Spectral Characterization of Sums of Commutators I

Suppose J is a two-sided quasi-Banach ideal of compact operators on a separable innnite-dimensional Hilbert space H. We show that an operator T 2 J can be expressed as nite linear combination of commutators A;B] where A 2 J and B 2 B(H) if and only its eigenvalues (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 f 1 n (1 + + n)g is a member of J : This answers (for quasi-Banach ideals) a question raised by Dykema, Figiel, Weiss and Wodzicki.