On the Brown--Shields conjecture for cyclicity in the Dirichlet space

2008/09/26 by Omar El-Fallah, El-Fallah, Omar, Karim Kellay +3 · 4 citations
Mathematics · #30H05 (Primary) #46E20 #47A15 (Secondary) #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · doi:10.48550/arxiv.0809.4557

openalex publication_date 2008/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \cD be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function f∈\cD to be \em cyclic, i.e. for \pf: pa polynomial\ to be dense in \cD. This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in \cD iff it is outer and its zero set (defined appropriately) is of capacity zero.

Cited by

• Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk
• Douglas Weighted Dirichlet Spaces: Analytic and Probabilistic Aspects
• Cyclicity of Multipliers on the Unit Ball of ℂⁿ: A Corona-Based Approach
• Cyclicity in Poletsky-Stessin Weighted Bergman Spaces

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