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值技术应用于巴拿赫空间几何的研究。主要研究结果如下：关于克拉克森型不等式和Rademacher型-共型：(1)用Rademacher型和共型描述了一系列多维克拉克森型不等式、广义克拉克森不等式、随机克拉克森不等式及其变体。这些不等式在Lebesgue-Bochner空间中是等价的。(2)在参数和元素数量上推广了q-一致凸性和p-一致光滑性不等式。关于Banach空间的几何常数：(1)澄清了von Neumann-Jordan （NJ-）常数与James常数之间的一些关系。，为法向结构系数。特别地，如果X的nj常数小于5/4，则X和对偶空间X^*具有一致法向结构，因此具有不动点性质。对高、刘提出的关于詹姆斯常数的问题也作了答复。(2)确定了二维洛伦兹序列空间d （w, q）的nj常数和James常数。(3)在L_1的子空间X是L_p的子空间X时，用X.3的nj常数确定了p， 1 <小于等于p <小于等于p的上极值。输出绝对范数：(1)确定并估计了C^2上绝对归一化范数的nj常数。我们还证明了除了l_1和l_∞范数之外所有这些范数都是一致非平方的。(2)将C^2上的绝对归一化范数与凸函数ψ[0,1]在一定条件下的对应关系推广到n维情况。(3)利用绝对范数引入有限个Banach空间的ψ-直和的概念，推广了有关严格正则的Banach空间的l_p和的已知事实。统一的凸性。

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.