Backward shift invariant subspaces in the bidisc. II

For every invariant subspace M in the Hardy spaces H-2(Gamma(2)), let V-z and V-w be multiplication operators on M. Then it is known that the condition VzVw* = V-w*V-z on M holds if and only if M is a Beurling type invariant subspace. For a backward shift invariant subspace N in H-2(Gamma(2)), two operators S-z and S-w on N are defined by S-z = PNLzPN and S-w = PNLwPN, where P-N is the orthogonal projection from L-2(Gamma(2)) onto N. It is given a characterization of N satisfying SzSw* = S-w*S-z on N.