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PreprintCC BY-NC-ND 4.0arXiv

The weak maximizing property is not inherited by subspaces of the domain

  • Alessandro Costa
  • Vinícius Miranda
  • Geivison Ribeiro
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Abstract

We construct a separable reflexive Banach lattice X and a closed sublattice E⊆ X such that every bounded linear operator from X into ℓ2 is compact, whereas there exists a positive operator from E into ℓ2 that does not attain its norm and admits a positive maximizing sequence converging weakly to a nonzero vector. This answers negatively the question posed by Dantas, Jung and Martínez-Cervantes in [4] concerning the inheritance of the weak maximizing property by closed subspaces of the domain. It also answers negatively the corresponding question for the positive weak maximizing property and closed sublattices of reflexive Banach lattices. Canonical lattice complexification yields a counterexample to the original question over the complex field as well.

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Cite

BibTeXRIS
BibTeX
@misc{costa2026weak,
  title = {{The weak maximizing property is not inherited by subspaces of the domain}},
  author = {Alessandro Costa and Vinícius Miranda and Geivison Ribeiro},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38410},
  eprint = {2609.38410},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38410},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - The weak maximizing property is not inherited by subspaces of the domain
AU  - Alessandro Costa
AU  - Vinícius Miranda
AU  - Geivison Ribeiro
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38410
UR  - https://arxiv.org/abs/2609.38410
AB  - We construct a separable reflexive Banach lattice X and a closed sublattice E⊆ X such that every bounded linear operator from X into ℓ_2 is compact, whereas there exists a positive operator from E into ℓ_2 that does not attain its norm and admits a positive maximizing sequence converging weakly to a nonzero vector. This answers negatively the question posed by Dantas, Jung and Martínez-Cervantes in [4] concerning the inheritance of the weak maximizing property by closed subspaces of the domain. It also answers negatively the corresponding question for the positive weak maximizing property and closed sublattices of reflexive Banach lattices. Canonical lattice complexification yields a counterexample to the original question over the complex field as well.
ER  -