The weak maximizing property, compact perturbations and duality

@misc{garcia2026weak,
  title = {{The weak maximizing property, compact perturbations and duality}},
  author = {Domingo García and Manuel Maestre and Damián Pinasco and Ignacio Zalduendo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23198},
  eprint = {2609.23198},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23198},
}

TY  - UNPB
TI  - The weak maximizing property, compact perturbations and duality
AU  - Domingo García
AU  - Manuel Maestre
AU  - Damián Pinasco
AU  - Ignacio Zalduendo
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23198
UR  - https://arxiv.org/abs/2609.23198
AB  - We construct a one-parameter family (X_c)_0<c<1 of real reflexive Banach spaces, isomorphic to ℓ_2 and satisfying d_BM(X_c,ℓ_2)≤ c^−1, such that (X_c,X_c) fails the weak maximizing property while (X_c^*,X_c^*) has it. The failure of the WMP for X_c is witnessed by a diagonal operator which does not attain its norm but admits a non-weakly null maximizing sequence. On the dual side, a separated-block estimate yields, after dualization, a reverse Pythagorean inequality and a quantitative form of the Opial property; together with property (M), this gives the weak maximizing property for X_c^*. We also show that the compact perturbation property is self-dual for reflexive pairs. Since the weak maximizing property implies the compact perturbation property, each (X_c,X_c) has the compact perturbation property. Consequently, the compact perturbation property does not imply the weak maximizing property even arbitrarily close to Hilbert space, and the weak maximizing property is not stable under duality in the reflexive setting.
ER  -