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.^{[clarification needed]} 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