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.