Multipliers of a wandering subspace for a shift invariant subspace

For a shift-invariant subspace M of the two variable Hardy space H 2 , we consider the associated wandering subspace M 0 = M ⊖ z M . Then there exists a nonconstant function ϕ in H ∞ such that ϕ M 0 ⊆ M 0 if and only if M = q H 2 for some inner function q.