OPERATOR THEORETICAL RESEARCH ON GEOMETRY OF BANACH SPACES AND APPLICATIONS

从算子理论的角度研究了Banach空间的几何性质及其相关的范数不等式。这种方法使我们能够将插补技术应用到Banach空间几何的研究中。主要研究结果如下：1.关于Clarkson型不等式和Rademacher型-余型：(1)利用Rademacher型和余型刻画了一系列多维Clarkson型不等式、广义Clarkson、随机Clarkson不等式及其变种。(2)推广了q-一致凸性和p-一致光滑性在参数和元素个数上的性质。关于Banach空间的几何常数：(1)阐明了von Neumann-Jordan(NJ-)常数与James常数，即正规结构系数之间的一些关系。特别地，如果X具有小于5/4的nj-常数，则X以及对偶空间X^*具有一致正规结构，从而具有不动点性质。(2)我们确定了二维Lorentz序列空间d(w，q)的Nj-常数和James常数.(3)由X.3的Nj-常数决定了p，1[小于等于]p[小于等于]p[小于等于]p[其中L_1的子空间X是L_p的上确界.绝对范数外：(1)确定并估计了C^2上绝对正规化范数的NJ-常数。证明了除了L_1范数和L_∞范数外，所有这些范数都是一致非方的。(2)将C^2上的绝对正规范数与[0，1]上的凸函数ψ在一定条件下的对应关系推广到n维Banach空间。(3)利用绝对范数，我们引入了有限个Banach空间的ψ-直和的概念，推广了关于Banach空间的L_p-和严格正规性的一个广为人知的事实。一致凸性。

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.