AI 中文总结
通过构造反例，否定了弱极大性性质及正弱极大性性质在定义域闭子空间（闭子格）上的继承性，并扩展至复数域。
AI 中文摘要
我们构造了一个可分自反巴拿赫格$X$和一个闭子格$E\subseteq X$，使得从$X$到$\ell_2$的每个有界线性算子都是紧算子，而存在一个从$E$到$\ell_2$的正算子不达到其范数，并且具有一个弱收敛到非零向量的正极大化序列。这否定了Dantas、Jung和Martínez-Cervantes在文献[4]中提出的关于弱极大性性质由定义域的闭子空间继承的问题。这也否定了关于正弱极大性性质和自反巴拿赫格的闭子格的相应问题。典范格复化在复数域上也给出了原始问题的一个反例。
英文摘要
We construct a separable reflexive Banach lattice $X$ and a closed sublattice $E\subseteq X$ such that every bounded linear operator from $X$ into $\ell_2$ is compact, whereas there exists a positive operator from $E$ into $\ell_2$ that does not attain its norm and admits a positive maximizing sequence converging weakly to a nonzero vector. This answers negatively the question posed by Dantas, Jung and Martínez-Cervantes in [4] concerning the inheritance of the weak maximizing property by closed subspaces of the domain. It also answers negatively the corresponding question for the positive weak maximizing property and closed sublattices of reflexive Banach lattices. Canonical lattice complexification yields a counterexample to the original question over the complex field as well.
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