Namespaces
Variants
Actions

Difference between revisions of "Resolution of the identity"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A one-parameter family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815902.png" />, of orthogonal projection operators acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815903.png" />, such that
+
<!--
 +
r0815901.png
 +
$#A+1 = 51 n = 0
 +
$#C+1 = 51 : ~/encyclopedia/old_files/data/R081/R.0801590 Resolution of the identity
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815904.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815905.png" />;
+
{{TEX|auto}}
 +
{{TEX|done}}
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815906.png" /> is strongly left continuous, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815907.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815908.png" />;
+
A one-parameter family  $  \{ E _  \lambda  \} $,  
 +
$  - \infty < \lambda < \infty $,
 +
of orthogonal projection operators acting on a Hilbert space  $  {\mathcal H} $,
 +
such that
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r0815909.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159011.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159012.png" />; here 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159013.png" /> are the zero and the identity operator on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159014.png" />.
+
1) $  E _  \lambda  \leq  E _  \mu  $
 +
if  $  \lambda < \mu $;
  
Condition 2) can be replaced by the condition of strong right continuity at every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159015.png" />.
+
2) $  E _  \lambda  $
 +
is strongly left continuous, i.e.  $  E _ {\lambda - 0 = E _  \lambda  $
 +
for every  $  \lambda \in ( - \infty , \infty ) $;
  
Every [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159016.png" /> acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159017.png" /> generates in a unique way a resolution of the identity. Here, in addition to 1)–3), the following conditions also hold:
+
3)  $  E _  \lambda  \rightarrow 0 $
 +
as  $  \lambda \rightarrow - \infty $
 +
and  $  E _  \lambda  \rightarrow E $
 +
as  $  \lambda \rightarrow \infty $;
 +
here 0 and  $  E $
 +
are the zero and the identity operator on the space  $  {\mathcal H} $.
  
4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159018.png" /> is a bounded operator such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159019.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159020.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159021.png" />;
+
Condition 2) can be replaced by the condition of strong right continuity at every point  $  \lambda \in ( - \infty , \infty ) $.
  
5) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159022.png" /> is a bounded operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159024.png" /> are its greatest lower and least upper bounds, respectively, then
+
Every [[Self-adjoint operator|self-adjoint operator]]  $  A $
 +
acting on  $  {\mathcal H} $
 +
generates in a unique way a resolution of the identity. Here, in addition to 1)–3), the following conditions also hold:
  
<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/r/r081/r081590/r08159025.png" /></td> </tr></table>
+
4) if  $  B $
 +
is a bounded operator such that  $  B A = A B $,
 +
then  $  B E _  \lambda  = E _  \lambda  B $
 +
for any  $  \lambda $;
  
The resolution of the identity given by the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159026.png" /> completely determines the spectral properties of that operator, namely:
+
5) if  $  A $
 +
is a bounded operator and  $  m $,
 +
$  M $
 +
are its greatest lower and least upper bounds, respectively, then
  
a) a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159027.png" /> is a regular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159028.png" /> if and only if it is a point of constancy, that is, if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159031.png" />;
+
$$
 +
E _  \lambda  = 0 \textrm{ for }  - \infty < \lambda < m \ \
 +
\textrm{ and } \  E _  \lambda  = E  \textrm{ for }  M < \lambda < \infty .
 +
$$
  
b) a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159032.png" /> is an eigenvalue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159033.png" /> if and only if at this point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159034.png" /> has a jump, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159035.png" />;
+
The resolution of the identity given by the operator  $  A $
 +
completely determines the spectral properties of that operator, namely:
  
g) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159037.png" /> is an invariant subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159038.png" />.
+
a) a point  $  \lambda $
 +
is a regular point of  $  A $
 +
if and only if it is a point of constancy, that is, if there is a  $  \delta > 0 $
 +
such that  $  E _  \mu  = E _  \lambda  $
 +
for  $  \mu \in ( \lambda - \delta , \lambda + \delta ) $;
  
Hence the resolution of the identity determined by the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159039.png" /> is also called the spectral function of this operator (cf. [[Spectral resolution|Spectral resolution]]).
+
b) a point  $  \lambda _ {0} $
 +
is an eigenvalue of $  A $
 +
if and only if at this point  $  E _  \lambda  $
 +
has a jump, that is,  $  E _ {\lambda _ {0}  + 0 } - E _ {\lambda _ {0}  } > 0 $;
  
Conversely, every resolution of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159040.png" /> uniquely determines a self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159041.png" /> for which this resolution is the spectral function. The domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159043.png" /> consists exactly of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159044.png" /> for which
+
g) if  $  E ( \Delta ) = E _  \mu  - E _  \lambda  $,
 +
then  $  L _ {E ( \Delta ) }  = E ( \Delta ) {\mathcal H} $
 +
is an invariant subspace of $  A $.
  
<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/r/r081/r081590/r08159045.png" /></td> </tr></table>
+
Hence the resolution of the identity determined by the operator  $  A $
 +
is also called the spectral function of this operator (cf. [[Spectral resolution|Spectral resolution]]).
  
and there is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159046.png" /> as an operator Stieltjes integral:
+
Conversely, every resolution of the identity  $  \{ E _  \lambda  \} $
 +
uniquely determines a self-adjoint operator  $  A $
 +
for which this resolution is the spectral function. The domain of definition  $  D ( A) $
 +
of  $  A $
 +
consists exactly of those  $  x \in {\mathcal H} $
 +
for which
  
<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/r/r081/r081590/r08159047.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {- \infty } ^  \infty  \lambda  ^ {2}  d \langle  E _  \lambda  x , x \rangle
 +
< \infty ,
 +
$$
 +
 
 +
and there is a representation of  $  A $
 +
as an operator Stieltjes integral:
 +
 
 +
$$
 +
= \int\limits _ {- \infty } ^  \infty  \lambda  d E _  \lambda  .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in a Hilbert space" , '''1–2''' , F. Ungar  (1961–1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis in normed spaces" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in a Hilbert space" , '''1–2''' , F. Ungar  (1961–1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,  G.P. Akilov,  "Functional analysis in normed spaces" , Pergamon  (1964)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
To the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159048.png" />) mentioned above one may add that the spectrum of the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159049.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159050.png" /> is contained in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081590/r08159051.png" />.
+
To the property $  \gamma $)  
 +
mentioned above one may add that the spectrum of the restriction of $  A $
 +
to $  L _ {E( \Delta ) }  $
 +
is contained in the set $  \Delta $.

Latest revision as of 08:11, 6 June 2020


A one-parameter family $ \{ E _ \lambda \} $, $ - \infty < \lambda < \infty $, of orthogonal projection operators acting on a Hilbert space $ {\mathcal H} $, such that

1) $ E _ \lambda \leq E _ \mu $ if $ \lambda < \mu $;

2) $ E _ \lambda $ is strongly left continuous, i.e. $ E _ {\lambda - 0 } = E _ \lambda $ for every $ \lambda \in ( - \infty , \infty ) $;

3) $ E _ \lambda \rightarrow 0 $ as $ \lambda \rightarrow - \infty $ and $ E _ \lambda \rightarrow E $ as $ \lambda \rightarrow \infty $; here 0 and $ E $ are the zero and the identity operator on the space $ {\mathcal H} $.

Condition 2) can be replaced by the condition of strong right continuity at every point $ \lambda \in ( - \infty , \infty ) $.

Every self-adjoint operator $ A $ acting on $ {\mathcal H} $ generates in a unique way a resolution of the identity. Here, in addition to 1)–3), the following conditions also hold:

4) if $ B $ is a bounded operator such that $ B A = A B $, then $ B E _ \lambda = E _ \lambda B $ for any $ \lambda $;

5) if $ A $ is a bounded operator and $ m $, $ M $ are its greatest lower and least upper bounds, respectively, then

$$ E _ \lambda = 0 \textrm{ for } - \infty < \lambda < m \ \ \textrm{ and } \ E _ \lambda = E \textrm{ for } M < \lambda < \infty . $$

The resolution of the identity given by the operator $ A $ completely determines the spectral properties of that operator, namely:

a) a point $ \lambda $ is a regular point of $ A $ if and only if it is a point of constancy, that is, if there is a $ \delta > 0 $ such that $ E _ \mu = E _ \lambda $ for $ \mu \in ( \lambda - \delta , \lambda + \delta ) $;

b) a point $ \lambda _ {0} $ is an eigenvalue of $ A $ if and only if at this point $ E _ \lambda $ has a jump, that is, $ E _ {\lambda _ {0} + 0 } - E _ {\lambda _ {0} } > 0 $;

g) if $ E ( \Delta ) = E _ \mu - E _ \lambda $, then $ L _ {E ( \Delta ) } = E ( \Delta ) {\mathcal H} $ is an invariant subspace of $ A $.

Hence the resolution of the identity determined by the operator $ A $ is also called the spectral function of this operator (cf. Spectral resolution).

Conversely, every resolution of the identity $ \{ E _ \lambda \} $ uniquely determines a self-adjoint operator $ A $ for which this resolution is the spectral function. The domain of definition $ D ( A) $ of $ A $ consists exactly of those $ x \in {\mathcal H} $ for which

$$ \int\limits _ {- \infty } ^ \infty \lambda ^ {2} d \langle E _ \lambda x , x \rangle < \infty , $$

and there is a representation of $ A $ as an operator Stieltjes integral:

$$ A = \int\limits _ {- \infty } ^ \infty \lambda d E _ \lambda . $$

References

[1] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
[2] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in a Hilbert space" , 1–2 , F. Ungar (1961–1963) (Translated from Russian)
[3] L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)

Comments

To the property $ \gamma $) mentioned above one may add that the spectrum of the restriction of $ A $ to $ L _ {E( \Delta ) } $ is contained in the set $ \Delta $.

How to Cite This Entry:
Resolution of the identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolution_of_the_identity&oldid=48529
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article