Namespaces
Variants
Actions

Difference between revisions of "Spectral operator"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
s0864501.png
 +
$#A+1 = 43 n = 0
 +
$#C+1 = 43 : ~/encyclopedia/old_files/data/S086/S.0806450 Spectral operator,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''spectral measure''
 
''spectral measure''
  
A bounded [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864501.png" /> mapping a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864502.png" /> into itself and such that for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864503.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864504.png" /> of Borel subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864505.png" /> in the plane there is a [[Resolution of the identity|resolution of the identity]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864506.png" /> with the following properties: 1) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864507.png" /> the projector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864508.png" /> reduces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s0864509.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645010.png" /> and the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645011.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645013.png" /> is the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645014.png" /> to the invariant subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645015.png" />; 2) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645016.png" /> is a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645017.png" /> into the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645018.png" />; 3) all projectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645019.png" /> are bounded, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645021.png" />; and 4) the resolution of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645022.png" /> is countably additive in the strong topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645023.png" />, that is, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645024.png" /> and any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645025.png" /> of pairwise disjoint sets,
+
A bounded [[Linear operator|linear operator]] $  A $
 +
mapping a [[Banach space|Banach space]] $  X $
 +
into itself and such that for the $  \sigma $-
 +
algebra $  {\mathcal B} $
 +
of Borel subsets $  \delta $
 +
in the plane there is a [[Resolution of the identity|resolution of the identity]] $  E ( \delta ) $
 +
with the following properties: 1) for any $  \delta \in {\mathcal B} $
 +
the projector $  E ( \delta ) $
 +
reduces $  A $,  
 +
that is, $  E ( \delta ) A = A E ( \delta ) $
 +
and the spectrum $  \sigma ( A _  \delta  ) $
 +
lies in $  \overline \delta \; $,  
 +
where $  A _  \delta  $
 +
is the restriction of $  A $
 +
to the invariant subspace $  E ( \delta ) X $;  
 +
2) the mapping $  \delta \mapsto E ( \delta ) $
 +
is a homeomorphism of $  {\mathcal B} = \{ \delta \} $
 +
into the Boolean algebra $  \{ E ( \delta ) \} $;  
 +
3) all projectors $  E ( \delta ) $
 +
are bounded, that is, $  \| E ( \delta ) \| \leq  M $,  
 +
$  \delta \in {\mathcal B} $;  
 +
and 4) the resolution of the identity $  E ( \delta ) $
 +
is countably additive in the strong topology of $  X $,  
 +
that is, for any $  x \in X $
 +
and any sequence $  \{ \delta _ {n} \} \subset  {\mathcal B} $
 +
of pairwise disjoint sets,
  
<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/s/s086/s086450/s08645026.png" /></td> </tr></table>
+
$$
 +
E \left ( \cup _ { n= } 1 ^  \infty  \delta _ {n} \right ) x  = \sum _ { n= } 1 ^  \infty  E ( \delta _ {n} ) x .
 +
$$
  
The concept of a spectral operator can be generalized to the case of closed unbounded operators. In 1), the additional requirement is then that the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645027.png" /> holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645028.png" /> is the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645029.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645030.png" /> for bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645031.png" />.
+
The concept of a spectral operator can be generalized to the case of closed unbounded operators. In 1), the additional requirement is then that the inclusion $  E ( \delta ) D ( A) \subset  D ( A) $
 +
holds, where $  D ( A) $
 +
is the domain of definition of $  A $,  
 +
and $  E ( \delta ) X \subset  D ( A) $
 +
for bounded $  \delta $.
  
 
All linear operators on a finite-dimensional space and all self-adjoint and normal operators on a Hilbert space are spectral operators. For example, the operator
 
All linear operators on a finite-dimensional space and all self-adjoint and normal operators on a Hilbert space are spectral operators. For example, the operator
  
<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/s/s086/s086450/s08645032.png" /></td> </tr></table>
+
$$
 +
A x ( t)  = t x( t) + \int\limits _ {- \infty } ^  \infty 
 +
K ( t , s ) x ( s)  d s
 +
$$
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645034.png" />, is spectral on
+
on $  L _ {p} ( - \infty , \infty ) $,
 +
$  1 < p < \infty $,  
 +
is spectral on
  
<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/s/s086/s086450/s08645035.png" /></td> </tr></table>
+
$$
 +
D ( A)  = \left \{ {x ( t) } : {
 +
\int\limits _ {- \infty } ^  \infty  | t x ( t) |  ^ {2}  d t
 +
< \infty } \right \}
 +
$$
  
if the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645036.png" /> is the Fourier transform of a Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645037.png" /> on the plane of total variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645038.png" /> and is such that
+
if the kernel $  K ( t , s ) $
 +
is the Fourier transform of a Borel measure $  \mu $
 +
on the plane of total variation $  \mathop{\rm var}  \mu < 1 / 2 \pi $
 +
and is such that
  
<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/s/s086/s086450/s08645039.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {- \infty } ^  \infty  K ( t , s )
 +
x ( s)  d s ,\  \int\limits _ {- \infty } ^  \infty 
 +
K ( t , s ) x ( t)  d t
 +
$$
  
are bounded linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645040.png" />.
+
are bounded linear operators on $  L _ {p} ( - \infty , \infty ) $.
  
 
Spectral operators have many important properties, such as:
 
Spectral operators have many important properties, such as:
  
<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/s/s086/s086450/s08645041.png" /></td> </tr></table>
+
$$
 +
\lambda \in \delta ( A)  \iff  \exists \
 +
\{ x _ {n} \} \subset  X , \| x _ {n} \|
 +
= 1 , ( A - \lambda I ) x _ {n} \rightarrow 0 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645042.png" /> is separable, the point and residual spectra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086450/s08645043.png" /> are at most countable.
+
If $  X $
 +
is separable, the point and residual spectra of $  A $
 +
are at most countable.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral operators" , '''3''' , Interscience  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  "A survey of the theory of spectral operators"  ''Bull. Amer. Math. Soc.'' , '''64'''  (1958)  pp. 217–274</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral operators" , '''3''' , Interscience  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Dunford,  "A survey of the theory of spectral operators"  ''Bull. Amer. Math. Soc.'' , '''64'''  (1958)  pp. 217–274</TD></TR></table>

Revision as of 08:22, 6 June 2020


spectral measure

A bounded linear operator $ A $ mapping a Banach space $ X $ into itself and such that for the $ \sigma $- algebra $ {\mathcal B} $ of Borel subsets $ \delta $ in the plane there is a resolution of the identity $ E ( \delta ) $ with the following properties: 1) for any $ \delta \in {\mathcal B} $ the projector $ E ( \delta ) $ reduces $ A $, that is, $ E ( \delta ) A = A E ( \delta ) $ and the spectrum $ \sigma ( A _ \delta ) $ lies in $ \overline \delta \; $, where $ A _ \delta $ is the restriction of $ A $ to the invariant subspace $ E ( \delta ) X $; 2) the mapping $ \delta \mapsto E ( \delta ) $ is a homeomorphism of $ {\mathcal B} = \{ \delta \} $ into the Boolean algebra $ \{ E ( \delta ) \} $; 3) all projectors $ E ( \delta ) $ are bounded, that is, $ \| E ( \delta ) \| \leq M $, $ \delta \in {\mathcal B} $; and 4) the resolution of the identity $ E ( \delta ) $ is countably additive in the strong topology of $ X $, that is, for any $ x \in X $ and any sequence $ \{ \delta _ {n} \} \subset {\mathcal B} $ of pairwise disjoint sets,

$$ E \left ( \cup _ { n= } 1 ^ \infty \delta _ {n} \right ) x = \sum _ { n= } 1 ^ \infty E ( \delta _ {n} ) x . $$

The concept of a spectral operator can be generalized to the case of closed unbounded operators. In 1), the additional requirement is then that the inclusion $ E ( \delta ) D ( A) \subset D ( A) $ holds, where $ D ( A) $ is the domain of definition of $ A $, and $ E ( \delta ) X \subset D ( A) $ for bounded $ \delta $.

All linear operators on a finite-dimensional space and all self-adjoint and normal operators on a Hilbert space are spectral operators. For example, the operator

$$ A x ( t) = t x( t) + \int\limits _ {- \infty } ^ \infty K ( t , s ) x ( s) d s $$

on $ L _ {p} ( - \infty , \infty ) $, $ 1 < p < \infty $, is spectral on

$$ D ( A) = \left \{ {x ( t) } : { \int\limits _ {- \infty } ^ \infty | t x ( t) | ^ {2} d t < \infty } \right \} $$

if the kernel $ K ( t , s ) $ is the Fourier transform of a Borel measure $ \mu $ on the plane of total variation $ \mathop{\rm var} \mu < 1 / 2 \pi $ and is such that

$$ \int\limits _ {- \infty } ^ \infty K ( t , s ) x ( s) d s ,\ \int\limits _ {- \infty } ^ \infty K ( t , s ) x ( t) d t $$

are bounded linear operators on $ L _ {p} ( - \infty , \infty ) $.

Spectral operators have many important properties, such as:

$$ \lambda \in \delta ( A) \iff \exists \ \{ x _ {n} \} \subset X , \| x _ {n} \| = 1 , ( A - \lambda I ) x _ {n} \rightarrow 0 . $$

If $ X $ is separable, the point and residual spectra of $ A $ are at most countable.

References

[1] N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971)
[2] N. Dunford, "A survey of the theory of spectral operators" Bull. Amer. Math. Soc. , 64 (1958) pp. 217–274
How to Cite This Entry:
Spectral operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_operator&oldid=48761
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article