Cyclicity of Multipliers on the Unit Ball of $\mathbb{C}^n$: A Corona-Based Approach

TL;DR

This paper characterizes when certain multipliers are cyclic in Dirichlet-type spaces on the unit ball in complex n-space, using a novel corona-based approach involving peak sets and a two-generator corona theorem.

Contribution

It provides a precise criterion for cyclicity of multipliers based on the geometry of their zero sets and introduces a corona-based method to analyze this problem.

Findings

Cyclicity depends on the dimension of the zero set and the parameter  in the space.

Established a corona theorem with two generators for the multiplier algebra.

Connected classical peak set results with modern cyclicity criteria.

Abstract

We study the cyclicity of multipliers in Dirichlet-type spaces \( D_\alpha(\mathbb{B}_n) \). Specifically, we show that a multiplier \( f \) analytic on a neighborhood of $\overline{\mathbb{B}}_n$, whose zero set on the unit sphere is a compact, smooth, complex tangential submanifold of real dimension \( m \leq n - 1 \), is cyclic in \( D_\alpha(\mathbb{B}_n) \) if and only if \( \alpha \leq \frac{2n - m}{2} \). Our approach combines classical results on peak sets in \( A^\infty(\mathbb{B}_n) \) due to Chaumat and Chollet with a Corona-type theorem with two generators for the multiplier algebra.