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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：
11640172
负责人：
KATO Mikio
金额：
$ 2.24万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640172/
关键词：
von Neumann-Jordan constant James constant uniform normal structure Clarkson-type inequality Rademacher type-cotype absolute norm direct sum of Banach spaces Geometry of Banach spaces convex function uniformly convex space Jordan-von N

项目摘要

Geometric properties of Banach spaces, as well as their related norm inequalities, are investigated from an operator theoretical point of view. This approach enables us to apply interpolation techniques to research on the Banach space geometry. Major results are as follows.1. On Clarkson-type inequalities and Rademacher type-cotype :(1) A sequence of multi-dimensional Clarkson-type inequalities, generalized Clarkson, random Clarkson inequalities and thier variants, are characterized in terms of Rademacher type and cotype. These inequalities are equivalent in a Lebesgue-Bochner space.(2) We extended q-uniform convexity and p-uniform smoothness inequalities in parameters and in number of elements.2. On geometric constants of Banach spaces :(1) We clarified some relations between the von Neumann-Jordan (NJ-) constant and James constant, resp., the normal structure coefficient. In particular, if X has the NJ-constant less than 5/4, then X, as well as the dual space X^*, has the uniform normal structure and hence the fixed point property. An answer was also presented to the question of Gao and Lau concerning James constant.(2) We determined the NJ-and James constants of 2-dimensional Lorentz sequence spaces d (w, q).(3) The supremum of p, 1【less than or equal】p【less than or equal】, for which a subspace X of L_1 is that of L_p was determined by the NJ-constant of X.3. Out absolute norms :(1) The NJ-constant of absolute normalized norms on C^2 was determined and estimated. Also we showed that all these norms are uniformly non-square except the l_1-and l_∞-norms.(2) The correspondence between the absolute normalized norms on C^2 and the convex functions ψon [0, 1] with certain conditions was extended to the n-dimensional case.(3) By using absolute norms we introduced the notion of ψ-direct sum of a finite number of Banach spaces, and extended the well-known facts for the l_p-sums of Banach spaces concerning strict resp. uniform convexity.

从算子理论的角度研究了Banach空间的几何性质及其相关的范数不等式。这种方法使我们能够将插值技术应用于Banach空间几何的研究。主要研究结果如下：1.关于Clarkson型不等式和Rademacher型-余型：（1）利用Rademacher型和余型刻画了一列多维Clarkson型不等式，广义克拉克森不等式，随机克拉克森不等式及其变式.这些不等式在Lebesgue-Bochner空间中是等价的。(2)推广了q-一致凸性和p-一致光滑性不等式的参数和元素个数.关于Banach空间的几何常数：（1）分别阐明了von Neumann-Jordan（NJ-）常数与James常数之间的一些关系，正常结构系数。特别地，如果X的NJ常数小于5/4，则X以及对偶空间X^* 具有一致正规结构，因此具有不动点性质。对高、刘关于詹姆斯常数的问题也作了回答。(2)我们确定了二维Lorentz序列空间d（w，q）的NJ常数和James常数. (3)p，1[小于或等于]p[小于或等于]的上确界，其中L_1的子空间X是L_p的子空间，由X的NJ常数决定。绝对范数：（1）确定并估计了C^2上绝对归一化范数的NJ常数。证明了除l_1-模和l_∞-模外，所有这些模都是一致非方的. (2)将C^2上的绝对正规范数与[0，1]上的凸函数在一定条件下的对应关系推广到n维情形. (3)本文利用绝对范数引入了有限个Banach空间的l_p-直和的概念，推广了Banach空间的l_p-和关于严格相应的著名事实。一致凸性