Cyclic polynomials in anisotropic Dirichlet spaces

Abstract

Consider the Dirichlet-type space on the bidisk consisting of holomorphic functions $f(z_1,z_2):=\sum_{k,l\geq0}a_{kl}z_1^kz_2^l$ such that

$$\sum_{^{k,l\geq0}}(k+1)^{\alpha_1}(l+1)^{\alpha_2}|a_{kl}|^{2}<\infty.$$

Here the parameters α₁, α₂ are arbitrary real numbers. We characterize the polynomials that are cyclic for the shift operators on this space. More precisely, we show that, given an irreducible polynomial p(z₁, z₂) depending on both z₁ and z₂ and having no zeros in the bidisk:

• if α₁ + α₂ ≤ 1, then p is cyclic;

• if α₁ + α₂ > 1 and min{α₁, α₂} ≤ 1, then p is cyclic if and only if it has finitely many zeros in the two-torus $\mathbb{T}^2$;

• if min{α₁, α₂} > 1, then p is cyclic if and only if it has no zeros in $\mathbb{T}^2$.