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Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency

Limani, Adem LU and Malman, Bartosz LU (2021) In International Mathematics Research Notices 2021(23). p.17695-17707
Abstract
For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator Tg is unchanged if the symbol g is perturbed to g+h by an analytic function h inducing a quasi-nilpotent operator Th⁠, that is, spectrum of Th equals {0}⁠. We also show that any Tg operator that can be approximated in the operator norm by an operator Th with bounded symbol h is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function g∈BMOA to be in the BMOA norm closure of H∞⁠. This condition turns out to be equivalent to quasi-nilpotency of the operator Tg on the Hardy spaces. This raises the question whether similar statement is true in the context of... (More)
For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator Tg is unchanged if the symbol g is perturbed to g+h by an analytic function h inducing a quasi-nilpotent operator Th⁠, that is, spectrum of Th equals {0}⁠. We also show that any Tg operator that can be approximated in the operator norm by an operator Th with bounded symbol h is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function g∈BMOA to be in the BMOA norm closure of H∞⁠. This condition turns out to be equivalent to quasi-nilpotency of the operator Tg on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of Tg operators. (Less)
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type
Contribution to journal
publication status
published
subject
in
International Mathematics Research Notices
volume
2021
issue
23
pages
17695 - 17707
publisher
Oxford University Press
external identifiers
  • scopus:85122335925
ISSN
1687-0247
DOI
10.1093/imrn/rnaa070
language
English
LU publication?
yes
id
b49f6ce8-482b-47e1-82d1-d13ef9baf484
date added to LUP
2020-11-25 00:42:57
date last changed
2023-06-29 04:17:15
@article{b49f6ce8-482b-47e1-82d1-d13ef9baf484,
  abstract     = {{For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator Tg is unchanged if the symbol g is perturbed to g+h by an analytic function h inducing a quasi-nilpotent operator Th⁠, that is, spectrum of Th equals {0}⁠. We also show that any Tg operator that can be approximated in the operator norm by an operator Th with bounded symbol h is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function g∈BMOA to be in the BMOA norm closure of H∞⁠. This condition turns out to be equivalent to quasi-nilpotency of the operator Tg on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of Tg operators.}},
  author       = {{Limani, Adem and Malman, Bartosz}},
  issn         = {{1687-0247}},
  language     = {{eng}},
  number       = {{23}},
  pages        = {{17695--17707}},
  publisher    = {{Oxford University Press}},
  series       = {{International Mathematics Research Notices}},
  title        = {{Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency}},
  url          = {{http://dx.doi.org/10.1093/imrn/rnaa070}},
  doi          = {{10.1093/imrn/rnaa070}},
  volume       = {{2021}},
  year         = {{2021}},
}