eigen values, $ P _ {i} $ are the projection operators onto the corresponding eigen spaces, and the series converges in the operator norm. The norm of $ A $ coincides with the maximum modulus of the eigen values and with $ \max \{ {| ( A x , x ) | } : {x \in H, | x | = 1 } \} $; the maximum is attained at the corresponding eigen vector.

Let $ \lambda _ {1} ^ {+} \geq \lambda _ {2} ^ {+} \geq \dots $ be the positive eigen values of $ A $, where each eigen value is repeated as often as its multiplicity. Then

$$ \tag{1 } \left . \begin{array}{c} \lambda _ {1} ^ {+} = \ \max _ {x \in H } \frac{( A x , x ) }{| x | ^ {2} } , \\ \lambda _ {n+} 1 ^ {+} = \ \min _ {y _ {1} \dots y _ {n} } \ \max _ { {( x , y _ {i} ) = 0 } {i = 1 \dots n } } \ \frac{( A x , x ) }{| x | ^ {2} } ,\ \ n > 1 , \end{array} \right \} $$

where $ x , y _ {1} \dots y _ {n} $ are arbitrary non-zero vectors in $ H $. Similar relations hold for the negative eigen values $ \lambda _ {1} ^ {-} \geq \lambda _ {2} ^ {-} \geq \dots $:

$$ \tag{2 } \left . \begin{array}{c} \lambda _ {1} ^ {-} = \ \min _ {x \in H } \frac{( A x , x ) }{| x | ^ {2} } , \\ \lambda _ {n+} 1 ^ {-} = \ \max _ {y _ {1} \dots y _ {n} } \ \min _ { {( x , y _ {i} ) = 0 } {i = 1 \dots n } } \ \frac{( A x , x ) }{| x | ^ {2} } ,\ \ n > 1 . \end{array} \right \} $$

Relations (1) and (2) are applied for finding the eigen values of integral operators with a symmetric kernel. If $ A $ and $ B $ are completely-continuous self-adjoint operators, $ A \leq B $( that is, $ ( Ax , x) \leq ( B x , x ) $), $ \lambda _ {n} $ and $ \mu _ {n} $ the sequences of their positive eigen values, listed in decreasing order, where each value is repeated as often as its multiplicity, then $ \lambda _ {n} \leq \mu _ {n} $.

References

[1] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Wiley (1988)
How to Cite This Entry:
Minimax property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimax_property&oldid=19194
This article was adapted from an original article by A.I. Loginov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article