Factorizations of contractions

Article Type

Research Article

Publication Title

Advances in Mathematics


The celebrated Sz.-Nagy and Foias theorem asserts that every pure contraction is unitarily equivalent to an operator of the form PQMz|Q where Q is a Mz⁎-invariant subspace of a D-valued Hardy space HD2(D), for some Hilbert space D. On the other hand, the celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry V on a Hilbert space H is a product of two commuting isometries V1 and V2 in B(H) if and only if there exist a Hilbert space E, a unitary U in B(E) and an orthogonal projection P in B(E) such that (V,V1,V2) and (Mz,MΦ,MΨ) on HE2(D) are unitarily equivalent, where Φ(z)=(P+zP⊥)U⁎andΨ(z)=U(P⊥+zP)(z∈D). 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 that φ(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.

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