Characterizations of weakly compact sets and new fixed point free maps in c₀

Abstract

Abstract. We give a basic sequence characterization of relative weak compactness in c 0 and we construct new examples of closed, bounded, convex subsets of c 0 failing the fixed point property for nonexpansive self-maps. Combining these results, we derive the following characterization of weak compactness for closed, bounded, convex subsets C of c 0 : such a C is weakly compact if and only if all of its closed, convex, nonempty subsets have the fixed point property for nonexpansive mappings.

Weak compactness is equivalent to the fixed point property in $c_0$

“…Partial results along these lines have been obtained in Domínguez Benavides, Japón Pineda and Prus [3] and Dowling, Lennard and Turett [4]. In both of these articles, the authors provide characterizations of the weakly compact convex subsets of c 0 in terms of the fixed point property for certain classes of mappings.…”

“…In both of these articles, the authors provide characterizations of the weakly compact convex subsets of c 0 in terms of the fixed point property for certain classes of mappings. For example, in [4], it is shown that a closed, bounded, convex subset of c 0 is weakly compact if and only if all of its nonempty, closed, convex subsets have the fixed point property for nonexpansive mappings. However, this result is not strong enough to prove the conjecture of Llorens-Fuster and Sims.…”

“…As in [4], a sequence (w n ) n∈N satisfying ( * ) is called an asymptotically isometric c 0 -basic sequence. For information concerning asymptotically isometric c 0 -basic sequences and asymptotically isometric c 0 -summing basic sequences, see [4] and the references therein.…”

“…For information concerning asymptotically isometric c 0 -basic sequences and asymptotically isometric c 0 -summing basic sequences, see [4] and the references therein. The following result (Theorem 4 in [4]) as well as certain technical details in its proof will prove crucial.…”

“…Recall from [4] that a sequence (y n ) n∈N in a Banach space X is an asymptotically isometric c 0 -summing basic sequence if there exists a null sequence (ε n ) n∈N in (0, ∞) such that…”