Factorizations of Contractions

TL;DR

This paper extends classical factorization results from isometries to pure contractions, characterizing when a contraction can be expressed as a product of two commuting contractions via polynomial invariance conditions.

Contribution

It provides a new characterization of pure contractions as products of commuting contractions using polynomial invariance and Sz.-Nagy and Foias model theory.

Findings

Characterization of pure contractions as products of commuting contractions.

Use of polynomial invariance in the model space.

Extension of classical isometry factorization results.

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 $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where \[ \Phi(z)=(P+zP^{\perp})U^*\;\text{and}\; \Psi(z)=U(P^{\perp}+zP) \;;(z \in \mathbb{D}). \] 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 $\mathcal{Q} \subseteq H^2_{\mathcal{D}}(\mathbb{D})$. 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 $\varphi$ and $\psi$ of degree $ \leq 1$ such that $\mathcal{Q}$ is a joint $(M_{\varphi}^*, M_{\psi}^*)$-invariant subspace, \[P_{\mathcal{Q}} M_z|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\varphi \psi}|_{\mathcal{Q}} = P_{\mathcal{Q}} M_{\psi \varphi}|_{\mathcal{Q}} \; \mbox{and} \;(T_1, T_2) \cong (P_{\mathcal{Q}} M_{\varphi}|_{\mathcal{Q}}, P_{\mathcal{Q}} M_{\psi}|_{\mathcal{Q}}).\]