Weak trace-class operator

Short description: Mathematical concept

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,

[math]\displaystyle{ L_{1,\infty} = \{ A \in K(H) : \mu(n,A) = O(n^{-1}) \}. }[/math]

where [math]\displaystyle{ K(H) }[/math] 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

[math]\displaystyle{ \| A \|_{w} = \sup_{n \geq 0} (1+n)\mu(n,A), }[/math]

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

• B. Simon (2005). Trace ideals and their applications. Providence, RI: Amer. Math. Soc.. ISBN 978-0-82-183581-4. 
• A. Pietsch (1987). Eigenvalues and s-numbers. Cambridge, UK: Cambridge University Press. ISBN 978-0-52-132532-5. 
• A. Connes (1994). Noncommutative geometry. Boston, MA: Academic Press. ISBN 978-0-12-185860-5. https://archive.org/details/noncommutativege0000conn. 
• S. Lord, F. A. Sukochev. D. Zanin (2012). Singular traces: theory and applications. Berlin: De Gruyter. ISBN 978-3-11-026255-1. http://www.degruyter.com/view/product/177778.