A p-summability approach to the weak maximizing property
Abstract
Motivated by the weak maximizing property (WMP), we investigate p-summability conditions on maximizing sequences of bounded linear operators. Since the naive p-summability formulation is independent of p and collapses to norm attainment for all operators, we introduce the weakly p-singular maximizing property (SMPp), based on maximizing sequences with no weakly p-summable subsequence. We completely characterize when the pairs (ℓp,ℓq) and (ℓp,c0) have the SMPr, revealing sharp contrasts with the WMP. We also characterize the Schur property of order p via the WMP and the SMPr. Under relative weak p-precompactness and suitable Dunford-Pettis-type assumptions, we characterize universal norm attainment for operators from X to Y in terms of either the SMPq for adjoint operators or the weak*-to-weak* maximizing property, and derive corresponding duality consequences for the SMPp. Finally, we introduce p-convergent perturbation properties for operators and their adjoints, characterize them for classical sequence spaces, and show that the p-convergent perturbation property is strictly weaker than the SMPp.
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BibTeX
@misc{ardila2026psummability,
title = {{A $p$-summability approach to the weak maximizing property}},
author = {Oscar Iván Blanco Ardila and Vinícius C. C. Miranda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03988},
eprint = {2609.03988},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03988},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A p-summability approach to the weak maximizing property
AU - Oscar Iván Blanco Ardila
AU - Vinícius C. C. Miranda
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03988
UR - https://arxiv.org/abs/2609.03988
AB - Motivated by the weak maximizing property (WMP), we investigate p-summability conditions on maximizing sequences of bounded linear operators. Since the naive p-summability formulation is independent of p and collapses to norm attainment for all operators, we introduce the weakly p-singular maximizing property (SMP_p), based on maximizing sequences with no weakly p-summable subsequence. We completely characterize when the pairs (ℓ_p,ℓ_q) and (ℓ_p,c_0) have the SMP_r, revealing sharp contrasts with the WMP. We also characterize the Schur property of order p via the WMP and the SMP_r. Under relative weak p-precompactness and suitable Dunford-Pettis-type assumptions, we characterize universal norm attainment for operators from X to Y in terms of either the SMP_q for adjoint operators or the weak^*-to-weak^* maximizing property, and derive corresponding duality consequences for the SMP_p. Finally, we introduce p-convergent perturbation properties for operators and their adjoints, characterize them for classical sequence spaces, and show that the p-convergent perturbation property is strictly weaker than the SMP_p.
ER -