Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency

Adem Limani, Bartosz Malman

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Pages (from-to)17695-17707
JournalInternational Mathematics Research Notices
Volume2021
Issue number23
Early online date2020 Jul 8
DOIs
Publication statusPublished - 2021

Subject classification (UKÄ)

  • Mathematical Analysis

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