Banach Spaces and their Applications

在这个提议中，目的是研究与Banach空间理论及其应用有关的一些问题。该项目的一部分涉及一个Banach空间对另一个Banach空间的扩张理论。例如，计划研究如何对这些Banach空间进行分类，使得每个极小扩张都是平凡的。这个问题可以表述为近似理论中的一个问题。它还计划研究在Banach空间的非线性理论中的应用，这些问题涉及两个Banach空间之间或它们的单位球之间存在一致同胚的问题。在一个有点不同的方向上，提出者计划继续他正在进行的关于Rademacher有界算子族的研究，并将其应用于扇形算子和半群。扩张理论可以用以下术语来考虑。假设我们有一个三维或更多维的中心对称实体，我们只被允许通过某些方向上的切片和物体在垂直方向上投射的阴影来计算它的横截面。这给了我们一些关于实体的信息，但不允许完全重建。研究延伸的想法是在某些附加条件下获得关于身体的更完整的信息。数学分析和应用中的许多重要问题，虽然不是以这种方式正式表达的，但可以用扩展的方式来可视化。扇形算子在发展型偏微分方程组的基本理论中具有重要意义，在物理应用中也是非常重要的。通常，扇形运算符是作用于某些适当的函数空间的微分运算符。通过了解扇形算子及其生成的半群的性质，可以更好地理解某些偏微分方程解的性质。

AbstractKaltonIn this proposal, the aim is to study a number of problemsrelated to Banach space theory and its applications. Partof the project deals with the theory of extensions of oneBanach space by another. For example it is planned to studythe problem of classifying those Banach spaces such thatevery minimal extension is trivial. This problem can bereformulated as a problem in approximation theory. It isalso planned to study applications to problems in thenonlinear theory of Banach spaces, related to problems ofexistence of uniform homeomorphisms between two Banachspaces or between their unit balls. In a somewhat differentdirection the proposer plans to continue his ongoingresearch on Rademacher-bounded families of operators withapplications to sectorial operators and semigroups.The theory of extensions can be considered in the followingterms. Suppose we are given a centrally symmetric solid inthree or more dimensions and we are allowed only to computeits cross-section by a slice in some directions and theshadow cast by the body in the perpendicular directions.This gives us some information about the solid, but does notpermit complete reconstruction. The idea of studyingextensions is to obtain more complete information about thebody under certain additional conditions. Many questions ofimportance in mathematical analysis and applications,although not formally expressed in this way, can bevisualized in terms of extensions. Sectorial operators areof importance in the basic theory of partial differentialequations of evolution type; such equations are veryimportant in physical applications. Typically a sectorialoperator is a differential operator acting on some suitablespace of functions. By understanding the properties ofsectorial operators and the semigroups they generate onegains a better understanding of the behavior of solutions ofcertain partial differential equations.