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

Abstract

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 $\boldsymbol{\mu}\in\mathcal{P}\mathcal{M}_{+}(\mathbb T^2).$