Stability of the Fixed Point Property of Hilbert Spaces

We prove that any Banach space X whose Banach-Mazur distance to a Hilbert space is less than √ 5+ √ 13 2 has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of a Banach space X . A mapping T : C → C is said to be nonexpansive if ‖T x− T y‖ ≤ ‖x− y‖ for any x, y ∈ C. A nonempty weakly compact convex set C is said to have the fixed point property if every nonexpansive T : C → C has a fixed point. X is said to have the fixed point property if every nonempty weakly compact convex subset C of X has the fixed point property. Let C be a nonempty weakly compact convex subset of X and T : C → C be nonexpansive. A closed convex nonempty subset K of C is said to be minimal for T if T (K) ⊆ K and for any nonempty closed convex subset K ′ of K, T (K ′) ⊆ K ′ implies K ′ = K. Since C is weakly compact, C has a minimal subset. Hence we can assume that C is minimal for T . Recall that a sequence {xn} in C is called an approximate fixed point sequence (afps in short) for T if lim n→∞ |xn − T xn| = 0. It is known that if T is a nonexpansive mapping on a bounded convex set, then T has an afps. Karlovitz [Ka] proved the following theorem. Theorem 1. Let (K, | · |) be a minimal weakly compact convex set for a nonexpansive mapping T . For any apfs {xn} of T and any y ∈ K, lim n→∞ |y − xn| = diam(K). Using Theorem 1, one can easily prove that the `2 with the norm ‖x‖ = max{‖x‖2, 2‖x‖∞} has the fixed point property. In [A], Alspach showed that L1 does not have the fixed point property. In [M], Maurey introduced the ultraproduct technique and Received by the editors January 28, 1997 and, in revised form, February 16, 1998. 1991 Mathematics Subject Classification. Primary 47H09, 47H10. The work was done while the author was visiting the University of Texas at Austin. The author wishes to thank V. Mascioni, E. Odell and H. Rosenthal for their hospitality, particularly to V. Mascioni and E. Odell for their valuable discussion. c ©1999 American Mathematical Society