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Cyclicity and Iterated Logarithms in the Drury–Arveson Space

Aleman, Alexandru LU ; Perfekt, Karl Mikael LU ; Richter, Stefan LU ; Sundberg, Carl LU and Sunkes, James (2026) In Springer INdAM Series 66. p.37-53
Abstract

Let Hd2 be the Drury–Arveson space, and let f∈Hd2 have bounded argument and no zeros in Bd. We show that f is cyclic in Hd2 if and only if logf belongs to the Pick-Smirnov class N+(Hd2). Furthermore, for non-vanishing functions f∈Hd2 with bounded argument and H∞-norm less than 1, cyclicity can also be tested via iterated logarithms. For example, we show that f is cyclic if and only if log(1+log(1∕f))∈N+(Hd2). Thus, a sufficient condition for cyclicity is that log(1+log(1∕f))∈Hd2. More generally, our results hold for all radially weighted... (More)

Let Hd2 be the Drury–Arveson space, and let f∈Hd2 have bounded argument and no zeros in Bd. We show that f is cyclic in Hd2 if and only if logf belongs to the Pick-Smirnov class N+(Hd2). Furthermore, for non-vanishing functions f∈Hd2 with bounded argument and H∞-norm less than 1, cyclicity can also be tested via iterated logarithms. For example, we show that f is cyclic if and only if log(1+log(1∕f))∈N+(Hd2). Thus, a sufficient condition for cyclicity is that log(1+log(1∕f))∈Hd2. More generally, our results hold for all radially weighted Besov spaces that also are complete Pick spaces.

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Please use this url to cite or link to this publication:
author
; ; ; and
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
47A16; Secondary 30H25, Cyclic vectors, Drury-Arveson space, Primary 47B32
host publication
New Trends in Complex Analysis, Fourier Analysis, and Operator Theory
series title
Springer INdAM Series
volume
66
pages
17 pages
publisher
Springer-Verlag Italia s.r.l.
external identifiers
  • scopus:105036202693
ISSN
2281-518X
2281-5198
DOI
10.1007/978-981-95-5280-1_2
language
English
LU publication?
yes
id
484e7c9a-abd5-4335-a7a6-4f2a392457da
date added to LUP
2026-05-29 13:44:49
date last changed
2026-07-26 00:36:08
@inbook{484e7c9a-abd5-4335-a7a6-4f2a392457da,
  abstract     = {{<p>Let H<sub>d</sub><sup>2</sup> be the Drury–Arveson space, and let f∈H<sub>d</sub><sup>2</sup> have bounded argument and no zeros in B<sub>d</sub>. We show that f is cyclic in H<sub>d</sub><sup>2</sup> if and only if logf belongs to the Pick-Smirnov class N<sup>+</sup>(H<sub>d</sub><sup>2</sup>). Furthermore, for non-vanishing functions f∈H<sub>d</sub><sup>2</sup> with bounded argument and H<sup>∞</sup>-norm less than 1, cyclicity can also be tested via iterated logarithms. For example, we show that f is cyclic if and only if log(1+log(1∕f))∈N<sup>+</sup>(H<sub>d</sub><sup>2</sup>). Thus, a sufficient condition for cyclicity is that log(1+log(1∕f))∈H<sub>d</sub><sup>2</sup>. More generally, our results hold for all radially weighted Besov spaces that also are complete Pick spaces.</p>}},
  author       = {{Aleman, Alexandru and Perfekt, Karl Mikael and Richter, Stefan and Sundberg, Carl and Sunkes, James}},
  booktitle    = {{New Trends in Complex Analysis, Fourier Analysis, and Operator Theory}},
  issn         = {{2281-518X}},
  keywords     = {{47A16; Secondary 30H25; Cyclic vectors; Drury-Arveson space; Primary 47B32}},
  language     = {{eng}},
  pages        = {{37--53}},
  publisher    = {{Springer-Verlag Italia s.r.l.}},
  series       = {{Springer INdAM Series}},
  title        = {{Cyclicity and Iterated Logarithms in the Drury–Arveson Space}},
  url          = {{http://dx.doi.org/10.1007/978-981-95-5280-1_2}},
  doi          = {{10.1007/978-981-95-5280-1_2}},
  volume       = {{66}},
  year         = {{2026}},
}