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

研究了Banach空间和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n元尖角三角形不等式及其逆不等式，给出了几个等式成立的条件。我们利用它们研究了Banach空间的一致非L、n_1-性等性质。我们推广了这些不等式，并得到了一些关于带参数的这类不等式的结果。(2)自2002年作者等人提出-ψ-直和以来，一直是人们关注的焦点。在这项研究中，我们得到了关于-ψ-直和的一系列结果，这些结果涉及弱近一致光滑性、价值性、舒尔性、一致非L^n_1-和等.作为极端情况，我们得到了关于L_1和L_∞-和的一些有趣的结果：特别地，我们构造了一致非L^3_1Banach空间，它们不是一致非方的，但具有非扩张映象的不动点性质。引入了von Neumann-Jordan和James类型的常数，并利用这些常数得到了一致非正方形、一致正规结构等方面的一些结果。2007年7月，中国主任参加了在哈尔滨举行的关于Banach空间、算子理论和非线性分析应用的国际研讨会，并作了特邀讲座和研究生专题讲座，介绍了关于Banach空间的-ψ-直和的一些最新结果。2006年9月，我们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.