Weak trace-class operator: Difference between revisions

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{{Short description|Mathematical concept}}

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In mathematics, a '''weak trace class''' operator is a [[compact operator]] on a [[separable space|separable]] [[Hilbert space]] ''H'' with [[singular value|singular values]]s the same order as the [[harmonic series (mathematics)|harmonic sequence]].

When the dimension of ''H'' is infinite, the ideal of weak trace-class operators hasis fundamentallystrictly different propertieslarger than the ideal of [[trace class operator|trace]]s, classand operators]]has fundamentally different properties. The usual [[trace class#Definition|operator trace]] on the trace-class operators does not extend to the weak trace class. Instead the ideal of weak trace-class operators admits an infinite number of linearly independent quasi-continuous traces, and it is the smallest two-sided ideal for which all traces on it are [[Singularsingular trace|singular traces]]s.

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

In mathematics, a ''weak trace class'' operator is a [[compact operator]] on a [[separable space|separable]] [[Hilbert space]] ''H'' with [[singular value|singular values]] the same order as the [[harmonic series|harmonic sequence]].

When the dimension of ''H'' is infinite the ideal of weak trace-class operators has fundamentally different properties than the ideal of [[trace class operator|trace class operators]]. The usual [[trace class#Definition|operator trace]] on the trace-class operators does not extend to the weak trace class. Instead the ideal of weak trace-class operators admits an infinite number of linearly independent quasi-continuous traces, and it is the smallest two-sided ideal for which all traces on it are [[Singular trace|singular traces]].

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''^-1−1), where μ(''A'') is the sequence of [[singular value|singular values]]s. In mathematical notation the two-sided [[ideal (ring theory)|ideal]] of all weak trace-class operators is denoted,

::::<math> L_{1,\infty} = \{ A \in K(H) : \mu(n,A) = O(n^{-1}) \}. </math>

where <math>K(H) </math> are the compact operators.{{what|reason= This definition disagrees with the definition in the article on the [[Dixmier trace]].|date=December 2016}} 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 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]].

== See also ==

==Traces on weak trace-class operators==

{{seealso|Singular trace}}

{{seealso|Dixmier trace}}

* [[Lp space]]

==References==

* [[Spectral triple]]

{{seealso|* [[Singular trace}}]]

{{seealso|* [[Dixmier trace}}]]

== References ==

{{reflist}}

* {{cite book

| isbn=978-0-82-183581-4

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| publisher=Amer. Math. Soc.

| ___location=Providence, RI }}

* {{cite book

| isbn=978-0-52-132532-5

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| publisher=Cambridge University Press

| ___location=Cambridge, UK }}

*{{cite book

| author=A. Connes

| title=Noncommutative geometry

| url=httphttps://www.alainconnesarchive.org/docsdetails/book94bigpdf.pdfnoncommutativege0000conn

| publisher=Academic Press

| ___location=Boston, MA

| isbn=978-0-12-185860-5

| year=1994 }}

| url-access=registration

}}

* {{cite book

| isbn=978-3-11-026255-1

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| ___location=Berlin }}

[[Category:Operator algebras]]

==See also==

[[Category:Hilbert spaces]]

* [[Lp space]]

[[Category:Von Neumann algebras]]

* [[Spectral triple]]

* [[Singular trace]]

* [[Dixmier trace]]

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