Weak trace-class operator: Difference between revisions

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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]] 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]].

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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 ==

* [[Lp space]]

{{seealso|Singular trace}}

* [[Spectral triple]]

{{seealso|Dixmier trace}}

{{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

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

| year=1994 }}

* {{cite book

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

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| ___location=Berlin }}

== See also ==

* [[Lp space]]

* [[Spectral triple]]

* [[Singular trace]]

* [[Dixmier trace]]

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