Dirichlet Type Spaces in the Unit Bidisc and Wandering Subspace Property for Operator Tuples

In this article, we define Dirichlet-type space 𝒟^2(μ) over the bidisc 𝔻^2 for any measure μ∈𝒫ℳ_+(𝕋^2). We show that the set of polynomials is dense in 𝒟^2(μ) and the pair (M_z_1, M_z_2) of multiplication operator by co-ordinate functions on 𝒟^2(μ) 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_jL_M_z_i for 1⩽ i≠ j⩽ n, where L_M_z_i is the left inverse of M_z_i with L_M_z_i = M_z_i^*, 1⩽ i ⩽ n. Furthermore, it turns out that, for the class of left-inverse commuting tuple T=(T_1, …, T_n) acting on a Hilbert space 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 𝒟^2(μ) for some μ∈𝒫ℳ_+(𝕋^2).