Self-improvement for absolutely Cesáro bounded operators

@misc{arnold2026selfimprovement,
  title = {{Self-improvement for absolutely Cesáro bounded operators}},
  author = {Loris Arnold},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00271},
  eprint = {2610.00271},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00271},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Self-improvement for absolutely Cesáro bounded operators
AU  - Loris Arnold
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00271
UR  - https://arxiv.org/abs/2610.00271
AB  - We prove that every p-absolutely Cesàro bounded operator with constant K>1 is q-absolutely Cesàro bounded for 1≤ q<pK/(K−1). This range is optimal, even for positive weighted shifts. On Banach spaces of type s∈ (1,2], we prove that every absolutely Cesàro bounded operator is q-absolutely Cesàro bounded for every 1≤ q<s, with a logarithmic estimate at q=s. We also prove the polynomial growth bound | T^n | =O(n^1/s−ε) for some ε>0. On L^p-spaces, 1<p<∞, we extend the implication from absolute Cesàro boundedness to p-absolute Cesàro boundedness from positive to individually eventually positive operators. Finally, for every 1≤ p<∞, we construct a p-absolutely Cesàro bounded operator T on ℓ^1 such that σ(T)⊆𝔻∪{1} but limsup_n→∞|T^n(I−T)|=∞. We also observe that the construction yields a single operator which is p-absolutely Cesàro bounded for every 1≤ p<∞.
ER  -