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

Pouriya Torkinejad Ziarati

Abstract

We study the cyclicity of multipliers in Dirichlet-type spaces \( D_α(\mathbb{B}_n) \). 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 \( D_α(\mathbb{B}_n) \) if and only if \( α\leq \frac{2n - m}{2} \). 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.

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

Abstract

We study the cyclicity of multipliers in Dirichlet-type spaces \( D_α(\mathbb{B}_n) \). Specifically, we show that a multiplier analytic on a neighborhood of , whose zero set on the unit sphere is a compact, smooth, complex tangential submanifold of real dimension , is cyclic in \( D_α(\mathbb{B}_n) \) if and only if . 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.

Paper Structure

This paper contains 6 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminary Tools
  3. Basics and Notations
  4. Proof of Theorem \ref{['Subspace_Inclusion']}
  5. Peak Sets of $A^{\infty}(\mathbb{B}_n)$
  6. Proof of the Main Result

Key Result

Theorem 1

Let $f \in \mathcal{O}(\overline{\mathbb{B}}_n)$ be nonvanishing in $\mathbb{B}_n$, and suppose that $\mathcal{Z}(f) \cap \overline{\mathbb{B}}_n = M$, where $M$ is a compact complex tangential smooth manifold. Let $m \coloneqq \dim_{\mathbb R} M$, if $\alpha \leq \frac{2n - m}{2}$, then $f$ is cycl

Theorems & Definitions (30)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Lemma 10
  • Definition 11
  • Corollary 12
  • ...and 20 more