RESEARCH ON GEOMETRIC STRUCTURES OF BANACH AND FUNCTION SPACES AND ψ-DIRECT SUMS

研究了Banach空间与函数空间的几何结构以及Banach空间的ψ直和。主要结果如下。(1)得到了一个有n个元素的尖锐三角形不等式及其逆不等式，并给出了等式成立的若干条件。利用它们研究了Banach空间的一致非1 ^n_1等性质。我们扩展了这些不等式我们也得到了一些关于带参数的类型不等式的结果。(2)自2002年作者等人提出-ψ直接和以来，一直受到广泛关注。在本研究项目中，我们得到了关于弱近一致光滑性、WORTH性质、Schur性质和一致非1 ^n_1性等一系列-ψ-直接和的结果。作为极端情况，我们在l_1和l_∞和上得到了一些有趣的结果：特别地，我们构造了一致非l^3_1 Banach空间，它不是一致非平方的，但对于非扩张映射具有不动点性质（见第2章）。但都是超反射性的)。(3)引入了von Neumann-Jordan型常数和James型常数，利用这些常数得到了均匀非方度和均匀正构等几个结果。同时，我们也得到了一些近似均匀光滑的弱模的结果。(4) 2007年7月，首席研究员参加了在中国哈尔滨举行的巴拿赫空间、算子理论及其在非线性分析中的应用国际研讨会，并作了专题演讲和研究生专题讲座，介绍了巴拿赫空间的-ψ-直和的一些最新成果。我们于2006年9月在日本北九州组织了第二届Banach与功能空间国际研讨会，其论文集《Banach与功能空间II》（第467页）于2008年由横滨出版社出版。

Geometric structures of Banach and function spaces and also ψ-direct sums of Banach spaces are investigated. Major results are as follows.(1)A sharp triangle inequality and its reverse inequality with n elements were obtained, where some conditions for equality attainedness were presented. We used them to investigate the uniform non-l^n_1-ness etc. of Banach spaces. We extended these inequalities and also we had some results on such type inequalities with a parameter.(2)Since it was introduced by the author etc. in 2002, the-ψ-direct sums have been attracting a good deal of attention. In this research project we obtained a sequence of results on the-ψ-direct sums concerning weak nearly uniform smoothness, WORTH property, Schur property, and uniform non-l^n_1-ness etc. As extreme cases some interesting results on l_1 and l_∞-sums are included: In particular we constructed uniformly non-l^3_1 Banach spaces which are not uniformly non-square, but have the fixed point property for non-expansive mappings (resp. but are super-reflexive).(3)We introduced the von Neumann-Jordan type and the James type constants and obtained several results on the uniform non-squareness and uniform normal structure etc. with these constants. Also we obtained some results on the weak modules of nearly uniform smoothness.(4)In July, 2007 the head investigator participated in the International Workshop on Banach space, Operator Theory and Applications to Nonlinear Analysis held in Harbin, China and presented an invited talk and a special lecture for graduate students, where he introduced some recent results on the-ψ-direct sums of Banach spaces. We organized the Second International Symposium on Banach and Function Spaces 2006in September, 2006, Kitakyushu, Japan, and its proceedings Banach and Function Spaces II (pp.467)was published by Yokohama Publishers in 2008.