Rank-one cross commutators on backward shift invariant subspaces on the bidisk

For a backward shift invariant subspace N in H 2(.2), the operators S z and S w on N are defined by S z = P N T z | N and S w = P N T w | N , where P N is the orthogonal projection from L 2(.2) onto N. We give a characterization of N satisfying rank [S z , S * w ] = 1.