OPERATOR THEORETICAL RESEARCH ON GEOMETRY OF BANACH SPACES AND APPLICATIONS

Banach空间几何算子理论研究及应用

基本信息

批准号：
09640203
负责人：
KATO Mikio
金额：
$ 1.92万
依托单位：
KYUSHU INSTITUTE OF TECHNOLOGY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640203/
关键词：
Von Neumann-Jordan constant Clarkson-type inequality Rademacher type, cotype Banach space Geometry of Banach spaces normal structure matrix operator interpolation Littlewood matrix Lebesgue-Bochner space

项目摘要

Geometrical properties of Banach spaces as well as related norm inequalities are investigated from an operator theoretical point of view. This approach allows a unifying treatment of them and also enables us to apply interpolation techniques in research of the Banach space geometry. Not only they have their own beauty and significance, but also they provide essential or useful notions and tools in various branches of analysis including applicable areas, which indicates the fundamental importance of this subject.Major results are as follows.1. On Clarkson-type inequalities :(1) A sequence of Clarkson-type inequalities are characterized in the general Banach space setting by the notions of Rademacher type and cotype which are of great importance in "Probability in Banach Spaces".(2) It is shown how Clarkson's and related inequalities are inherited by the Lebesgue-Bochner space L_r (X) from a given Banach space X, by which most of these inequalities known for various spaces are derived unifyingly.2. On the von Neumann-Jordan (NJ-) constant of a Banach space a sort of modulus of skewness of the norm :(1) A systematic way to calculate NJ-constant is given, by which all the previous results for various spaces and some new ones as well are obtained.(2) A sequence of informations NJ-constant gives is presented, especially about type and cotype, uniform convexity, uniform non-squareness, super-reflexivity, normal structure and fixed point property, etc.3. Several geometrical properties are charcterized unifyingly via behavior of operator norms of 1 matrices between finite dimensional X-valued l_p-spaces. In particular, a sequence of characterizations of uniformly non-square spaces is given, some of which are similar to the well-known one for uniform convexity.

从算子理论的角度研究了 Banach 空间的几何性质以及相关的范数不等式。这种方法可以对它们进行统一处理，并使我们能够在巴拿赫空间几何的研究中应用插值技术。它们不仅有其自身的美感和意义，而且为包括应用领域在内的各个分析分支提供了必要的或有用的概念和工具，这表明了本学科的根本重要性。主要结果如下： 1．关于克拉克森型不等式：（1）克拉克森型不等式序列在一般巴拿赫空间设置中通过拉德马赫型和余型的概念来表征，这在“巴拿赫空间中的概率”中非常重要。（2）显示了克拉克森和相关的不等式是如何由 Lebesgue-Bochner 空间 L_r (X) 从给定的 Banach 空间 X 继承的，其中大多数不等式已知各种空间统一导出。2．关于 Banach 空间的 von Neumann-Jordan (NJ-) 常数，一种范数的偏度模：(1) 给出了计算 NJ-常数的系统方法，通过该方法，给出了各种空间和某些空间的所有先前结果(2)给出了NJ-constant给出的一系列信息，特别是关于类型和共型、均匀凸性、均匀非方形性、超自反性、正常结构和固定的信息。点属性等 3．一些几何性质通过有限维 X 值 l_p 空间之间 1 矩阵的算子范数的行为来统一表征。特别是，给出了一致非正方形空间的一系列表征，其中一些类似于众所周知的一致凸性。