Beurling quotient modules on the polydisc

Abstract

Let H2(Dn) denote the Hardy space over the polydisc Dn, n≥2. A closed subspace Q⊆H2(Dn) is called Beurling quotient module if there exists an inner function θ∈H∞(Dn) such that Q=H2(Dn)/θH2(Dn). We present a complete characterization of Beurling quotient modules of H2(Dn): Let Q⊆H2(Dn) be a closed subspace, and let Czi=PQMzi|Q, i=1,…,n. Then Q is a Beurling quotient module if and only if(IQ−Czi⁎Czi)(IQ−Czj⁎Czj)=0(i≠j). We present two applications: first, we obtain a dilation theorem for Brehmer n-tuples of commuting contractions, and, second, we relate joint invariant subspaces with factorizations of inner functions. All results work equally well for general vector-valued Hardy spaces.