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Difference between revisions of "Spectral resolution"

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''spectral function, resolution of the identity''
 
''spectral function, resolution of the identity''
  
A monotone mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864701.png" /> which is left-continuous in the strong operator topology, from the real line into the set of orthogonal projectors on a Hilbert space, and satisfying the conditions
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A [[monotone mapping]] $P(\cdot)$ from the real line into the set of [[orthogonal projector]]s on a [[Hilbert space]], which is left-continuous in the strong operator topology and satisfies the conditions
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$$
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\lim_{t \rightarrow -\infty} P(t) = 0 \ ;\ \ \ \lim_{t \rightarrow +\infty} P(t) = I \ .
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$$
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Every self-adjoint (i.e. taking self-adjoint values) strongly countably-additive Borel [[spectral measure]] $E(\cdot)$ on the line defines a spectral resolution by the formula $P(t) = E((-\infty,t))$, and for every spectral resolution there is a unique spectral measure defining it.
  
<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/s086470/s0864702.png" /></td> </tr></table>
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The concept of a spectral resolution is fundamental in the spectral theory of self-adjoint operators: By the spectral decomposition theorem (cf. [[Spectral decomposition of a linear operator]]), every such operator has an integral representation $\int_{-\infty}^{\infty} t dP(t)$, where $P(t)$ is some spectral resolution. An analogous role in the theory of symmetric operators is played by the concept of a generalized spectral resolution, which is a mapping from the real line into the set of non-negative operators that satisfies all the conditions imposed on spectral resolutions, except that the values need not be projectors. Every generalized spectral resolution can be extended to a spectral resolution on a larger space (Naimark's theorem).
  
Every self-adjoint (i.e. taking self-adjoint values) strongly countably-additive Borel [[Spectral measure|spectral measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864703.png" /> on the line defines a spectral resolution by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864704.png" />, and for every spectral resolution there is a unique spectral measure defining it.
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian) {{ZBL|0467.47001}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,  "Self-adjoint extensions of the second kind of a symmetric operator" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''4''' : 1  (1940)  pp. 53–104  (In Russian)  (English abstract) {{ZBL|0025.06402}}</TD></TR>
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</table>
  
The concept of a spectral resolution is fundamental in the spectral theory of self-adjoint operators: By the spectral decomposition theorem (cf. [[Spectral decomposition of a linear operator|Spectral decomposition of a linear operator]]), every such operator has an integral representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864705.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086470/s0864706.png" /> is some spectral resolution. An analogous role in the theory of symmetric operators is played by the concept of a generalized spectral resolution, which is a mapping from the real line into the set of non-negative operators that satisfies all the conditions imposed on spectral resolutions, except that the values need not be projectors. Every generalized spectral resolution can be extended to a spectral resolution on a larger space (Naimark's theorem).
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.I. Akhiezer,  I.M. Glazman,  "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman  (1981)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,  "Self-adjoint extensions of the second kind of a symmetric operator"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''4''' :  1  (1940)  pp. 53–104  (In Russian)  (English abstract)</TD></TR></table>
 

Latest revision as of 18:24, 18 October 2016

spectral function, resolution of the identity

A monotone mapping $P(\cdot)$ from the real line into the set of orthogonal projectors on a Hilbert space, which is left-continuous in the strong operator topology and satisfies the conditions $$ \lim_{t \rightarrow -\infty} P(t) = 0 \ ;\ \ \ \lim_{t \rightarrow +\infty} P(t) = I \ . $$ Every self-adjoint (i.e. taking self-adjoint values) strongly countably-additive Borel spectral measure $E(\cdot)$ on the line defines a spectral resolution by the formula $P(t) = E((-\infty,t))$, and for every spectral resolution there is a unique spectral measure defining it.

The concept of a spectral resolution is fundamental in the spectral theory of self-adjoint operators: By the spectral decomposition theorem (cf. Spectral decomposition of a linear operator), every such operator has an integral representation $\int_{-\infty}^{\infty} t dP(t)$, where $P(t)$ is some spectral resolution. An analogous role in the theory of symmetric operators is played by the concept of a generalized spectral resolution, which is a mapping from the real line into the set of non-negative operators that satisfies all the conditions imposed on spectral resolutions, except that the values need not be projectors. Every generalized spectral resolution can be extended to a spectral resolution on a larger space (Naimark's theorem).

References

[1] N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian) Zbl 0467.47001
[2] M.A. Naimark, "Self-adjoint extensions of the second kind of a symmetric operator" Izv. Akad. Nauk SSSR Ser. Mat. , 4 : 1 (1940) pp. 53–104 (In Russian) (English abstract) Zbl 0025.06402
How to Cite This Entry:
Spectral resolution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_resolution&oldid=39435
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