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]]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 hasis fundamentallystrictly different propertieslarger than the ideal of [[trace class operator]]s, and has fundamentally different properties. 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.

Weak trace-class operators feature in the [[noncommutative geometry]] of French mathematician [[Alain Connes]].