Operator theory on Banach spaces

1980年，Bourain和Delbaen设计了一种革命性的构建剧本L空间的新方法。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淡出了人们的视线，直到大约15年前，Argyros和Haydon成功地将它与遗传不可分解Banach空间的构造融合在一起，最初是由于Gowers和Maurey，产生了一个Banach空间，其中每个有界算子都是紧算子和恒等式的标量倍数的和。这解决了所谓的“标量+紧致”问题，这是当时Banach空间理论中最重要的公开问题。因此，更深入地研究通过Bourain-Delbaen方法(或其某种变体)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：通过Bourain-Delbaen方法构造的Banach空间何时同构于它的笛卡尔正方形？它何时同构于它的超平面？(Argyros和Haydon构造的空间都没有通过其他非常“自然”的性质。)

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.)