Abstract. Nakazi, Seto, and the first author gave a characterization of backward shift invariant subspaces of the Hardy space over the bidisk on which the cross commutators $[S_z,S^*_w]$ and $[S_{z^2},S^*_w]$ vanish, respectively. A characterization is given of backward shift invariant subspaces on which $[S_{z^n},S^*_w] = 0$ for a positive integer $n\ge2$.

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

Keyword(s): Hardy space, backward shift invariant subspace, cross commutator

Received December 13, 2005, and in revised form March 22, 2006. (Registered under 5920/2009.)