Traces on irregular ideals. (English) Zbl 0692.47036
Let B(H) denote the ring of all bounded linear operators acting on a separable complex Hilbert space H. Given any compact operator \(A\in B(H)\), then I(A) and N(A), respectively, are the smallest ideal and the smallest norm ideal containing A. It is shown that I(A) and N(A) support a nontrivial trace (positive unitary invariant linear functional), if and only if the s-numbers of A fail to satisfy a relation
\[
\sum^{n}_{k=1}s_ k(A)=O(ns_ n(A)).
\]
Reviewer: A.Pietsch
MSC:
| 47L30 | Abstract operator algebras on Hilbert spaces |
Keywords:
irregular ideals; operator ideal; norm ideal; nontrivial trace; positive unitary invariant linear functional; s-numbersReferences:
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