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 , ∞ = { A ∈ K ( H ) : μ ( n , A ) = O ( n − 1 ) } . {\displaystyle L_{1,\infty }=\{A\in K(H):\mu (n,A)=O(n^{-1})\}.}where K ( H ) {\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
See also
- 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.