Dirichlet-type spaces of the bidisc and Toral $2$-isometries

We introduce and study Dirichlet-type spaces $\mathcal{D}({\mu }_1,{\mu }_2)$ of the unit bidisc ${\mathbb{D}}^2,$ where ${\mu }_1,{\mu }_2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal{D}({\mu }_1,{\mu }_2)$ and the complex polynomials are dense in $\mathcal{D}({\mu }_1,{\mu }_2)\mathrm{.}$ Further, we obtain the division property and solve Gleason’s problem for $\mathcal{D}({\mu }_1,{\mu }_2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair ${\mathcal{M}}_z$ of the multiplication operators ${\mathcal{M}}_{z_1},$ ${\mathcal{M}}_{z_2}$ on $\mathcal{D}({\mu }_1,{\mu }_2)$ defines a cyclic toral $2$-isometry and ${\mathcal{M}}_z^{\ast }$ belongs to the Cowen-Douglas class ${\mathbf{B}}_1({\mathbb{D}}_r^2)$ for some $r>0.$ Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter’s representation theorem for cyclic analytic $2$-isometries. In particular, we show that a cyclic analytic toral $2$-isometric pair $T$ with cyclic vector $f_0$ is unitarily equivalent to ${\mathcal{M}}_z$ on $\mathcal{D}({\mu }_1,{\mu }_2)$ if and only if $\ker T^{\ast },$ spanned by $f_0,$ is a wandering subspace for $T\mathrm{.}$