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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 (mathematics)|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.
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== 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]]{{dn|date=July 2014}} of all weak trace-class operators is denoted,
::::<math> L_{1,\infty} = \{ A \in K(H) : \mu(n,A) = O(n^{-1}) \}. </math>
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