Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces
Abstract
For every 0<p<1, we establish maximal spaceability of the bounded operators on ℓp which fail to be (r,s)-absolutely summing for all 1≤ s≤ r<∞, using the coordinate-disjointization construction and Maddox's theorem. We give detailed isometric estimates and prove a corresponding maximal spaceability theorem inside the compact operators. The compact result is based on an explicit diagonal multiplier and a single weakly $1$-summable sequence witnessing the simultaneous failure of all admissible absolute summability conditions. A diagonal characterization, a logarithmic compact example, and extensions to operators with values in quasi-Banach spaces containing a copy of ℓp are also obtained. The constructed subspaces are isomorphic to ℓp and have maximal algebraic dimension whenever the ambient operator spaces have dimension c.
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BibTeX
@misc{araujo2026maximal,
title = {{Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces}},
author = {Daniel Araújo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33999},
eprint = {2610.33999},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33999},
}
RIS
TY - UNPB
TI - Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces
AU - Daniel Araújo
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.33999
UR - https://arxiv.org/abs/2610.33999
AB - For every 0<p<1, we establish maximal spaceability of the bounded operators on ℓ_p which fail to be (r,s)-absolutely summing for all 1≤ s≤ r<∞, using the coordinate-disjointization construction and Maddox's theorem. We give detailed isometric estimates and prove a corresponding maximal spaceability theorem inside the compact operators. The compact result is based on an explicit diagonal multiplier and a single weakly $1$-summable sequence witnessing the simultaneous failure of all admissible absolute summability conditions. A diagonal characterization, a logarithmic compact example, and extensions to operators with values in quasi-Banach spaces containing a copy of ℓ_p are also obtained. The constructed subspaces are isomorphic to ℓ_p and have maximal algebraic dimension whenever the ambient operator spaces have dimension c.
ER -