Cyclicity and iterated logarithms in the Drury-Arveson space

TL;DR

This paper characterizes cyclic functions in the Drury-Arveson space using logarithmic conditions and extends these results to weighted Besov spaces, providing new criteria for cyclicity based on iterated logarithms.

Contribution

It introduces a novel characterization of cyclic functions via logarithmic conditions and extends the analysis to a broader class of weighted Besov spaces with Pick space properties.

Findings

Cyclicity of functions is characterized by their logarithms belonging to the Pick-Smirnov class.

Cyclicity can be tested using iterated logarithms for certain bounded functions.

Results apply to all radially weighted Besov spaces that are complete Pick spaces.

Abstract

Let $H^2_d$ be the Drury-Arveson space, and let $f\in H^2_d$ have bounded argument and no zeros in $\mathbb{B}_d$. We show that $f$ is cyclic in $H^2_d$ if and only if $\log f$ belongs to the Pick-Smirnov class $N^+(H^2_d)$. Furthermore, for non-vanishing functions $f\in H^2_d$ with bounded argument and $H^\infty$-norm less than 1, cyclicity can also be tested via iterated logarithms. For example, we show that $f$ is cyclic if and only if $\log(1+\log (1/f))\in N^+(H^2_d)$. Thus, a sufficient condition for cyclicity is that $\log(1+\log (1/f))\in H^2_d$. More generally, our results hold for all radially weighted Besov spaces that also are complete Pick spaces.