Factorizations of Contractions

Abstract

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 $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exists a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V,V_1,V_2)$ and $(M_z,M_{\Phi },M_{\Psi })$ on $H_{\mathcal{E}}^2(\mathbb{D})$ are unitarily equivalent, where \[ Φ(z)=(P+zP^{\perp})U^*\;\text{and}\; Ψ(z)=U(P^{\perp}+zP) \;;(z \in \mathbb{D}). \] Here we prove a similar factorization result for pure contractions. More particularly, let $T$ be a pure contraction on a Hilbert space $\mathcal{H}$ and let $P_{\mathcal{Q}}{M}_z{|}_{\mathcal{Q}}$ be the Sz.-Nagy and Foias representation of $T$ for some canonical $\mathcal{Q}\subseteq H_{\mathcal{D}}^2(\mathbb{D})$. Then $T=T_1{T}_2$, for some commuting contractions $T_1$ and $T_2$ on $\mathcal{H}$, if and only if there exists $\mathcal{B}(\mathcal{D})$-valued polynomials $\phi$ and $\psi$ of degree $\le 1$ such that $\mathcal{Q}$ is a joint $(M_{\phi }^{\ast },M_{\psi }^{\ast })$-invariant subspace, \[P_{\mathcal{Q}} M_z|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{φψ}|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{ψφ}|_{\mathcal{Q}} \; \mbox{and} \;(T_1, T_2) \cong (P_{\mathcal{Q}} M_φ|_{\mathcal{Q}}, P_{\mathcal{Q}} M_ψ|_{\mathcal{Q}}).\]

12 pages. Some corrections. To appear in Adv. in Math