Abstract: 

Consider the weak-trace class ideal $\mathcal L_{1,\infty}$ of bounded, compact operators $A$ on a separable  Hilbert space $H$ such that $\sup_{n\in\mathbb N}(n+1)\mu(n,A)<\infty,$ where $\mu(A)$ is a  singular value function of $A$.

In 1966 J. Dixmier have considered the functional $${\rm Tr}_{\omega}(A) = \omega\left(n \mapsto \frac{1}{\log(1+n)}\sum_{k=0}^n\mu(k,A) \right), \ 0\le A \in \mathcal L_{1,\infty}$$ and have proved that for generalised limiting procedure $\omega$ this functional extends to a linear functional on the whole $\mathcal L_{1,\infty}$. In general, one has no access to the singular value function of an operator. This makes the computation of Dixmier traces a difficult problem. 

In the present talk we discuss the computation of Dixmier traces of the commutators of the form $$P[P, W_{1/2, \gamma, c}][P, W_{1/2, \gamma, d}],$$ where $P:L^2(S^1)\to H^2(S^1)$ is the Szeg\"o projection to the Hardy space and $W_{1/2, \gamma, c}$ ($\gamma\in \mathbb N$, $\gamma>1$, $c$ is a bounded sequence) is a generalised Weierstrass function. 

In particular, we show that the class of such commutators is large enough to separate all Dixmier traces on $\mathcal L_{1,\infty}$.

This talk is based on the joint work with M. Goffeng.