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

Author:

Aleman Alexandru,Perfekt Karl-Mikael,Richter Stefan,Sundberg Carl,Sunkes James

Abstract

Let H \mathcal {H} be a space of analytic functions on the unit ball B d {\mathbb {B}_d} in C d \mathbb {C}^d with multiplier algebra M u l t ( H ) \mathrm {Mult}(\mathcal {H}) . A function f H f\in \mathcal {H} is called cyclic if the set [ f ] [f] , the closure of { φ f : φ M u l t ( H ) } \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\} , equals H \mathcal {H} . For multipliers we also consider a weakened form of the cyclicity concept. Namely for n N 0 n\in \mathbb {N}_0 we consider the classes C n ( H ) = { φ M u l t ( H ) : φ 0 , [ φ n ] = [ φ n + 1 ] } . \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 N :th order radially weighted Besov spaces on B d {\mathbb {B}_d} , H = B ω N \mathcal {H}= B^N_\omega , but we describe our results only for the Drury-Arveson space H d 2 H^2_d here.

Letting C s t a b l e [ z ] \mathbb {C}_{stable}[z] denote the stable polynomials for B d {\mathbb {B}_d} , i.e. the d d -variable complex polynomials without zeros in B d {\mathbb {B}_d} , we show that a m p ;  if  d  is odd, then  C s t a b l e [ z ] C d 1 2 ( H d 2 ) ,  and  a m p ;  if  d  is even, then  C s t a b l e [ z ] C d 2 1 ( H d 2 ) . \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 d=2 and d = 4 d=4 these inclusions are the best possible, but in general we can only show that if 0 n d 4 1 0\le n\le \frac {d}{4}-1 , then C s t a b l e [ z ] C n ( H d 2 ) \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d) .

For functions other than polynomials we show that if f , g H d 2 f,g\in H^2_d such that f / g H f/g\in H^\infty and f f is cyclic, then g g is cyclic. We use this to prove that if f , g f,g extend to be analytic in a neighborhood of B d ¯ \overline {{\mathbb {B}_d}} , have no zeros in B d {\mathbb {B}_d} , and the same zero sets on the boundary, then f f is cyclic in H d 2 \in H^2_d if and only if g g is. Furthermore, if the boundary zero set of f H d 2 C ( B d ¯ ) f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension 3 \ge 3 , then f f is not cyclic in the Drury-Arveson space.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference45 articles.

1. Free outer functions in complete Pick spaces;Aleman, Alexandru;Trans. Amer. Math. Soc.,2023

2. The Smirnov class for spaces with the complete Pick property;Aleman, Alexandru;J. Lond. Math. Soc. (2),2017

3. Factorizations induced by complete Nevanlinna-Pick factors;Aleman, Alexandru;Adv. Math.,2018

4. Radially weighted Besov spaces and the Pick property;Aleman, Alexandru,2019

5. Weak products of complete Pick spaces;Aleman, Alexandru;Indiana Univ. Math. J.,2021

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3