Dirichlet-type spaces of the unit bidisc and toral 2-isometries

We introduce and study Dirichlet-type spaces D(μ ₁, μ ₂) of the unit bidisc D², where μ ₁, μ ₂ are finite positive Borel measures on the unit circle. We show that the coordinate functions z₁ and z₂ are multipliers for D(μ ₁, μ ₂) and the complex polynomials are dense in D(μ ₁, μ ₂). Further, we obtain the division property and solve Gleason’s problem for D(μ ₁, μ ₂) over a bidisc centered at the origin. In particular, we show that the commuting pair Mz of the multiplication operators Mz₁, Mz₂ on D(μ ₁, μ ₂) defines a cyclic toral $2$ -isometry and M^*z belongs to the Cowen–Douglas class B₁( D²ᵣ) 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 f₀ is unitarily equivalent to Mz on D(μ ₁, μ ₂) for some μ ₁,μ ₂ if and only if T^*, spanned by f₀, is a wandering subspace for $T.$