Mathematical concept
In mathematics, a weak trace class operator is a compact operator on a separable Hilbert space "H" with singular values the same order as the harmonic sequence.
When the dimension of "H" is infinite, the ideal of weak trace-class operators is strictly larger than the ideal of trace class operators, and has fundamentally different properties. The usual 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 traces.
Weak trace-class operators feature in the noncommutative geometry of French mathematician Alain Connes.
Definition.
A compact operator "A" on an infinite dimensional separable Hilbert space "H" is "weak trace class" if μ("n","A") 
 O("n"−1), where μ("A") is the sequence of singular values. In mathematical notation the two-sided ideal of all weak trace-class operators is denoted,
formula_1
where formula_2 are the compact operators. The term weak trace-class, or weak-"L"1, is used because the operator ideal corresponds, in J. W. Calkin's correspondence between two-sided ideals of bounded linear operators and rearrangement invariant sequence spaces, to the weak-"l"1 sequence space.
formula_3
making "L"1,∞ a quasi-Banach operator ideal, that is an ideal that is also a quasi-Banach space.
References.
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