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.

从操作员的理论角度研究了Banach空间的几何特性及其相关规范的不平等。这种方法使我们能够应用插值技术来研究Banach空间几何形状。主要结果如下1。在克拉克森类型的不等式和Rademacher Type-cotype上：（1）多维克拉克森型不平等的序列，普遍的克拉克森，随机的克拉克森不等式和THIER变体，以Rademacher类型和Cotype为特征。这些不等式在Lebesgue-Bochner空间中是等效的。（2）我们扩展了参数和元素数量的Q-均匀的凸度和P-均匀的平滑度不等式。2。在Banach空间的几何常数上：（1）我们阐明了冯·诺伊曼 - 约旦（NJ-）常数与詹姆斯恒定的某些关系，正常结构系数。特别是，如果X的NJ构恒定小于5/4，则X以及双空间X^*具有统一的正常结构，因此具有固定点特性。（2）我们确定了二维Lorentz序列空间D（W，Q）的NJ和James Stonstant的答案。 X.3的NJ-COSTANT。绝对规范：（1）确定并估算了C^2上绝对归一化规范的NJ恒定体。我们还表明，所有这些规范除了L_1和L_∞-NORMS外，所有这些规范都是统一的。（2）C^2上的绝对归一化规范与凸函数ψon[0，1]之间的对应关系扩展到n维情况。对于Banach空间的L_P-sums，涉及严格的resp。均匀的凸度。