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Commutator structure of operator ideals. (English) Zbl 1103.47054

This important paper, based on the authors’ works of the preceding ten years, is centered around the proof and consequences of the following result.
Theorem. Let \(I,J\) be two ideals in \(B(H)\) and assume that at least one of them is proper. Then a normal operator \(T\in IJ\) belongs to the commutator space \([I,J]\) if and only if \[ \Big|\frac{\lambda_{1}(T)+\dots+\lambda_n(T)}{n}\Big|=O(\mu_{n}) \] for some sequence \(\mu\in\Sigma(IJ)\).

MSC:

47L20 Operator ideals
46B99 Normed linear spaces and Banach spaces; Banach lattices
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B47 Commutators, derivations, elementary operators, etc.
47L30 Abstract operator algebras on Hilbert spaces
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