L-tauberian operators and subspaces with equi-integrable unit ball
Abstract
We introduce the class LW+ of L-tauberian operators and its dual counterpart, the class MW- of M-cotauberian operators. We obtain sequential, perturbative, and algebraic characterizations of these classes. As a consequence, every T∈ LW+(E,Y) is disjointly non-singular, and the converse implication holds if and only if every dispersed subspace of E has equi-integrable unit ball. For E=Lp, the latter property depends on p. For 1≤ p<2, equi-integrability of the unit ball is equivalent to dispersedness, and L-tauberian operators coincide with disjointly non-singular operators, while for p≥ 2 these equivalences fail. Moreover, LW+(Lp) is neither open nor stable under composition. Applying a result of Haydon, Odell, and Schlumprecht, we show that for 2<p<∞ every infinite dimensional dispersed subspace of Lp contains an infinite dimensional subspace with equi-integrable unit ball. However, we give an example showing that this is not true for p=2.
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BibTeX
@misc{caldera2026ltauberian,
title = {{$L$-tauberian operators and subspaces with equi-integrable unit ball}},
author = {Alberto Caldera and Manuel González},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32432},
eprint = {2610.32432},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32432},
}
RIS
TY - UNPB
TI - L-tauberian operators and subspaces with equi-integrable unit ball
AU - Alberto Caldera
AU - Manuel González
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32432
UR - https://arxiv.org/abs/2610.32432
AB - We introduce the class LW_+ of L-tauberian operators and its dual counterpart, the class MW_- of M-cotauberian operators. We obtain sequential, perturbative, and algebraic characterizations of these classes. As a consequence, every T∈ LW_+(E,Y) is disjointly non-singular, and the converse implication holds if and only if every dispersed subspace of E has equi-integrable unit ball. For E=L_p, the latter property depends on p. For 1≤ p<2, equi-integrability of the unit ball is equivalent to dispersedness, and L-tauberian operators coincide with disjointly non-singular operators, while for p≥ 2 these equivalences fail. Moreover, LW_+(L_p) is neither open nor stable under composition. Applying a result of Haydon, Odell, and Schlumprecht, we show that for 2<p<∞ every infinite dimensional dispersed subspace of L_p contains an infinite dimensional subspace with equi-integrable unit ball. However, we give an example showing that this is not true for p=2.
ER -