Commutators of two compressed shifts and the Hardy space on the bidisc

Abstract

For a subset $E$ of the bidisc $D^2, M=\{f\in H^2(D^2)~:~f=0$ on $E\}$ and $N$ is the orthogonal complement of $M$ in $H^2(D^2)$ where $H^2(D^2)$ is the two variable Hardy space on $D^2$. We describe the finite rank commutants of the restricted shifts $S_z$ and $S_w$ on $N$ when $E$ satisfies some natural condition. Moreover we give a sufficient condition for that the Pick interpolation is possible.