Vidhya Venugopal | Searcharxiv

arXiv2025

Multivariable Wold-Type Decomposition and Analytic Models for a class of left-inverse commuting pairs

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 $\mathcal D_{\mathcal E}(\mu_1, \mu_2)$ associated with two finite positive operator-valued Borel measures $\mu_1$ and $\mu_2$ on the unit circle. An explicit functional model is further derived for the non-analytic case.

math.FA↗

arXiv2025

Wold-type decomposition for Doubly commuting two-isometries

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.

math.FA↗

arXiv2024

Dirichlet type spaces in the unit bidisc and Wandering Subspace Property for operator tuples

In this article, we define Dirichlet-type space $\mathcal{D}^{2}(\boldsymbol{\mu})$ over the bidisc $\mathbb D^2$ for any measure $\boldsymbol{\mu}\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$ We show that the set of polynomials is dense in $\mathcal{D}^{2}(\boldsymbol{\mu})$ and the pair $(M_{z_1}, M_{z_2})$ of multiplication operator by co-ordinate functions on $\mathcal{D}^{2}(\boldsymbol{\mu})$ is a pair of commuting $2$-isometries. Moreover, the pair $(M_{z_1}, M_{z_2})$ is a left-inverse commuting pair in the following sense: $L_{M_{z_i}} M_{z_j}=M_{z_j}L_{M_{z_i}}$ for $1\leqslant i\neq j\leqslant n,$ where $L_{M_{z_i}}$ is the left inverse of $M_{z_i}$ with $\ker L_{M_{z_i}} =\ker M_{z_i}^*$, $1\leqslant i \leqslant n$. Furthermore, it turns out that, for the class of left-inverse commuting tuple $\boldsymbol T=(T_1, \ldots, T_n)$ acting on a Hilbert space $\mathcal{H}$, the joint wandering subspace property is equivalent to the individual wandering subspace property. As an application of this, the article shows that the class of left-inverse commuting pair with certain splitting property is modelled by the pair of multiplication by co-ordinate functions $(M_{z_1}, M_{z_2})$ on $\mathcal{D}^{2}(\boldsymbol{\mu})$ for some $\boldsymbol{\mu}\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$

math.FA↗