Cyclicity in the Drury-Arveson space and other weighted Besov spaces

Abstract

Let $\mathcal{H}$ be a space of analytic functions on the unit ball ${\mathbb{B}}_d$ in ${\mathbb{C}}^d$ with multiplier algebra $\mathrm{Mult}(\mathcal{H})$. A function $f\in \mathcal{H}$ is called cyclic if the set $[f]$, the closure of $\{\phi f:\phi \in \mathrm{Mult}(\mathcal{H})\}$, equals $\mathcal{H}$. For multipliers we also consider a weakened form of the cyclicity concept. Namely for $n\in {\mathbb{N}}_0$ we consider the classes $${\mathcal{C}}_n(\mathcal{H})=\{\phi \in \mathrm{Mult}(\mathcal{H}):\phi \ne 0,[{\phi }^n]=[{\phi }^{n+1}]\}.$$ Many of our results hold for $N$:th order radially weighted Besov spaces on ${\mathbb{B}}_d$, but we describe our results only for the Drury-Arveson space $H_d^2$ here. Letting ${\mathbb{C}}_{stable}[z]$ denote the stable polynomials for ${\mathbb{B}}_d$, i.e. the $d$-variable complex polynomials without zeros in ${\mathbb{B}}_d$, we show that \begin{align*} &\text{ if }d \text{ is odd, then } \mathbb C_{stable}[z]\subseteq \mathcal C_{\frac{d-1}{2}}(H^2_d), \text{ and }\\ &\text{ if }d \text{ is even, then } \mathbb C_{stable}[z]\subseteq \mathcal C_{\frac{d}{2}-1}(H^2_d).\end{align*} For $d=2$ and $d=4$ these inclusions are the best possible, but in general we can only show that if $0\le n\le \frac{d}{4}-1$, then ${\mathbb{C}}_{stable}[z]⊈{\mathcal{C}}_n(H_d^2)$. For functions other than polynomials we show that if $f,g\in H_d^2$ such that $f/g\in H^{\infty }$ and $f$ is cyclic, then $g$ is cyclic. We use this to prove that if $f,g\in H_d^2$ extend to be analytic in a neighborhood of $\overline{{\mathbb{B}}_d}$, have no zeros in ${\mathbb{B}}_d$, and their zero sets coincide on the boundary, then $f$ is cyclic if and only if $g$ is cyclic. Furthermore, if for $f\in H_d^2\cap C(\overline{{\mathbb{B}}_d})$ the set $Z(f)\cap \partial {\mathbb{B}}_d$ embeds a cube of real dimension $\ge 3$, then $f$ is not cyclic in the Drury-Arveson space.

32 pages