In mathematics, aweak trace class operator is acompact operator on aseparableHilbert spaceH withsingular values the same order as theharmonic sequence.When the dimension ofH is infinite, the ideal of weak trace-class operators is strictly larger than the ideal oftrace class operators, and has fundamentally different properties. The usualoperator 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 aresingular traces.

Weak trace-class operators feature in thenoncommutative geometry of French mathematicianAlain Connes.

Acompact operatorA on an infinite dimensionalseparableHilbert spaceH isweak trace class if μ(n,A) = O(n⁻¹), where μ(A) is the sequence ofsingular values. In mathematical notation the two-sidedideal of all weak trace-class operators is denoted,

${\displaystyle L_{1,\mathrm{\infty }}=\{A\in K(H):\mu (n,A)=O(n^{-1})\}.}$

where${\displaystyle K(H)}$ are the compact operators.^{[clarification needed]} The term weak trace-class, or weak-L₁, is used because the operator ideal corresponds, in J. W. Calkin'scorrespondence between two-sided ideals of bounded linear operators and rearrangement invariant sequence spaces, to theweak-l₁ sequence space.

• the weak trace-class operators admit aquasi-norm defined by

${\displaystyle \Vert A{\Vert }_w=\underset{n\ge 0}{sup}(1+n)\mu (n,A),}$

makingL_1,∞ a quasi-Banach operator ideal, that is an ideal that is also aquasi-Banach space.

• Lp space
• Spectral triple
• Singular trace
• Dixmier trace

• B. Simon (2005).Trace ideals and their applications. Providence, RI: Amer. Math. Soc.ISBN 978-0-82-183581-4.
• A. Pietsch (1987).Eigenvalues and s-numbers. Cambridge, UK: Cambridge University Press.ISBN 978-0-52-132532-5.
• A. Connes (1994).Noncommutative geometry. Boston, MA: Academic Press.ISBN 978-0-12-185860-5.
• S. Lord, F. A. Sukochev. D. Zanin (2012).Singular traces: theory and applications. Berlin: De Gruyter.ISBN 978-3-11-026255-1.

• Operator algebras
• Hilbert spaces
• Von Neumann algebras

• Articles with short description
• Short description matches Wikidata
• Wikipedia articles needing clarification from December 2016