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，该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． -type 不等式和 Rademacher type-cotype ：(1) 一系列多维 Clarkson 型不等式、广义 Clarkson、随机 Clarkson 不等式及其变体，用以下形式表征Rademacher 型和余型。这些不等式在 Lebesgue-Bochner 空间中是等价的。(2) 我们在参数和元素数量上扩展了 q-均匀凸性和 p-均匀平滑不等式。2. 关于 Banach 空间的几何常数：(1) ）我们澄清了冯·诺依曼-乔丹（NJ-）常数和詹姆斯常数之间的一些关系，特别是，如果 X 具有： NJ 常数小于 5/4，则 X 以及对偶空间 X^* 具有一致的正规结构，因此也具有不动点性质。 (2) 确定了二维洛伦兹序列空间 d(w, q) 的 NJ-和 James 常数。 (3) p, 1【小于等于】p【小于等于】的上界，其中一个子空间L_1 的 X 是 L_p 的 X.3 的绝对范数：(1) C^2 上的绝对归一化范数的 NJ 常数是确定和估计的。除l_1-和l_∞-范数外，均一致非方。 (2) 将C^2上的绝对归一化范数与一定条件下的凸函数ψon [0, 1]之间的对应关系推广为(3) 通过使用绝对范数，我们引入了有限数量的 Banach 空间的 ψ-直和的概念，并扩展了 Banach 空间的 l_p-和有关严格一致的众所周知的事实。凸性。