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.
A compact operator A on an infinite dimensional separable Hilbert space H is weak trace class if ü(n,A) O(n<sup>âÂÂ1</sup>), where ü(A) is the sequence of singular values. In mathematical notation the two-sided ideal of all weak trace-class operators is denoted,
where are the compact operators. The term weak trace-class, or weak-L<sub>1</sub>, 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<sub>1</sub> sequence space.