On the fixed point property in direct sums of Banach spaces with strictly monotone norms

TL;DR

This paper proves that the weak fixed point property is preserved in certain direct sums of Banach spaces with specific properties, under strictly monotone norms, expanding understanding of fixed point theory in Banach spaces.

Contribution

It establishes that the weak fixed point property holds for direct sums of Banach spaces with particular properties when equipped with strictly monotone norms, including cases with 1-unconditional bases.

Findings

Weak fixed point property preserved in direct sums with strictly monotone norms

Applicable to spaces with the weak Banach-Saks property and property (P)

Results extend fixed point theory in Banach space sums

Abstract

It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.