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

Spectral characterization of sums of commutators II

For countably generated ideals, \Jc, of B(\Hil), geometric stability is necessary for the canonical spectral characterization of sums of (\Jc,B(\Hil))--commutators to hold. This answers a question raised by Dykema, Figiel, Weiss and Wodzicki. There are some ideals, \Jc, having quasi--nilpotent elements that are not sums of (\Jc,B(\Hil))--commutators. Also, every trace on every geometrically stable ideal is a spectral trace