Isometric dilations and Beurling quotient modules

Abstract

In \cite{Dan-Guo}, the authors provided a characterization of all sequences of doubly commuting pure contractions that have the sequence ${\text{M}}_z,$ the sequence of the multiplication by the coordinate functions on the vector-valued Hardy space $H_{\mathcal{E}}^2({\mathbb{D}}_2^{(\infty )})$ over the Hilbert's multidisk ${\mathbb{D}}_2^{(\infty )},$ as their minimal isometric dilation for some Hilbert space $\mathcal{E}$. In this article, we remove the hypothesis of doubly commutativity and characterize all sequences of commuting contractions that are unitarily equivalent to $(P_{\mathcal{Q}}{M}_{z_i}{|}_{\mathcal{Q}}{)}_{i\ge 1}$ for some quotient module $\mathcal{Q}$ of $H_{\mathcal{E}}^2({\mathbb{D}}_2^{(\infty )})$. We further characterize those sequences for which $\mathcal{Q}$ is a Beurling quotient module. This generalizes Theorem 3.2 from \cite{BKD} to a sequence of commuting contractions.

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