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

Abstract We introduce and study Dirichlet-type spaces D (₁, ₂) of the unit bidisc D², where ₁, ₂ are finite positive Borel measures on the unit circle. We show that the coordinate functions z₁ and z₂ are multipliers for D (₁, ₂) and the complex polynomials are dense in D (₁, ₂). Further, we obtain the division property and solve Gleason’s problem for D (₁, ₂) over a bidisc centered at the origin. In particular, we show that the commuting pair Mᵦ of the multiplication operators Mₙ䃑, Mₙ䃒 on D (₁, ₂) defines a cyclic toral 2 -isometry and M^*ᵦ 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 Mᵦ on D (₁, ₂) for some ₁, ₂ if and only if T^*, spanned by f₀, is a wandering subspace for T.