Banach space theory and its applications

这个项目涉及一些关于巴拿赫空间理论和应用到其他数学领域的问题。项目的一部分讨论了Banach空间的可拓理论和Banach空间间线性算子的可拓存在性的相关问题。例如，一个中心问题是如何确定经典勒贝格空间的可积函数的子空间，这些子空间具有希尔伯特空间中的每个算子都可以扩展到整个空间的性质。这相当于找到验证格罗滕迪克定理的子空间，相当于找到具有希尔伯特空间的每一个扩展都是平凡的性质的空间的问题。用连续函数空间代替希尔伯特空间的类似问题也很有趣。项目的第二部分涉及巴拿赫空间的非线性理论；一个典型的开放问题是两个Lipschitz同构可分巴拿赫空间是否总是线性同构的。第三部分讨论了扇区算子的研究以及该研究在偏微分方程中的可能应用。Banach空间的扩展在有限维上具有几何解释。如果给定一个中心对称凸集，只有某些方向上的切片信息和垂直方向上的阴影信息，则无法重建原点集。然而，希望是尽可能准确地重建身体，即使这些信息是不完整的。这是一种具体的方法来看待分析中以不同形式出现的许多问题。Banach空间的非线性理论和度量空间的相关Lipschitz结构在处理大数据集的理论计算机科学中具有潜在的应用。扇区算子和半群是研究演化型偏微分方程的基本工具。

0555670 KaltonAbstractThis project deals with a number of problems concerning Banach space theory and applications to other areas of mathematics. Part of the project deals with the theory of extensions of Banach spaces and related problems about the existence of extensions of linear operators between Banach spaces. For example, one central problem concerns identifying those subspaces of the classical Lebesgue space of integrable functions that have the property that every operator into a Hilbert space can be extended to the whole space. This amounts to finding subspaces that verify Grothendieck's theorem and is equivalent to the problem of finding spaces with the property that every extension by a Hilbert space is trivial. Similar problems for spaces of continuous functions in place of Hilbert space are also of interest. A second part of the project deals with the nonlinear theory of Banach spaces; a typical open problem here is whether two Lipschitz isomorphic separable Banach spaces are always linearly isomorphic. A third part deals with the study of sectorial operators and possible applications of that study to partial differential equations.Extensions of Banach spaces have a geometrical interpretation in finite dimensions. If, given a centrally symmetric convex set, one only has information about a slice in some directions and the shadow cast in the perpendicular directions, one cannot reconstruct the origin set. However, the hope is to reconstruct the body as accurately as possible, even though this information is incomplete. This is a concrete way of looking at many questions in analysis that arise in quite different forms. The nonlinear theory of Banach spaces and the related Lipschitz structure of metric spaces has potential applications in theoretical computer science in the handling of large data sets. Sectorial operators and semigroups are a basic tool in the study of partial differential equations of evolution type.