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

In this article, we define Dirichlet-type space ${\mathcal{D}}^2(\mathit{\mu })$ over the bidisc ${\mathbb{D}}^2$ for any measure $\mathit{\mu }\in \mathcal{P}{\mathcal{M}}_{+}({\mathbb{T}}^2)\mathrm{.}$ We show that the set of polynomials is dense in ${\mathcal{D}}^2(\mathit{\mu })$ and the pair $(M_{z_1},M_{z_2})$ of multiplication operator by co-ordinate functions on ${\mathcal{D}}^2(\mathit{\mu })$ 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 $1⩽i\mathrm{\ne }j⩽n,$ where $L_{M_{z_i}}$ is the left inverse of $M_{z_i}$ with $\ker L_{M_{z_i}}=\ker M_{z_i}^{\ast }$, $1⩽i⩽n$. Furthermore, it turns out that, for the class of left-inverse commuting tuple $\mathit{T}=(T_1,\dots ,T_n)$ 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 ${\mathcal{D}}^2(\mathit{\mu })$ for some $\mathit{\mu }\in \mathcal{P}{\mathcal{M}}_{+}({\mathbb{T}}^2)\mathrm{.}$