Remarks on the FPP in Banach spaces with unconditional Schauder basis

TL;DR

This paper investigates the fixed point property in Banach spaces with Schauder bases, providing new conditions for the existence of fixed points and exploring the weak-FPP under various basis conditions.

Contribution

It offers new results on the fixed point property in Banach spaces with Schauder bases, including conditions for the existence of fixed points and invariance of weak-FPP.

Findings

Banach space isomorphic to c0 with the FPP cannot contain a pre-monotone basic sequence equivalent to c0 basis.

Certain unconditional bases and failure of PCP influence fixed point properties.

Weak-FPP is invariant under Banach-Mazur distance one under specific conditions.

Abstract

This paper brings new results on the FPP in Banach spaces $X$ with a Schauder basis. We first deal with the problem of whether there is a Banach space isomorphic to $\co$ having the FPP. We show that the answer is negative if $X$ contains a pre-monotone basic sequence equivalent to the unit basis of $\co$. We then study sufficient conditions to ensure the existence of such sequences. Interesting results are obtained, including the case when $X$ has a $1$-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 $1$-suppression unconditional or $1$-spreading then $X$ has the weak-FPP provided that a Rosenthal's type property on block basis is verified.