Geometry of Banach spaces and their spaces of operators

该研究项目包括Banach空间几何和算子理论中的广泛一类问题。巴拿赫空间，特别是函数空间和序列空间，是直接或间接用于科学领域如工程和物理的重要工具。希尔伯特空间是一种特殊类型的巴拿赫空间，以及作用在这种空间上的算子，提供了一个理论框架，可用于研究量子力学和计算机科学中的问题。因此，研究Banach空间及其算子空间这一相当抽象的概念具有潜在的实用价值。该项目的重点是研究旧问题，以及从理论的最新发展中出现的问题。虽然它是制定在巴拿赫空间理论，许多问题的研究涉及到其他领域的数学，如描述集理论和算子理论。这些问题的解决方案将需要组合学、集合论和拓扑学等领域的技术相结合。该项目围绕着各种问题的调查。重点是基本序列的局部和渐近行为，这实际上可能对某些空间上的算子性质产生深远的影响。研究的一些概念是扩散模型，有限块表示性和脚本L-无限空间。如前所述，主要目的之一是推导运算符的属性。有两种主要的方法可以在这种情况下使用。第一个是构造具有遗传非均匀渐近结构的空间。该方法以前由主要研究者和S。Argyros构造了具有不变子空间性质的可分自反Banach空间的第一个例子。关于可分希尔伯特空间是否也具有这种性质的问题，是算子理论的中心问题之一，实际上是数学的中心问题。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空间的进一步例子，例如一致凸空间，具有不变子空间性质。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构造了第一个已知的具有标量加紧性质的Banach空间。这意味着这个空间上的每个有界线性算子都是单位元的标量倍的紧扰动。这种方法起源于J. Bourgain和F.德尔班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特征化。

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.