Geometry of Banach spaces and their spaces of operators

该研究项目包括巴拿赫空间几何和算子理论中的一系列问题。巴拿赫空间，特别是函数空间和序列空间，是直接或间接应用于工程和物理等科学领域的重要工具。希尔伯特空间是巴拿赫空间的一种特殊类型，它和作用于这种空间上的算子为研究量子力学和计算机科学中的问题提供了一个理论框架。因此，研究相当抽象的巴拿赫空间及其算子空间具有潜在的实用价值。这个项目的中心是对老问题的研究，以及从理论的最新发展中出现的问题。虽然它是根据巴拿赫空间理论制定的，但许多研究的问题与其他数学领域有关，如描述性集合论和算子论。这些问题的解决需要结合组合学、集合论和拓扑学等领域的技术。这个项目围绕着对各种问题的调查展开。重点讨论了基本序列的局部性和渐近性，这实际上对某些空间上算子的性质具有深远的意义。研究了扩展模型、有限块可表示性和无限空间的概念。如前所述，主要目的之一是推导算子的性质。在此设置中可以使用两种主要方法。第一类是具有遗传异质渐近结构的空间构造。该方法已被主要研究者和S. Argyros用于构造具有不变子空间性质的可分离自反Banach空间的第一个例子。关于可分希尔伯特空间是否也具有这种性质的问题，是算符理论的核心问题之一，实际上是数学的核心问题之一。对该方法的进一步研究可能会得到更多类似于希尔伯特空间的例子，例如具有不变子空间性质的一致凸空间。第二种方法与脚本- l -无穷空间的研究有关。该方法是基于S. Argyros和R. Haydon开发的构造已知的第一个具有标量加紧性的巴拿赫空间的方法。这意味着这个空间上的每一个有界线性算子都是单位元的标量倍的紧摄动。这种方法源于J.布尔甘和F.德尔巴恩的构造。对这种方法的进一步研究可能会导致具有上述性质的空间的表征。

The research project is comprised of a broad class of problems in the geometry of Banach spaces and operator theory. Banach spaces, in particular function spaces and sequence spaces, are an important tool used either directly or indirectly in scientific fields such as engineering and physics. Hilbert spaces, which are a special type of Banach spaces, and operators acting on such spaces, provide a theoretical framework that can be used to study problems in quantum mechanics and computer science. Therefore, there exists potential practical value in studying the fairly abstract notion of a Banach space and its space of operators. The project is centered around the study of old problems, as well as ones that have emerged from recent developments in the theory. Although it is formulated in terms of Banach space theory, many of the problems studied are related to other areas of mathematics, such as descriptive set theory and operator theory. The solution of those problems will require the combination of techniques from such areas as combinatorics, set theory and topology.The project revolves around the investigation of a variety of problems. Emphasis is given on the local and asymptotic behavior of basic sequences, which can in fact have far reaching implications in the properties of operators on certain spaces. Some notions that are studied are those of spreading models, finite block representability, and script-L-infinity spaces. As already mentioned, one of the main intentions is to deduce properties of operators. There are two main approaches that can be used in this setting. The first one is the construction of spaces with hereditary heterogeneous asymptotic structure. This method was previously used by the principal investigator and S. Argyros to construct the first example of a separable reflexive Banach space with the invariant subspace property. The question as to whether the separable Hilbert space has this property as well, is one of the central problems of operator theory, in fact of mathematics. Further study of this method can possibly yield further examples of spaces which resemble Hilbert spaces, for example uniformly convex spaces, with the invariant subspace property. The second approach is related to the study of script-L-infinity spaces. This approach is based on the method developed by S. Argyros and R. Haydon to construct the first known Banach space with the scalar-plus-compact property. This means that every bounded linear operator on this space is a compact perturbation of a scalar multiple of the identity. This method has its roots in a construction of J. Bourgain and F. Delbaen. Further study of this method may lead to the characterization of spaces with the aforementioned property.