Cyclicity and iterated logarithms in the Drury-Arveson space

Abstract

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

17 pages