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，Bourgain和Delbaen设计了一种革命性的新方法来构建脚本L空间。当时，它引起了很多关注，但在接下来的几十年里，它可能在某种程度上逐渐淡出了人们的视线，直到大约 15 年前，阿吉罗斯和海顿成功地将它与遗传性不可分解的巴拿赫空间的构造融合在一起，这最初是由高尔斯和Maurey，产生一个巴纳赫空间，其中每个有界运算符都是紧致运算符和恒等式标量倍数之和。这解决了所谓的“标量加紧”问题，这是当时 Banach 空间理论中最突出的开放问题。因此，研究通过 Bourgain-Delbaen 构造的 Banach 空间上的有界算子的时机已经成熟。这是 Acuaviva 博士项目的目标，自然问题包括：何时通过 Bourgain-Delbaen 方法构造巴纳赫空间与其同构。笛卡尔平方什么时候与其超平面同构？（阿吉罗斯和海顿构建的空间不符合这两个非常“自然”的属性。）