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Spectral characterization of sums of commutators I

Suppose $\Cal J$ is a two-sided quasi-Banach ideal of compact operators on a separable infinite-dimensional Hilbert space $\Cal H$. We show that an operator $T\in\Cal J$ can be expressed as finite linear combination of commutators $[A,B]$ where $A\in\Cal J$ and $B\in\Cal B(\Cal H)$ if and only its eigenvalues $(λ_n)$ (arranged in decreasing order of absolute value, repeated according to algebraic multiplicity and augmented by zeros if necessary) satisfy the condition that the diagonal operator $\diag\{\frac1n(λ_1+\cdots +λ_n)\}$ is a member of $\Cal J.$ This answers (for quasi-Banach ideals) a question raised by Dykema, Figiel, Weiss and Wodzicki.