A note on cyclic vectors in Dirichlet-type spaces in the unit ball of $\mathbb{C}^n$

TL;DR

This paper characterizes cyclic polynomials and provides capacity-based criteria for non-cyclic vectors in Dirichlet-type spaces within the unit ball of complex n-space.

Contribution

It offers a characterization of cyclic polynomials and introduces a capacity condition to identify non-cyclic vectors in these function spaces.

Findings

Model polynomials are characterized as cyclic in Dirichlet-type spaces.

A capacity condition is established to determine non-cyclic vectors.

The results extend understanding of function behavior in multivariable Dirichlet spaces.

Abstract

We characterize model polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb{C}^n$, and we give a sufficient capacity condition in order to identify non-cyclic vectors.