Operator theory on Banach spaces

Banach空间的算子理论

• 批准号: 2886073
• 金额: --
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2886073
• 关键词: Operator; theory; Banach; spaces

In 1980, Bourgain and Delbaen devised a revolutionary new way of constructing script L spaces. At the time, it attracted much attention, but over the following decades perhaps it faded somewhat into the background until about 15 years ago, when Argyros and Haydon successfully managed to fuse it with the construction of hereditarily indecomposable Banach spaces, originally due to Gowers and Maurey, to produce a Banach space on which every bounded operator is the sum of a compact operator and a scalar multiple of the identity. This solved the so-called "scalar-plus-compact'' problem, which was the pre-eminent open problem in Banach space theory at the time. The time is therefore ripe to investigate bounded operators on Banach spaces constructed via the Bourgain-Delbaen method (or some variant thereof) in more depth. This is the aim of Acuaviva's PhD project. Natural questions include: when is a Banach space constructed via the Bourgain-Delbaen method isomorphic to its Cartesian square? When is it isomorphic to its hyperplanes? (The space constructed by Argyros and Haydon fail both of these otherwise very "natural" properties.)

1980年，布尔甘和德尔贝恩设计了一种革命性的新方法来构造字母L空间。当时，它引起了很大的关注，但在接下来的几十年里，它可能在某种程度上淡出了人们的视线，直到大约15年前，当Argyros和Haydon成功地将它与遗传不可分解巴拿赫空间的构造融合在一起，最初是由于Gowers和Maurey，产生了一个巴拿赫空间，其中每个有界算子都是紧算子和单位元的标量倍数的和。这解决了所谓的“标量+紧化”问题，这是当时巴拿赫空间理论中最突出的开放问题。因此，更深入地研究通过Bourgain-Delbaen方法（或其变体）构造的Banach空间上的有界算子的时机已经成熟。这就是Acuaviva博士项目的目标。自然的问题包括：用Bourgain-Delbaen方法构造的Banach空间何时同构于它的笛卡尔平方？什么时候它与它的超平面同构？（阿吉罗斯和海顿构建的空间没有这些非常“自然”的属性。）