Fixed Point Properties for c₀-like Spaces
Veysel Nezir · D-Scholarship@Pitt (University of Pittsburgh) · 2012
In 1981, Maurey proved that every weakly compact, convex subset C of c₀ is such that every nonexpansive (n.e.) mapping T:C→C has a fixed point; i.e., C has the fixed point property (FPP). Dowling, Lennard, and Turett proved the converse of Maurey's result by showing each closed bounded convex non-weakly compact subset C of c₀ fails FPP for n.e. mappings. However, in general the mapping failing to have a fixed point is not affine. In Chapter 2 and Chapter 3, we prove that for certain classes of closed bounded convex non-weakly compact subsets C of c₀, there exists an affine nonexpansive mapping T:C→C that fails to have a fixed point. Our result depends on our main theorem: if a Banach space contains an asymptotically isometric (a.i.) c₀-summing basic sequence (xᵢ)i∈ℕ, then the closed convex hull of the sequence fails the FPP for affine nonexpansive mappings. In fact, in Chapter 3, we show that very large classes of c₀-summing basic sequences turn out to be L-scaled a.i. c₀-summing basic sequences. In Chapter 4, we work on Lorentz-Marcinkiewicz spaces and explore the FPP for lw,∞⁰ spaces. Using Borwein and Sims' technique we prove for certain classes of weight sequence w that X := lw,∞⁰ has the weak fixed point property (w-FPP) by using the Riesz angle concept. Furthermore, we find a formula for the Riesz angle of X for any weight sequence. Next, we show that X has the w-FPP for any w, but fails the FPP for n.e. mappings. In Chapter 5, we show that any closed non-reflexive vector subspace Y of lw,∞⁰ contains an isomorphic copy of c₀ and so Y fails the FPP for strongly asymptotically nonexpansive maps. Also, we show that l¹ cannot be renormed to have the FPP for semi-strongly asymptotically nonexpansive maps, and that c₀ cannot be renormed to have the FPP for strongly asymptotically nonexpansive maps. Finally, we show that reflexivity for Banach lattices is equivalent to the FPP for affine semi-strongly asymptotically nonexpansive mappings.