Weak trace-class operator: Difference between revisions

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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|singular values]] 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|trace class operators]]. 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|singular traces]].

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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''^-1−1), where μ(''A'') is the sequence of [[singular value|singular values]]. 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>