Weak trace-class operator: Difference between revisions

Revision as of 16:55, 19 September 2013

Revision as of 16:55, 19 September 2013

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{{AFC submission|||ts=20130827235344|u=Dr.steven.lord|ns=5}}

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Weak trace-class operators feature in the [[Noncommutative geometry|noncommutative geometry]] of French mathematician [[Alain Connes]].

== Definition ==

A [[compact operator]] ''A'' on an infinite dimensional [[separable space|separable]] [[Hilbert space]] ''H'' is ''weak trace class'' if μ(''n'',''A''){{=}} O(''n''^-1), where μ(''A'') is the sequence of [[singular value|singular values]]. In mathematical notation the two-sided [[ideal]] of all weak trace-class operators is denoted,

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The term weak trace-class, or weak-''L''₁, is used because the operator ideal corresponds, in J. W. Calkin's [[Calkin correspondence|correspondence]] between two-sided ideals of bounded linear operators and rearrangement invariant sequence spaces, to the [[Lp space|weak-''l''₁ sequence space]].

== Properties ==

* the weak trace-class operators admit a [[quasinorm|quasi-norm]] defined by

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:making ''L''_1,∞ a quasi-Banach operator ideal, that is an ideal that is also a [[quasi-Banach space]].

== Traces on weak trace-class operators ==

{{seealso|Singular trace}}

{{seealso|Dixmier trace}}

== References ==

* {{cite book

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| ___location=Berlin }}

== See also ==

* [[Lp space]]

* [[Spectral triple]]

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* [[Dixmier trace]]

[[:Category:Operator algebras]] [[:Category:Hilbert space]] [[:Category:von Neumann algebras]]

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