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The celebrated Sz.-Nagy and Foias theorem asserts that every pure contraction is unitarily equivalent to an operator of the form, 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 V 1 and V 2 in B(H) if and only if there exist a Hilbert space E, a unitary U in B(E) and an orthogonal projectionIn 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 P Q M z | Q be the Sz.-Nagy and Foias representation of T for some canonical Q ⊆ H 2 D (D). Then T = T 1 T 2 , for some commuting contractions T 1 and T 2 on H, if and only if there exist B(D)-valued polynomials ϕ and ψ of degree ≤ 1 such that Q is a joint (M * ϕ , M * ψ )-invariant subspace,Moreover, there exist a Hilbert space E and an isometry V ∈ B(D; E) such thatwhere the pair (Φ, Ψ), as defined above, is the Berger, Coburn and Lebow representation of a pure pair of commuting isometries on H 2 E (D). As an application, we obtain a sharper von Neumann inequality for commuting pairs of contractions.2010 Mathematics Subject Classification. 47A13, 47A20, 47A56, 47A68, 47B38, 46E20, 30H10.

Abstract. Let D denote the unit disc in the complex plane C and let D 2 = D × D be the unit bidisc in C 2 . Let (T 1 , T 2 ) be a pair of commuting contractions on a Hilbert space H. Let dim ran(I H − T j T * j ) < ∞, j = 1, 2, and let T 1 be a pure contraction. Then there exists a variety V ⊆ D 2 such that for any polynomial p ∈ C[z 1 , z 2 ], the inequalityIf, in addition, T 2 is pure, thenis a distinguished variety, where Ψ is a matrix-valued analytic function on D that is unitary on ∂D. Our results comprise a new proof, as well as a generalization, of Agler and McCarthy's sharper von Neumann inequality for pairs of commuting and strictly contractive matrices.

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