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。这种方法可以统一处理它们，也使我们能够应用插值技术的研究Banach空间几何。它们不仅有其自身的美和意义，而且在包括适用领域在内的各个分析分支中提供了必要的或有用的概念和工具，这表明了这一课题的根本重要性.关于Clarkson型不等式：（1）在一般的Banach空间中，利用“Probability in Banach Spaces”中重要的Rademacher型和余型的概念刻画了一列Clarkson型不等式。(2)证明了克拉克森不等式及相关不等式是如何从给定的Banach空间X继承到Lebesgue-Bochner空间L_r（X）的，从而统一地导出了各种空间中已知的大多数不等式.关于Banach空间的vonNeumann-Jordan（NJ-）常数，给出了一种模的偏度模：（1）给出了计算NJ-常数的一种系统方法，得到了各种空间的所有结果和一些新的结果。(2)给出了NJ常数给出的一系列信息，特别是关于型与余型、一致凸性、一致非方形性、超自反性、正规结构和不动点性质等。利用有限维X值l_p-空间之间1-矩阵的算子范数的性质统一刻画了几个几何性质。特别地，给出了一致非方空间的一系列特征，其中一些特征类似于一致凸性的特征。

项目成果

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。

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。

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)

高橋泰嗣: "Recent progress in Banach space theory---W.T.Gowersの業績をめぐって" 京都大学数理解析研究所講究録. (発表予定).

高桥靖：“巴拿赫空间理论的最新进展——关于W.T.高尔斯的成就”京都大学数学科学研究所Kokyuroku（待出版）。