Remarks on the FPP in Banach spaces with unconditional Schauder basis
Abstract
This paper brings new results on the FPP in Banach spaces with a Schauder basis. We first deal with the problem of whether there is a Banach space isomorphic to having the FPP. We show that the answer is negative if contains a pre-monotone basic sequence equivalent to the unit basis of . We then study sufficient conditions to ensure the existence of such sequences. Interesting results are obtained, including the case when has a -suppression unconditional basis and its unit ball fails the PCP. With regarding the weak-FPP, we establish two fixed-point results. First, we show that under certain conditions this property is invariant under Banach-Mazur distance one. Next, it is shown that when the basis is either -suppression unconditional or -spreading then has the weak-FPP provided that a Rosenthal's type property on block basis is verified.
Keywords
Cite
@article{arxiv.2302.11287,
title = {Remarks on the FPP in Banach spaces with unconditional Schauder basis},
author = {Cleon S. Barroso},
journal= {arXiv preprint arXiv:2302.11287},
year = {2023}
}
Comments
All comments welcome. This current version fix some mistakes in the proofs of some previous results. In addition, it is shorter and more polished. Theorem E of the previous version was excluded, because its proof contains an essential flaw and although it is possible to correct it, according to one of the referees, the result itself would not bring any significant information