Fixed point properties for a degenerate Lorentz-Marcinkiewicz space

Abstract

We construct an equivalent renorming of $\ell^1$, which turns out to produce a degenerate $\ell^1$-analog Lorentz-Marcinkiewicz space $\ell_{\delta,1}$, where the weight sequence $\delta={(\delta_n)}_{n\in\N}=(2,1,1,1,\cdots)$ is a decreasing positive sequence in $\ell^\infty\backslash c_0$, rather than in $c_0\backslash\ell^1$ (the usual Lorentz situation). Then we obtain its isometrically isomorphic predual $\ell^0_{\delta,\infty}$ and dual $\ell_{\delta,\infty}$, corresponding degenerate $c_0$-analog and $\ell^\infty$-analog Lorentz-Marcinkiewicz spaces, respectively. We prove that both spaces $\ell_{\delta,1}$ and $\ell^0_{\delta,\infty}$ enjoy the weak fixed point property (w-fpp) for nonexpansive mappings yet they fail to have the fixed point property (fpp) for nonexpansive mappings since they contain an asymptotically isometric copy of $\ell^1$ and $c_0$, respectively. In fact, we prove for both spaces that there exist nonempty, closed, bounded, and convex subsets with invariant fixed point-free affine, nonexpansive mappings on them and so they fail to have fpp for affine nonexpansive mappings. Also, we show that any nonreflexive subspace of $l_{\delta,\ii}^0$ contains an isomorphic copy of $c_0$ and so fails fpp for strongly asymptotically nonexpansive maps. Finally, we get a Goebel and Kuczumow analogy by proving that there exists an infinite dimensional subspace of $\ell_{\delta,1}$ with fpp for affine nonexpansive mappings.