Dirichlet type spaces in the unit bidisc

In this article, we prove that an analytic $2$-isometric left-inverse commuting pair on a Hilbert space always possess wandering subspace property. Along with this, we define Dirichlet-type space D²(μ) over the bidisc D² for any measure μ₊( T²). We prove that the pair (Mz₁,Mz₂) of multiplication by the coordinate functions on D²(μ) is an analytic $2$-isometric left-inverse commuting pair and satisfies z₁ᵐ z₂ⁿ, z₁ᵖ z₂q μ = z₁ᵐ, z₁ᵖ D(μ₁) z₂ⁿ, z₂q D(μ₂), for any m,n,p,q∈ Z₀. Furthermore, the unitary equivalent class of the commuting analytic $2$-isometric pair (Mz₁,Mz₂) on D²(μ) is identified using left-inverse commutativity and the above property of inner product.