The basic sequence problem

We construct a quasi-Banach space $X$ which contains no basic sequence

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 $(\lambda_n)$ (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(\lambda_1+\cdots +\lambda_n)\} is a member of \Cal J. This answers (for quasi-Banach ideals) a question raised by Dykema, Figiel, Weiss and Wodzicki