Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency
(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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https://lup.lub.lu.se/record/b49f6ce8-482b-47e1-82d1-d13ef9baf484
- author
- Limani, Adem LU and Malman, Bartosz LU
- organization
- publishing date
- 2021
- 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}}, }