发表机构
Universidade Estadual da Paraíba; Universidade Federal de Campina Grande; Universidade Federal da Paraíba; Universidade Federal Rural do Semi-Árido（帕拉伊巴州立大学; 坎皮纳格兰德联邦大学; 帕拉伊巴联邦大学; 半干旱地区联邦农村大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究弱紧生成Banach空间的Lotz性质，提出Calkin代数判据，结合提取定理构造出可分不可分解的Lotz空间，并证明不可分WCG空间均非Lotz。
AI 中文摘要
我们研究每个无限维弱紧生成（WCG）Banach空间是否都不具有Lotz性质。在不可分情形下答案是肯定的：WCG空间的一个经典分解定理提供了Schauder分解，且一个显式的对角$C_0$-半群具有无界生成元。无限制的断言是假的。我们的主要抽象结果是Calkin代数判据：若$X$不具有有界紧逼近性质，且对某个固定的$m$，$\Cal(X)=\Lop(X)/\Kop(X)$中每个不可逆元素的$m$次幂为零，则$X$具有Lotz性质。特别地，这适用于每个算子都是标量加严格奇异算子且每个严格奇异算子的某个固定幂是紧算子的情形。将该判据与Maurey--Pisier--Szankowski提取定理以及Argyros和Motakis的构造相结合，在两个标量域上得到可分不可分解的Lotz空间：自反的例子，以及在复数域上包含无无限维自反子空间的例子。此外，这里考虑的Argyros--Motakis环境空间是Lotz饱和的，尽管它们本身是非Lotz的。因此，每个不可分WCG空间都是非Lotz的，而可分WCG类同时包含Lotz和非Lotz空间。
英文摘要
We study whether every infinite-dimensional weakly compactly generated (WCG) Banach space fails the Lotz property. The answer is affirmative in the nonseparable case: a classical decomposition theorem for WCG spaces provides a Schauder decomposition, and an explicit diagonal $C_0$-semigroup has an unbounded generator. The unrestricted assertion is false. Our main abstract result is a Calkin-algebra criterion: if $X$ fails the bounded compact approximation property and, for some fixed $m$, every noninvertible element of $\Cal(X)=\Lop(X)/\Kop(X)$ has $m$th power zero, then $X$ has the Lotz property. In particular, this applies when every operator is scalar-plus-strictly-singular and one fixed power of every strictly singular operator is compact. Combining this criterion with the Maurey--Pisier--Szankowski extraction theorem and constructions of Argyros and Motakis yields separable hereditarily indecomposable Lotz spaces over both scalar fields: reflexive examples, and, over the complex field, an example containing no infinite-dimensional reflexive subspace. Moreover, the ambient Argyros--Motakis spaces considered here are Lotz-saturated, although they themselves are non-Lotz. Thus every nonseparable WCG space is non-Lotz, while the separable WCG class contains both Lotz and non-Lotz spaces.
