Geometry of Banach spaces and their spaces of operators

该研究项目由Banach空间几何和算子理论中的一大类问题组成。Banach空间，特别是函数空间和序列空间，是直接或间接用于工程和物理等科学领域的重要工具。希尔伯特空间是Banach空间的一种特殊类型，它提供了一种理论框架，可用于研究量子力学和计算机科学中的问题。因此，研究相当抽象的Banach空间及其算子空间的概念具有潜在的实用价值。该项目的中心是研究老问题，以及从理论的最新发展中出现的问题。虽然它是用Banach空间理论来表述的，但所研究的许多问题都与其他数学领域有关，如描述集合论和算子理论。这些问题的解决将需要组合数学、集合论和拓扑学等领域的技术。该项目围绕着对各种问题的调查。重点给出了基本序列的局部和渐近行为，这些行为实际上可以对某些空间上的算子的性质产生深远的影响。所研究的一些概念包括扩展模型、有限块表示和脚本-L-无限空间。正如已经提到的，主要目的之一是推导运算符的性质。在此设置中可以使用两种主要方法。第一个是具有遗传异质性渐近结构的空间的构造。这个方法以前被主要研究者和S.Argyros用来构造具有不变子空间性质的可分自反Banach空间的第一个例子。关于可分Hilbert空间是否也具有这一性质的问题，实际上是算子论的核心问题之一。对这种方法的进一步研究可能会产生类似于希尔伯特空间的更多例子，例如具有不变子空间性质的一致凸空间。第二种方法是关于剧本-L-无限空间的研究。该方法基于S.Argyros和R.Haydon提出的构造第一个具有标量加紧性质的Banach空间的方法。这意味着这个空间上的每个有界线性算子都是恒等式的标量倍数的紧扰动。这种方法植根于J.Bourain和F.Delbaen的构造。对该方法的进一步研究可能导致对具有上述性质的空间的刻画。

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.