One dimensional perturbation of invariant subspaces in the Hardy space over the bidisk II

Abstract

This paper is a continuation of the previous paper [9]. Let $M_1$ be an invariant subspace of $H^2$ over the bidisk. Then there exists a nonzero $f_0$ in $M_1$ such that $M_2:=M_1\ominus \mathbb{C} \cdot f_0$ is also an invariant subspace. A relationship is given the ranks of the cross commutators $[R^*_z,R_w]$ on $M_1$ and $M_2$. We also give a relationship of the ranks of the cross commutators $[S_w,S^*_z]$ on $H^2\ominus M_1$ and $H^2\ominus M_2$.