Cyclicity of Multipliers on the Unit Ball of ℂⁿ: A Corona-Based Approach

2025/08/07 by Ziarati, Pouriya Torkinejad
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2508.05189

Abstract

We study the cyclicity of multipliers in Dirichlet-type spaces \( D_α(\mathbbB_n) \). Specifically, we show that a multiplier \( f \) analytic on a neighborhood of the closed unit ball, whose zero set on the unit sphere is a compact, smooth, complex tangential submanifold of real dimension \( m ≤ n - 1 \), is cyclic in \( D_α(\mathbbB_n) \) if and only if \( α≤ (2n - m)/(2) \), where \( m \) is the real dimension of the zero set of \( f \) on the boundary. Our approach combines classical results on peak sets in \( A^∞(\mathbbB_n) \) due to Chaumat and Chollet with a Corona-type theorem with two generators for the multiplier algebra.

Citations

• Cyclicity in Poletsky-Stessin Weighted Bergman Spaces
• Cyclicity in Besov-Dirichlet spaces from the Corona Theorem
• Non-cyclicity and polynomials in Dirichlet-type spaces of the unit ball
• Cyclicity in the Drury-Arveson space and other weighted Besov spaces
• Cyclic polynomials in anisotropic Dirichlet~spaces
• Cyclicity and invariant subspaces in the Dirichlet spaces
• On the Brown--Shields conjecture for cyclicity in the Dirichlet space
• Theory of Bergman Spaces in the Unit Ball of Cⁿ
• On two problems concerning linear transformations in hilbert space

Related

• Non-cyclicity and polynomials in Dirichlet-type spaces of the unit ball
• A note on cyclic vectors in Dirichlet-type spaces in the unit ball of ℂⁿ
• Cyclic polynomials in Dirichlet-type spaces in the unit ball of \mathbb C²
• Cyclicity in the Drury-Arveson space and other weighted Besov spaces
• Multipliers on Dirichlet type spaces