| Abstrakt: |
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 \begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*} Many of our results hold for N:th order radially weighted Besov spaces on {\mathbb {B}_d}, \mathcal {H}= B^N_\omega, 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 extend to be analytic in a neighborhood of \overline {{\mathbb {B}_d}}, have no zeros in {\mathbb {B}_d}, and the same zero sets on the boundary, then f is cyclic in \in H^2_d if and only if g is. Furthermore, if the boundary zero set of f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension \ge 3, then f is not cyclic in the Drury-Arveson space. [ABSTRACT FROM AUTHOR] |
