Beurling quotient modules on the polydisc

Let H 2 ( D n ) denote the Hardy space over polydisc , ≥ . A closed subspace Q ⊆ is called Beurling quotient module if there exists an inner function θ ∈ ∞ such that = / We present a complete characterization of modules : be subspace, and let C z i P M | 1 … Then only I − ⁎ j 0 ≠ two applications: first, we obtain dilation theorem for Brehmer -tuples commuting contractions, and, second, relate joint invariant subspaces with factorizations functions. All results work equally well general vector-valued spaces.