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''Hellinger type, of a measure''
 
''Hellinger type, of a measure''
  
The equivalence class with respect to mutual [[Absolute continuity|absolute continuity]] in the set of all non-negative measures on a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865401.png" />-algebra containing this measure. The set of spectral types with the order relation induced by the relation of absolute continuity of measures is a complete distributive lattice in which every countable subset is bounded.
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The equivalence class with respect to mutual [[absolute continuity]] in the set of all non-negative measures on a given [[Sigma-algebra|$\sigma$-algebra]] containing this measure. The set of spectral types with the order relation induced by the relation of absolute continuity of measures is a [[Complete lattice|complete]] [[distributive lattice]] in which every countable subset is bounded.
  
The theory of spectral types is used in constructing a system of unitary invariants of normal operators. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865402.png" /> be an arbitrary [[Normal operator|normal operator]] on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865403.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865404.png" /> be the corresponding [[Spectral measure|spectral measure]] in the plane. The spectral type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865405.png" /> of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865406.png" /> is defined as the spectral type of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865407.png" />. All spectral types of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865408.png" /> are called subordinate to the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s0865409.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s08654010.png" /> is separable, then among the spectral types subordinate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086540/s08654011.png" /> there is a maximal one. In particular, all cyclic normal operators have a maximal spectral type. It turns out that a cyclic normal operator is determined by its maximal spectral type up to unitary equivalence. In the general case the system of unitary invariants includes the multiplicities of the homogeneous components of the maximal spectral type.
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The theory of spectral types is used in constructing a system of unitary invariants of [[normal operator]]s. Let $A$ be an arbitrary normal operator on a space $H$ and let $E_A({\cdot})$ be the corresponding [[spectral measure]] in the plane. The spectral type $\sigma_A(x)$ of a vector $x \in H$ is defined as the spectral type of the measure $(E_A({\cdot})x,\, x)$. All spectral types of the form $\sigma_A(x)$ are called subordinate to the operator $A$. If $H$ is separable, then among the spectral types subordinate to $A$ there is a maximal one. In particular, all cyclic normal operators have a maximal spectral type. It turns out that a cyclic normal operator is determined by its maximal spectral type up to unitary equivalence. In the general case the system of unitary invariants includes the multiplicities of the homogeneous components of the maximal spectral type.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Plesner,  "Spectral theory of linear operators" , F. Ungar  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Dixmier,  "Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann" , Gauthier-Villars  (1957)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Plesner,  "Spectral theory of linear operators" , F. Ungar  (1965)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J. Dixmier,  "Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann" , Gauthier-Villars  (1957)</TD></TR>
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</table>
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Latest revision as of 20:27, 18 November 2016

Hellinger type, of a measure

The equivalence class with respect to mutual absolute continuity in the set of all non-negative measures on a given $\sigma$-algebra containing this measure. The set of spectral types with the order relation induced by the relation of absolute continuity of measures is a complete distributive lattice in which every countable subset is bounded.

The theory of spectral types is used in constructing a system of unitary invariants of normal operators. Let $A$ be an arbitrary normal operator on a space $H$ and let $E_A({\cdot})$ be the corresponding spectral measure in the plane. The spectral type $\sigma_A(x)$ of a vector $x \in H$ is defined as the spectral type of the measure $(E_A({\cdot})x,\, x)$. All spectral types of the form $\sigma_A(x)$ are called subordinate to the operator $A$. If $H$ is separable, then among the spectral types subordinate to $A$ there is a maximal one. In particular, all cyclic normal operators have a maximal spectral type. It turns out that a cyclic normal operator is determined by its maximal spectral type up to unitary equivalence. In the general case the system of unitary invariants includes the multiplicities of the homogeneous components of the maximal spectral type.

References

[1] A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian)
[2] J. Dixmier, "Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann" , Gauthier-Villars (1957)
How to Cite This Entry:
Spectral type. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spectral_type&oldid=39760
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