Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk

Abstract

For $\alpha \in \mathbb{R},$ we consider the scale of function spaces, namely the Dirichlet-type space $D_{\alpha }$ consisting of holomorphic functions on the unit bidisk ${\mathbb{D}}^2$, $f(z,w)={\sum }_{k,l=0}^{\infty }{a}_{kl}{z}^k{w}^l$ such that $$\sum _{k,l=0}^{\infty }(k+l+1{)}^{\alpha }|a_{kl}{|}^2<\infty .$$ In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial $2-z_1-z_2$ in $D_{\alpha }$ for $\frac{3}{2}<\alpha \le 2$. We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $D_{\alpha }$.

arXiv comment17 pages, Major changes