Remarks on functional calculus for perturbed first order Dirac operators

We make some remarks on earlier works on $R-$bisectoriality in $L^p$ of perturbed first order differential operators by Hyt\"onen, McIntosh and Portal. They have shown that this is equivalent to bounded holomorphic functional calculus in $L^p$ for $p$ in any open interval when suitable hypotheses are made. Hyt\"onen and McIntosh then showed that $R$-bisectoriality in $L^p$ at one value of $p$ can be extrapolated in a neighborhood of $p$. We give a different proof of this extrapolation and observe that the first proof has impact on the splitting of the space by the kernel and range.Comment: 11 page

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