Journal of the
Korean Mathematical Society JKMS

In the previous paper, we gave a characterization of backward shift invariant subspaces of the Hardy space over the bidisk on which $[S_{z^n},S^*_w] = 0$ for a positive integer $n\ge 2$. In this case, it holds that $S_{z^n}=cI$ for some $c\in{\mathbb C}$. In this paper, it is proved that if $[S_{\varphi},S^*_w] = 0$ and ${\varphi}\in H^\infty(\Gamma_z)$, then $S_{\varphi}=cI$ for some $c\in{\mathbb C}$.

Keywords: backward shift invariant subspace, invariant subspace, Hardy space, cross commutator

MSC numbers: Primary 47A15, 32A35; Secondary 47B35