The main goal of this article is to show that for every (reflexive) infinite-dimensional Banach space X there exists a reflexive Banach space Y and \(T, R \in \mathcal {L}(X,Y)\) such that R is a rank-one operator, \(\Vert T+R\Vert >\Vert T\Vert \) but \(T+R\) does not attain its norm. This answers a question posed by Dantas and the first two authors. Furthermore, motivated by the parallelism exhibited in the literature between the V-property introduced by Khatskevich, Ostrovskii and Shulman and the weak maximizing property introduced by Aron, García, Pellegrino and Teixeira, we also study the relationship between these two properties and norm-attaining perturbations of operators.

摘要

本文的主要目标是证明对于每个（自反）无限维 Banach 空间X都存在一个自反 Banach 空间Y和\(T, R \in \mathcal {L}(X,Y)\)使得R是一个秩一运算符，\(\Vert T+R\Vert >\Vert T\Vert \)但\(T+R\)未达到其范数。这回答了丹塔斯和前两位作者提出的问题。此外，受文献中Khatskevich、Ostrovskii 和 Shulman 引入的V性质与 Aron、García、Pellegrino 和 Teixeira 引入的弱最大化性质之间的平行性的启发，我们还研究了这两个性质与规范达到之间的关系操作员的扰动。