Factorizations of contractions

In this context, it is natural to ask whether similar factorization results hold true for pure contractions. The purpose of this paper is to answer this question. More particularly, let T be a pure contraction on a Hilbert space H and let PQMz|Q be the Sz.-Nagy and Foias representation of T for some canonical Q⊆HD2(D). Then T=T1T2, for some commuting contractions T1 and T2 on H, if and only if there exist B(D)-valued polynomials φ and ψ of degree ≤1 such that Q is a joint (Mφ⁎,Mψ⁎)-invariant subspace,PQMz|Q=PQMφψ|Q=PQMψφ|Q and (T1,T2)≅(PQMφ|Q,PQMψ|Q). Moreover, there exist a Hilbert space E and an isometry V∈B(D;E) such thatφ(z)=V⁎Φ(z)V and ψ(z)=V⁎Ψ(z)V(z∈D), where the pair (Φ,Ψ), as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on HE2(D). As an application, we obtain a sharper von Neumann inequality for commuting pairs of contractions.