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Weak trace-class operator
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<p>In mathematics, a weak trace class operator is a <a href="/facts/Compact_operator/KNkcK4mE">compact operator</a> on a <a href="/facts/Separable_space/2bmU73bk">separable</a> <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> <i>H</i> with <a href="/facts/Singular_value/0QWHnqGG">singular values</a> the same order as the <a href="/facts/Harmonic_series_(mathematics)/8KGDHKTg">harmonic sequence</a>. When the dimension of <i>H</i> is infinite, the ideal of weak trace-class operators is strictly larger than the ideal of <a href="/facts/Trace_class_operator/2AmWENA3">trace class operators</a>, and has fundamentally different properties. The usual <a href="/facts/Trace_class/2AmWENA3">operator trace</a> 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 <a href="/facts/Singular_trace/S4xkdjCI">singular traces</a>. </p><p>Weak trace-class operators feature in the <a href="/facts/Noncommutative_geometry/WSwggL1p">noncommutative geometry</a> of French mathematician <a href="/facts/Alain_Connes/tjFJ9Iou">Alain Connes</a>. </p>