Commutators in model spaces

V. V. Kapustin

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 34, Tome 333 (2006), pp. 54-61

Résumé

Let $\theta$ be an inner function, let $K_\theta=H^2\ominus\theta H^2$, and let $S_\theta\colon K_\theta\to K_\theta$ be defined by the formula $S_\theta f=P_\theta zf$, $f\in K_\theta$, where $P_\theta$ is the orthogonal projection of $H^2$ onto $K_\theta$. Consider the set $A$ of all trace class operators $L\colon K_\theta\to K_\theta$, $L=\sum(\cdot,u_n)v_n$, $\sum\|u_n\|\|v_n\|\infty$ $(u_n,v_n\in K_\theta)$, such that $\sum\bar u_nv_n\in H^1_0$. It is shown that the trace class commutators of the form $XS_\theta-S_\theta X$ (where $X$ is a bounded linear operator on $K_\theta$) are dense in $A$ in the trace class norm.