Weak trace-class operator explained

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.

Definition

A compact operator A on an infinite dimensional separable Hilbert space H is weak trace class if μ(n,A) O(n⁻¹), where μ(A) is the sequence of singular values. In mathematical notation the two-sided ideal of all weak trace-class operators is denoted,

L _{1,infty} = \{ A \in K(H) : \mu(n,A) = O(n ^{-1} ) \}. where K(H) are the compact operators. The term weak trace-class, or weak-L₁, 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-l₁ sequence space.

Properties

• the weak trace-class operators admit a quasi-norm defined by

\| A \| _{w} = \sup _{n} ( 1+ n)\mu(n,A),

making L_1,∞ a quasi-Banach operator ideal, that is an ideal that is also a quasi-Banach space.

See also

• Lp space
• Spectral triple
• Singular trace
• Dixmier trace

References

• Book: 978-0-82-183581-4. B. Simon. 2005. Trace ideals and their applications. Amer. Math. Soc.. Providence, RI .
• Book: 978-0-52-132532-5. A. Pietsch. 1987. Eigenvalues and s-numbers. Cambridge University Press. Cambridge, UK .
• Book: A. Connes. Noncommutative geometry. Academic Press. Boston, MA. 978-0-12-185860-5. 1994. registration.
• Book: 978-3-11-026255-1. S. Lord, F. A. Sukochev. D. Zanin. 2012. Singular traces: theory and applications. De Gruyter. Berlin .