Banach Spaces and their Applications

本文的目的是研究与Banach空间理论及其应用有关的若干问题。 该项目的一部分涉及到一个Banach空间的另一个扩展理论。 例如，计划研究对这些Banach空间进行分类的问题，使得每个最小扩展都是微不足道的。 这个问题可以重新表述为逼近论中的一个问题。 它还计划研究应用到问题的thenonlinear理论的Banach空间，有关的问题ofexistence的一致同胚之间的两个Banach空间或其单位球。 在一个稍微不同的方向，提议者计划继续他的 Rademacher有界算子族的研究及其在扇形算子和半群中的应用.扩张理论可以从以下几个方面来考虑. 假设我们有一个三维或多维的中心对称固体，我们只允许计算它在某些方向上的截面和物体在垂直方向上的投影，这给了我们一些关于该固体的信息，但不允许完全重建。 研究扩张的思想是在某些附加条件下获得关于身体的更完整的信息。 许多重要的问题， 数学分析 和应用程序，虽然没有以这种方式正式表示， bevisualized可视化in terms条款of extensions扩展. 扇形算子在发展型偏微分方程的基本理论中占有重要地位， 在物理应用中非常重要。 典型的扇形算子是作用在适当空间的函数上的微分算子。 通过理解扇形算子和它们所生成的半群的性质，可以更好地理解某些偏微分方程解的行为.

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.