OPERATOR THEORETICAL RESEARCH ON GEOMETRY OF BANACH SPACES AND APPLICATIONS

Banach空间几何算子理论研究及应用

• 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

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.

从操作员的理论角度研究了Banach空间的几何特性以及相关规范的不平等。这种方法允许对它们进行统一的处理，还使我们能够在Banach空间几何形状的研究中应用插值技术。它们不仅具有自己的美丽和意义，而且还提供了包括适用领域的各个分支分支中的基本或有用的概念和工具，这表明该主题的基本重要性。结果如下1。在克拉克森型不平等上：（1）在整个Banach空间设置中以Rademacher类型和Cotype的概念为特征的一系列克拉克森型不平等，这在“ Banach空间中的概率”中非常重要。这些因各个空间而闻名的这些不平等的大多数都是统一得出的。2。在巴拉赫空间的von noumann-Jordan（NJ-）常数上，一种规范偏斜的模量：（1）给出了一种计算NJ-Contstant的系统方法，通过该方法，所有先前的结果和一些新空间的所有结果都得到了nj-constant的序列。非方格，超反射性，正常结构和固定点特性等。3。通过有限尺寸X值L_P空间之间的1个矩阵的操作员规范的行为，将几种几何属性统一地汇总。特别是，给出了一系列均匀非方面空间的特征，其中一些与均匀凸的众所周知的序列相似。

项目成果

Mikio KATO and Yasuji TAKAHASHI: "Type, cotype constants and Clarkson's inequalities for Banach spaces" Mathematische Nachrichten. 186. 187-196 (1997)

Mikio KATO 和 Yasuji TAKAHASHI：“Banach 空间的类型、共型常数和克拉克森不等式”Mathematicische Nachrichten。

Mikio KATO: "On the von Neumann-Jordan constant for Banach spaces" Proceedings of the American Mathematical Society. 125. 1055-1062 (1997)

加藤干雄：“关于巴拿赫空间的冯·诺依曼-乔丹常数”美国数学会论文集。

加藤幹雄: "Norm inequalities and some geometrical constants for Banach spaces" 第37回実函数論・函数解析学合同シンポジウム講演集録. 17-36 (1999)

加藤干雄：“巴纳赫空间的范数不等式和一些几何常数”第 37 届实函数理论和泛函分析联合研讨会论文集 17-36 (1999)。

Yasuji TAKAHASHI and Mikio KATO: "Von Neumann-Jordan constant and normal structure of Banach spaces" Proceedings of Real Analysis Symposium 1997. 112-116 (1998)

Yasuji TAKAHASHI 和 Mikio KATO：“Banach 空间的 Von Neumann-Jordan 常数和正规结构”Proceedings of Real Analysis Symposium 1997. 112-116 (1998)

Mikio Kato: "Von Neumann-Jordan constant and some geomerical constants of Banach spaces" 京都大学数理解析研究所講究録. (発表予定).

加藤干雄：“冯·诺依曼-乔丹常数和巴拿赫空间的一些几何常数”京都大学数学科学研究所 Kokyuroku（待提交）。