On absolutely Cesàro bounded operators
Abstract
We study p-absolutely Cesàro bounded operators, with particular emphasis on self-improvement, weighted shifts, and linear dynamics. Our first main result shows that, for 1<p<∞, every positive absolutely Cesàro bounded operator on Lp(Ω) is automatically p-absolutely Cesàro bounded. We then characterize q-absolute Cesàro boundedness of backward shifts on weighted ℓp-spaces for q≥ p and use this characterization to construct examples exhibiting a wide range of possible growth rates of the powers. In the dynamical direction, we obtain new obstructions to absolute and strong Cesàro boundedness; in particular, no strongly Cesàro bounded operator on a nonzero Banach space is chaotic or upper frequently hypercyclic. We also introduce Cesàro ratio-boundedness and compare it with absolute Cesàro boundedness. Finally, a general Baire-category principle yields a genericity theorem for the failure of p-absolute Cesàro boundedness.
Related
- Self-improvement for absolutely Cesáro bounded operators
- Integral Operators on Fractional Cesàro--Morrey Spaces over ℚp
- The Cesàro Operator, Rhaly operators, and the Toeplitz Algebra
- Cesàro-type operators on average radial integrability spaces induced by doubling weights
- Weighted Cesàro type operators on the weighted Bergman spaces in the unit ball
BibTeX
@misc{abbar2026absolutely,
title = {{On absolutely Cesàro bounded operators}},
author = {Arafat Abbar and Loris Arnold and Clément Coine},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24601},
eprint = {2609.24601},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24601},
}
RIS
TY - UNPB
TI - On absolutely Cesàro bounded operators
AU - Arafat Abbar
AU - Loris Arnold
AU - Clément Coine
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24601
UR - https://arxiv.org/abs/2609.24601
AB - We study p-absolutely Cesàro bounded operators, with particular emphasis on self-improvement, weighted shifts, and linear dynamics. Our first main result shows that, for 1<p<∞, every positive absolutely Cesàro bounded operator on L^p(Ω) is automatically p-absolutely Cesàro bounded. We then characterize q-absolute Cesàro boundedness of backward shifts on weighted ℓ^p-spaces for q≥ p and use this characterization to construct examples exhibiting a wide range of possible growth rates of the powers. In the dynamical direction, we obtain new obstructions to absolute and strong Cesàro boundedness; in particular, no strongly Cesàro bounded operator on a nonzero Banach space is chaotic or upper frequently hypercyclic. We also introduce Cesàro ratio-boundedness and compare it with absolute Cesàro boundedness. Finally, a general Baire-category principle yields a genericity theorem for the failure of p-absolute Cesàro boundedness.
ER -