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Characterizing the Daugavet property by squares of rank-one operators

  • Johann Langemets
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Abstract

We prove that, for real Banach spaces of dimension greater than one, each of the identities |Id+T2|=1+|T2| and |Id−T2|=1+|T2|, required for every rank-one operator T, characterizes the Daugavet property, thereby answering a question posed by Kadets, Martín, and Merí (2007). Consequently, every extremely non-complex Banach space of dimension greater than one has the Daugavet property, answering a question of Martín and Merí (2011). We also compare pointwise versions of the square Daugavet properties with Daugavet points, Δ-points, and ∇-points.

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Cite

BibTeXRIS
BibTeX
@misc{langemets2026characterizing,
  title = {{Characterizing the Daugavet property by squares of rank-one operators}},
  author = {Johann Langemets},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27693},
  eprint = {2609.27693},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27693},
}
RIS
TY  - UNPB
TI  - Characterizing the Daugavet property by squares of rank-one operators
AU  - Johann Langemets
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27693
UR  - https://arxiv.org/abs/2609.27693
AB  - We prove that, for real Banach spaces of dimension greater than one, each of the identities |Id+T^2|=1+|T^2| and |Id−T^2|=1+|T^2|, required for every rank-one operator T, characterizes the Daugavet property, thereby answering a question posed by Kadets, Martín, and Merí (2007). Consequently, every extremely non-complex Banach space of dimension greater than one has the Daugavet property, answering a question of Martín and Merí (2011). We also compare pointwise versions of the square Daugavet properties with Daugavet points, Δ-points, and ∇-points.
ER  -