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

TL;DR

This paper investigates cyclic functions in the Drury-Arveson space and weighted Besov spaces, characterizing when polynomials and certain functions are cyclic based on their zero sets and boundary behavior.

Contribution

It establishes new inclusion relations for stable polynomials in classes of multipliers and characterizes cyclicity for functions with specific boundary zero set structures.

Findings

For odd d, stable polynomials are in C_{(d-1)/2}(H^2_d).

For even d, stable polynomials are in C_{d/2 - 1}(H^2_d).

Functions with boundary zero sets of dimension ≥ 3 are not cyclic.

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 $\{\varphi f:\varphi \in \mathrm{Mult}(\mathcal{H})\}$, 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 $$\mathcal{C}_n(\mathcal{H})=\{\varphi \in \mathrm{Mult}(\mathcal H):\varphi\ne 0, [\varphi^n]=[\varphi^{n+1}]\}.$$ 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 $H^2_d$ 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 $\mathbb C_{stable}[z]\nsubseteq \mathcal C_n(H^2_d)$. For functions other than polynomials we show that if $f,g\in H^2_d$ such that $f/g\in H^\infty$ and $f$ is cyclic, then $g$ is cyclic. We use this to prove that if $f,g\in H^2_d$ 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 $f\in H^2_d\cap C(\overline{\mathbb B_d })$ 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.