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Difference between revisions of "Spectral measure"

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A unitary homomorphism from some Boolean algebra of sets into the Boolean algebra of projection operators on a Banach space. Every operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864401.png" /> on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864402.png" /> defines a spectral measure on the set of open-and-closed subsets of its spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864403.png" /> by the formula
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A unitary homomorphism from some [[Boolean algebra]] of sets into the Boolean algebra of projection operators on a [[Banach space]]. Every operator $T$ on a Banach space $X$ defines a spectral measure on the set of [[Open-closed set|open-and-closed subset]]s of its [[Spectrum of an operator|spectrum]] $\sigma(T)$ by the formula
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$$
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E(\alpha) = \frac{1}{2 \pi i} \int_\Gamma (zI-T)^{-1} dz \ ,
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$$
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where $\Gamma$ is a [[Jordan curve]] separating $\alpha$ from $\sigma(T) \setminus \alpha$. Here, $TE(\alpha) = E(\alpha)T$ and $\sigma\left(T \downharpoonright_{E(\alpha)X}\right) \subseteq \bar\alpha$. The construction of spectral measures satisfying these conditions on wider classes of Boolean algebras of sets is one of the basic problems in the [[spectral theory]] of linear operators.
  
<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/s086440/s0864404.png" /></td> </tr></table>
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====References====
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<table>
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<TR><TD valign="top">[1a]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral operators" , '''3''' , Interscience  (1971)</TD></TR>
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<TR><TD valign="top">[1b]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR>
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</table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864405.png" /> is a Jordan curve separating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864406.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864407.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086440/s0864409.png" />. The construction of spectral measures satisfying these conditions on wider classes of Boolean algebras of sets is one of the basic problems in the [[Spectral theory|spectral theory]] of linear operators.
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral operators" , '''3''' , Interscience  (1971)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR></table>
 

Latest revision as of 18:24, 22 April 2016

A unitary homomorphism from some Boolean algebra of sets into the Boolean algebra of projection operators on a Banach space. Every operator $T$ on a Banach space $X$ defines a spectral measure on the set of open-and-closed subsets of its spectrum $\sigma(T)$ by the formula $$ E(\alpha) = \frac{1}{2 \pi i} \int_\Gamma (zI-T)^{-1} dz \ , $$ where $\Gamma$ is a Jordan curve separating $\alpha$ from $\sigma(T) \setminus \alpha$. Here, $TE(\alpha) = E(\alpha)T$ and $\sigma\left(T \downharpoonright_{E(\alpha)X}\right) \subseteq \bar\alpha$. The construction of spectral measures satisfying these conditions on wider classes of Boolean algebras of sets is one of the basic problems in the spectral theory of linear operators.

References

[1a] N. Dunford, J.T. Schwartz, "Linear operators. Spectral operators" , 3 , Interscience (1971)
[1b] N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Interscience (1963)
How to Cite This Entry:
Spectral measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_measure&oldid=17065
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article