Multivariable Wold-Type Decomposition and Analytic Models for a class of left-inverse commuting pairs
Monojit Bhattacharjee, Rajeev Gupta, Vidhya Venugopal
TL;DR
The paper addresses multivariable generalizations of the Wold-type decomposition for left-inverse commuting tuples and constructs analytic models for a rich class of pairs. It develops a joint Wold-type decomposition for left-inverse commuting pairs, yielding a four-block reduction that clarifies unitary and analytic components. The main advancement is a complete analytic model: any cyclic analytic toral $2$-isometry pair is unitarily equivalent to $(M_{z_1}, M_{z_2})$ on a vector-valued Dirichlet-type space $\mathcal{D}_{\mathcal{E}}(\mu_1,\mu_2)$ determined by operator-valued measures on $\mathbb{T}$; an explicit non-analytic model is also provided. The framework is extended to higher ranks, yielding a canonical block decomposition involving $\mathcal{D}_{\mathcal{E}_{01}}(\mu_1)$, $\mathcal{D}_{\mathcal{E}_{10}}(\mu_2)$, and $\mathcal{D}_{\mathcal{E}}(\nu_1,\nu_2)$, situating the study at the intersection of operator theory, function theory on the bidisc, and multivariable dilation theory.
Abstract
This work establishes a multivariable Wold-type decomposition for left-inverse commuting $n$-tuples of bounded operators, built on the hypothesis that each component admits a Wold-type decomposition. For pairs of operators, we obtain a complete analytic model: every left-inverse commuting analytic toral $2$-isometric pair is unitarily equivalent to the pair of multiplication operator by co-ordinate functions $(M_{z_1}, M_{z_2})$ acting on some $\mathcal{E}$-valued Dirichlet-type space $\mathcal D_{\mathcal E}(μ_1, μ_2)$ associated with two finite positive operator-valued Borel measures $μ_1$ and $μ_2$ on the unit circle. An explicit functional model is further derived for the non-analytic case.