非自反Banach空间中的投影格式与不动点性质

中文摘要 AI 辅助

我们引入Banach空间上投影格式的概念，并建立研究非扩张映射不动点性质（FPP）的抽象框架。在该框架内，我们得到了一类广泛非自反Banach空间满足FPP的通用判据。特别地，每个一致凸Banach空间都可等距嵌入到一个具有FPP的非自反Banach空间中。该判据统一并推广了此前关于具有Schauder基的非自反Banach空间的若干已知不动点结果。作为通用结果的应用，我们刻画了变指数Lebesgue空间的FPP。更准确地说，我们证明：若$(\boldsymbol{\Omega},\boldsymbol{\Sigma},\boldsymbol{\mu})$是$\sigma$-有限测度空间，且$\boldsymbol{p}:\boldsymbol{\Omega}\to[1,\infty]$是可测函数，则$\boldsymbol{L}^{\boldsymbol{p}(\cdot)}(\boldsymbol{\Omega})$具有FPP当且仅当它不包含$\boldsymbol{\ell}_1$的等距拷贝。因此，如同经典Lebesgue空间框架，$\boldsymbol{\ell}_1$的等距拷贝的存在性完全刻画了变指数Lebesgue空间中FPP的失效。该结果给出了一类具有不动点性质的非自反Banach空间，其中包含无穷多个$\boldsymbol{L}^p$空间（$1<\boldsymbol{p}<\infty$）的等距拷贝。尽管已知具有FPP的非自反序列空间，但据我们所知，这些变指数Lebesgue空间是首批具有FPP的非自反Banach函数格的例子。

英文摘要

We introduce the notion of a projection scheme on a Banach space and develop an abstract framework for studying the fixed point property (FPP) for nonexpansive mappings. Within this framework, we obtain a general criterion ensuring the FPP for a broad class of nonreflexive Banach spaces. In particular, every uniformly convex Banach space admits an isometric embedding into a nonreflexive Banach space having the FPP. This criterion unifies and extends several previously known fixed point results for nonreflexive Banach spaces with Schauder bases. As an application of our general result, we characterize the FPP for variable Lebesgue spaces. More precisely, we prove that if $(Ω,Σ,μ)$ is a $σ$-finite measure space and $p:Ω\to [1,\infty]$ is a measurable function, then $L^{p(\cdot)}(Ω)$ has the FPP if and only if it does not contain an isometric copy of $\ell_1$. Consequently, as in the classical Lebesgue space framework, the presence of an isometric copy of $\ell_1$ completely characterizes the failure of the FPP in variable Lebesgue spaces. This result yields a class of nonreflexive Banach spaces with the fixed point property that contains isometric copies of infinitely many $L^p$-spaces, $1<p<\infty$. Although nonreflexive sequence spaces with the FPP are known, to the best of our knowledge, these variable Lebesgue spaces provide the first examples of nonreflexive Banach function lattices with the FPP.

发表机构

• Universidad de Sevilla（塞维利亚大学）
• University of Memphis（孟菲斯大学）

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