TY - JOUR
T1 - Generalized Cesàro Operators: Geometry of Spectra and Quasi-Nilpotency
AU - Limani, Adem
AU - Malman, Bartosz
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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.
U2 - 10.1093/imrn/rnaa070
DO - 10.1093/imrn/rnaa070
M3 - Article
SN - 1687-0247
VL - 2021
SP - 17695
EP - 17707
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 23
ER -