Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

CYCLICITY IN THE DRURY-ARVESON SPACE AND OTHER WEIGHTED BESOV SPACES

Aleman, Alexandru LU ; Perfekt, Karl Mikael LU ; Richter, Stefan LU ; Sundberg, Carl LU and Sunkes, James (2024) In Transactions of the American Mathematical Society 377(2). p.1273-1298
Abstract

Let H be a space of analytic functions on the unit ball Bd in Cd with multiplier algebra Mult(H). A function f ∈ H is called cyclic if the set [f], the closure of {ϕf : ϕ ∈ Mult(H)}, equals H. For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N0 we consider the classes Cn(H) = {ϕ ∈ Mult(H): ϕ /= 0, [ϕn] = [ϕn+1]}. Many of our results hold for N:th order radially weighted Besov spaces on Bd, H = BωN, but we describe our results only for the Drury-Arveson space Hd2 here. Letting Cstable[z] denote the stable polynomials for Bd, i.e. the d-variable complex polynomials without zeros... (More)

Let H be a space of analytic functions on the unit ball Bd in Cd with multiplier algebra Mult(H). A function f ∈ H is called cyclic if the set [f], the closure of {ϕf : ϕ ∈ Mult(H)}, equals H. For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N0 we consider the classes Cn(H) = {ϕ ∈ Mult(H): ϕ /= 0, [ϕn] = [ϕn+1]}. Many of our results hold for N:th order radially weighted Besov spaces on Bd, H = BωN, but we describe our results only for the Drury-Arveson space Hd2 here. Letting Cstable[z] denote the stable polynomials for Bd, i.e. the d-variable complex polynomials without zeros in Bd, we show that if d is odd, then Cstable[z] ⊆ Cd−1 (Hd2), and 2 if d is even, then Cstable[z] ⊆ Cd2 −1(Hd2). For d = 2 and d = 4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ d4 − 1, then Cstable[z] Cn(Hd2). For functions other than polynomials we show that if f, g ∈ Hd2 such that f/g ∈ H∞ 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 Bd, have no zeros in Bd, and the same zero sets on the boundary, then f is cyclic in ∈ Hd2 if and only if g is. Furthermore, if the boundary zero set of f ∈ Hd2 ∩ C(Bd) embeds a cube of real dimension ≥ 3, then f is not cyclic in the Drury-Arveson space.

(Less)
Please use this url to cite or link to this publication:
author
; ; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Transactions of the American Mathematical Society
volume
377
issue
2
pages
26 pages
publisher
American Mathematical Society (AMS)
external identifiers
  • scopus:85183959824
ISSN
0002-9947
DOI
10.1090/tran/9060
language
English
LU publication?
yes
id
a1ead3c3-be98-4a4b-81ae-74c6ccdba940
date added to LUP
2024-03-01 10:52:26
date last changed
2024-03-01 10:55:44
@article{a1ead3c3-be98-4a4b-81ae-74c6ccdba940,
  abstract     = {{<p>Let H be a space of analytic functions on the unit ball B<sub>d</sub> in C<sup>d</sup> with multiplier algebra Mult(H). A function f ∈ H is called cyclic if the set [f], the closure of {ϕf : ϕ ∈ Mult(H)}, equals H. For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N0 we consider the classes Cn(H) = {ϕ ∈ Mult(H): ϕ /= 0, [ϕ<sup>n</sup>] = [ϕ<sup>n+1</sup>]}. Many of our results hold for N:th order radially weighted Besov spaces on B<sub>d</sub>, H = B<sub>ω</sub><sup>N</sup>, but we describe our results only for the Drury-Arveson space H<sub>d</sub><sup>2</sup> here. Letting C<sub>stable</sub>[z] denote the stable polynomials for B<sub>d</sub>, i.e. the d-variable complex polynomials without zeros in B<sub>d</sub>, we show that if d is odd, then C<sub>stable</sub>[z] ⊆ Cd−1 (H<sub>d</sub><sup>2</sup>), and 2 if d is even, then C<sub>stable</sub>[z] ⊆ Cd<sub>2 −1</sub>(H<sub>d</sub><sup>2</sup>). For d = 2 and d = 4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ <sup>d</sup><sub>4</sub> − 1, then C<sub>stable</sub>[z] Cn(H<sub>d</sub><sup>2</sup>). For functions other than polynomials we show that if f, g ∈ H<sub>d</sub><sup>2</sup> such that f/g ∈ H<sup>∞</sup> 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 B<sub>d</sub>, have no zeros in B<sub>d</sub>, and the same zero sets on the boundary, then f is cyclic in ∈ H<sub>d</sub><sup>2</sup> if and only if g is. Furthermore, if the boundary zero set of f ∈ H<sub>d</sub><sup>2</sup> ∩ C(B<sub>d</sub>) embeds a cube of real dimension ≥ 3, then f is not cyclic in the Drury-Arveson space.</p>}},
  author       = {{Aleman, Alexandru and Perfekt, Karl Mikael and Richter, Stefan and Sundberg, Carl and Sunkes, James}},
  issn         = {{0002-9947}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{1273--1298}},
  publisher    = {{American Mathematical Society (AMS)}},
  series       = {{Transactions of the American Mathematical Society}},
  title        = {{CYCLICITY IN THE DRURY-ARVESON SPACE AND OTHER WEIGHTED BESOV SPACES}},
  url          = {{http://dx.doi.org/10.1090/tran/9060}},
  doi          = {{10.1090/tran/9060}},
  volume       = {{377}},
  year         = {{2024}},
}