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A term in [[Spectral theory|spectral theory]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579201.png" /> be a self-adjoint and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579202.png" /> a unitary operator acting in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579203.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579204.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579205.png" />, has a simple Lebesgue spectrum if it is unitarily equivalent to the operator of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579206.png" /> in a space of complex-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579207.png" /> that are defined on the real axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l0579208.png" />, respectively on the circle
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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/l/l057/l057920/l0579209.png" /></td> </tr></table>
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and for which
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A term in [[Spectral theory|spectral theory]]. Let  $  A $
 +
be a self-adjoint and $  U $
 +
a unitary operator acting in a Hilbert space  $  H $.
 +
The operator  $  A $,
 +
respectively  $  U $,
 +
has a simple Lebesgue spectrum if it is unitarily equivalent to the operator of multiplication by  $  \lambda $
 +
in a space of complex-valued functions  $  f ( \lambda ) $
 +
that are defined on the real axis  $  \mathbf R $,
 +
respectively on the circle
  
<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/l/l057/l057920/l05792010.png" /></td> </tr></table>
+
$$
 +
S  ^ {1}  = \{  \lambda  : {\lambda \in \mathbf C , | \lambda | = 1 } \}
 +
,
 +
$$
  
where the integration is carried out with respect to the ordinary [[Lebesgue measure|Lebesgue measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792011.png" />, respectively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792012.png" />; hence the name Lebesgue spectrum (see [[Unitarily-equivalent operators|Unitarily-equivalent operators]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792013.png" /> this definition is equivalent to the following: In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792014.png" /> there is an orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792017.png" />. Also, an operator has a Lebesgue spectrum if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792018.png" /> can be decomposed into an orthogonal direct sum of invariant subspaces in each of which the operator has a simple Lebesgue spectrum. Although for a given operator there can be many such decompositions, the number of  "summands"  in each of them is the same (it may be an infinite [[Cardinal number|cardinal number]]). This number is called the multiplicity of the Lebesgue spectrum. Finally, similar concepts can be introduced for one-parameter groups of unitary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792019.png" />, continuous in the weak (or strong, which is the same in the given case) operator topology. By Stone's theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792021.png" /> is a self-adjoint operator (cf. [[Semi-group of operators|Semi-group of operators]]; [[Generating operator of a semi-group|Generating operator of a semi-group]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792022.png" /> has a Lebesgue spectrum of a certain multiplicity, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792023.png" /> has the same properties. For example, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792024.png" /> has a simple Lebesgue spectrum if it is unitarily equivalent to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792026.png" />, and this group, in turn, is equivalent to the group of shifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792027.png" /> in the same space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057920/l05792028.png" />.
+
and for which
  
 +
$$
 +
\| f \|  ^ {2}  =  \int\limits | f ( \lambda ) |  ^ {2} \
 +
d \lambda  <  \infty ,
 +
$$
  
 +
where the integration is carried out with respect to the ordinary [[Lebesgue measure|Lebesgue measure]] on  $  \mathbf R $,
 +
respectively on  $  S  ^ {1} $;
 +
hence the name Lebesgue spectrum (see [[Unitarily-equivalent operators|Unitarily-equivalent operators]]). For  $  U $
 +
this definition is equivalent to the following: In  $  H $
 +
there is an orthonormal basis  $  e _ {j} $,
 +
$  j = 0 , \pm  1 , \pm  2, \dots $
 +
such that  $  U e _ {j} = e _ {j+1} $.
 +
Also, an operator has a Lebesgue spectrum if  $  H $
 +
can be decomposed into an orthogonal direct sum of invariant subspaces in each of which the operator has a simple Lebesgue spectrum. Although for a given operator there can be many such decompositions, the number of  "summands"  in each of them is the same (it may be an infinite [[Cardinal number|cardinal number]]). This number is called the multiplicity of the Lebesgue spectrum. Finally, similar concepts can be introduced for one-parameter groups of unitary operators  $  U ( t) $,
 +
continuous in the weak (or strong, which is the same in the given case) operator topology. By Stone's theorem,  $  U ( t) = e  ^ {iAt} $,
 +
where  $  A $
 +
is a self-adjoint operator (cf. [[Semi-group of operators|Semi-group of operators]]; [[Generating operator of a semi-group|Generating operator of a semi-group]]). If  $  A $
 +
has a Lebesgue spectrum of a certain multiplicity, one says that  $  U ( t) $
 +
has the same properties. For example, the group  $  U ( t) $
 +
has a simple Lebesgue spectrum if it is unitarily equivalent to the group  $  f ( \lambda ) \rightarrow e ^ {i \lambda t } f ( \lambda ) $
 +
in  $  L _ {2} ( \mathbf R ) $,
 +
and this group, in turn, is equivalent to the group of shifts  $  f ( \lambda ) \rightarrow f( \lambda + t ) $
 +
in the same space  $  L _ {2} ( \mathbf R ) $.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Helson,  "The spectral theorem" , Springer  (1986)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Helson,  "The spectral theorem" , Springer  (1986)</TD></TR></table>

Latest revision as of 15:28, 28 February 2022


A term in spectral theory. Let $ A $ be a self-adjoint and $ U $ a unitary operator acting in a Hilbert space $ H $. The operator $ A $, respectively $ U $, has a simple Lebesgue spectrum if it is unitarily equivalent to the operator of multiplication by $ \lambda $ in a space of complex-valued functions $ f ( \lambda ) $ that are defined on the real axis $ \mathbf R $, respectively on the circle

$$ S ^ {1} = \{ \lambda : {\lambda \in \mathbf C , | \lambda | = 1 } \} , $$

and for which

$$ \| f \| ^ {2} = \int\limits | f ( \lambda ) | ^ {2} \ d \lambda < \infty , $$

where the integration is carried out with respect to the ordinary Lebesgue measure on $ \mathbf R $, respectively on $ S ^ {1} $; hence the name Lebesgue spectrum (see Unitarily-equivalent operators). For $ U $ this definition is equivalent to the following: In $ H $ there is an orthonormal basis $ e _ {j} $, $ j = 0 , \pm 1 , \pm 2, \dots $ such that $ U e _ {j} = e _ {j+1} $. Also, an operator has a Lebesgue spectrum if $ H $ can be decomposed into an orthogonal direct sum of invariant subspaces in each of which the operator has a simple Lebesgue spectrum. Although for a given operator there can be many such decompositions, the number of "summands" in each of them is the same (it may be an infinite cardinal number). This number is called the multiplicity of the Lebesgue spectrum. Finally, similar concepts can be introduced for one-parameter groups of unitary operators $ U ( t) $, continuous in the weak (or strong, which is the same in the given case) operator topology. By Stone's theorem, $ U ( t) = e ^ {iAt} $, where $ A $ is a self-adjoint operator (cf. Semi-group of operators; Generating operator of a semi-group). If $ A $ has a Lebesgue spectrum of a certain multiplicity, one says that $ U ( t) $ has the same properties. For example, the group $ U ( t) $ has a simple Lebesgue spectrum if it is unitarily equivalent to the group $ f ( \lambda ) \rightarrow e ^ {i \lambda t } f ( \lambda ) $ in $ L _ {2} ( \mathbf R ) $, and this group, in turn, is equivalent to the group of shifts $ f ( \lambda ) \rightarrow f( \lambda + t ) $ in the same space $ L _ {2} ( \mathbf R ) $.

Comments

References

[a1] H. Helson, "The spectral theorem" , Springer (1986)
How to Cite This Entry:
Lebesgue spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebesgue_spectrum&oldid=14137
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article