Banach Spaces and their Applications

摘要本文的目的是研究与巴拿赫空间理论及其应用有关的一些问题。项目的一部分涉及一个巴拿赫空间被另一个巴拿赫空间扩展的理论。例如，计划研究那些Banach空间的分类问题，使得每个最小扩展都是平凡的。这个问题可以重新表述为近似理论中的问题。本文还计划研究在Banach空间非线性理论问题中的应用，涉及两个Banach空间之间或它们的单位球之间一致同态的存在性问题。在一个稍微不同的方向上，提议者计划继续他正在进行的关于Rademacher-bounded算子族的研究，并将其应用于扇区算子和半群。可拓理论可以用下列术语来考虑。假设我们有一个三维或多维的中心对称实体，我们只允许在某些方向上计算它的截面，在垂直方向上计算物体投射的阴影。这给了我们一些关于固体的信息，但不允许完全重建。研究扩展的目的是在一定的附加条件下获得关于身体的更完整的信息。在数学分析和应用中，许多重要的问题，虽然没有以这种方式正式表示，但可以用扩展的形式来可视化。扇区算子在演化型偏微分方程的基本理论中占有重要地位。这类方程在物理应用中非常重要。典型地，扇区算子是作用于某个适当的函数空间的微分算子。通过理解扇区算子和它们所产生的半群的性质，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.