Abstract. Let $M$ be an invariant subspace of the Hardy space $H^2$ over the bidisk and $N=H^2\ominus M$. It is revisited the study of $M$ on which $R_zR^*_w=R^*_wR_z$ and of $N$ on which $S_zS^*_w=S^*_wS_z$.

AMS Subject Classification (1991): 47A15, 32A35

Keyword(s): invariant subspace, backward shift invariant subspace, Hardy space, compression operator, two-variable Jordan blocks

Received January 18, 2011, and in revised form March 16, 2011. (Registered under 5/2011.)