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

Pouriya Torkinejad Ziarati

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arXiv · 2508.05189

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

Pouriya Torkinejad Ziarati

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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.

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Pouriya Torkinejad Ziarati. 2025-08-07. Cyclicity of Multipliers on the Unit Ball of $\mathbb{C}^n$: A Corona-Based Approach. https://arxiv.org/abs/2508.05189

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