Weak trace-class operator

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,

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

where ${\displaystyle K(H)}$ 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

${\displaystyle \|A\|_{w}=\sup _{n\geq 0}(1+n)\mu (n,A),}$

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

• 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.
• S. Lord, F. A. Sukochev. D. Zanin (2012). Singular traces: theory and applications. Berlin: De Gruyter. ISBN 978-3-11-026255-1.