Wold-type decomposition for Doubly commuting two-isometries

TL;DR

The paper addresses the problem of obtaining a Wold-type decomposition for pairs of doubly commuting $2$-isometries on a Hilbert space. It develops a multivariable functional-model framework: the analytic part of such a pair is unitarily equivalent to the pair $(M_{z_1},M_{z_2})$ on a bidisc Dirichlet-type space $\,\\mathcal{D}^2_{\\mathcal{E}}( ilde{\mu}_1, ilde{\mu}_2)$ defined by operator-valued measures $\tilde{\mu}_1,\tilde{\mu}_2$, and in the general case it yields a four-way decomposition into unitary and analytic components with explicit Dirichlet-model realizations on each piece. This provides a comprehensive multivariate extension of Wold theory for $2$-isometries, connecting to shifts on Dirichlet-type spaces and extending known results for isometries (e.g., Slocinski) and single $2$-isometries (e.g., Shimorin-Olofsson). The results offer a canonical functional-model description for doubly commuting $2$-isometries and have potential applications in multivariable operator theory and invariant-subspace problems on the bidisc.

Abstract

In this article, we prove that any pair of doubly commuting $2$-isometries on a Hilbert space has a Wold-type decomposition. Moreover, the analytic part of the pair is unitary equivalent to the pair of multiplication by coordinate function on a Dirichlet-type space on the bidisc.

Theorems & Definitions (16)

• Theorem 1.1: Richter2
• Theorem 1.2: Olofsson
• Theorem 1.3
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Proposition 3.2
• proof
• Theorem 3.3
• proof