Cyclicity and iterated logarithms in the Dirichlet space

2024/09/30 by Aleman, Alexandru, Richter, Stefan
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2409.20298

Abstract

Let D(μ) denote a harmonically weighted Dirichlet space on the unit disc \mathbb D. We show that outer functions f∈ D(μ) are cyclic in D(μ), whenever log f belongs to the Pick-Smirnov class N⁺(D(μ)). If f has H^∞-norm less than or equal to 1, then cyclicity can also be checked via iterated logarithms. For example, we show that such outer functions f are cyclic, whenever log(1+ log(1/f))∈ N⁺(D(μ)). This condition can be checked by verifying that log(1+ log(1/f))∈ D(μ). If f satisfies a mild extra condition, then the conditions also become necessary for cyclicity.

Related

• Cyclicity and iterated logarithms in the Drury-Arveson space
• Cyclicity in the harmonic Dirichlet space
• On the Brown--Shields conjecture for cyclicity in the Dirichlet space
• Cyclicity in reproducing kernel Hilbert spaces of analytic functions
• Cyclicity in de Branges--Rovnyak spaces