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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]]s 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]]s. 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]]s.

Weak trace-class operators feature in the [[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''<sup>−1</sup>), where μ(''A'') is the sequence of [[singular value]]s.  In mathematical notation the two-sided [[ideal]] of all weak trace-class operators is denoted,
::::<math> L_{1,\infty} = \{ A \in K(H) : \mu(n,A) = O(n^{-1}) \}. </math>

The term weak trace-class, or weak-''L''<sub>1</sub>, 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''<sub>1</sub> sequence space]].

== Properties ==

* the weak trace-class operators admit a [[quasinorm|quasi-norm]] defined by
::::<math> \| A \|_{w} = \sup_{n \geq 0} (1+n)\mu(n,A), </math>
:making ''L''<sub>1,∞</sub> a quasi-Banach operator ideal, that is an ideal that is also a [[quasi-Banach space]].

== See also ==

* [[Lp space]]
* [[Spectral triple]]
* [[Singular trace]]
* [[Dixmier trace]]

== References ==
{{reflist}}
* {{cite book
| isbn=978-0-82-183581-4
| author= B. Simon
| year=2005
| title=Trace ideals and their applications
| publisher=Amer. Math. Soc.
| location=Providence, RI }}
* {{cite book
| isbn=978-0-52-132532-5
| author= A. Pietsch
| year=1987
| title=Eigenvalues and s-numbers
| publisher=Cambridge University Press
| location=Cambridge, UK }}
*{{cite book
| author=A. Connes
| title=Noncommutative geometry
| url=http://www.alainconnes.org/docs/book94bigpdf.pdf
| publisher=Academic Press
| location=Boston, MA
| isbn=978-0-12-185860-5
| year=1994 }}
* {{cite book
| isbn=978-3-11-026255-1
| author= S. Lord, F. A. Sukochev. D. Zanin
| year=2012
| url=http://www.degruyter.com/view/product/177778
| title=Singular traces: theory and applications
| publisher=De Gruyter
| location=Berlin }}

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