Banach空间几何常数与正算子性质的研究

The researches of Banach space geometry constants and positive operator properties play an important role in the theory of functional analysis.The content and the aim of this project consists of four parts. First, by using modulus of convexity and smoothness, we study the relation among several important geometric constants and calcaluate their exact values in some concrete Banach spaces. Second, we construct several new constants in weak topology to study the relation between geometric structure and the existence of fixed point of some generalized nonexpansive mappings. Third, by using polar decomposition and inequalities about positive operators, we discuss many properties such as several spectrum properties, power properties of operators, characterizations of operator class, the relationship among some operator classes, and normal properties and so on. By having studied the p-w-hyponormal operator classs and wF(p,r,q)operator class, we will expansion our results. Finally, by the mean theory of positive operator, we study the order preseving operator inequalities, operator equalitions, the monotonicity of operator functions, the monotonicity and the best possibility of the Furuta type operator functions. In particular, we study the best possibility of the outside exponent value in the Furuta type inequality with the negative powers .

Banach空间几何常数和正算子性质的研究在泛函分析理论方面起着重要的作用。我们项目的研究内容和研究目标如下： 1)利用凸性模、光滑模等工具进一步研究一些重要几何常数如若当纽曼常数﹑一致非方常数等的计算方法和它们之间的关系，并计算一些具体空间的几何常数的值; 2)在弱拓扑下构造一些新的几何常数用于研究广义非扩张算子的不动点存在性与空间几何结构之间的关系； 3）利用算子的极分解和正算子不等式，来研究一系列算子类中算子的各种谱性质、算子的幂性质、算子类的特征、算子类之间的包含关系和正规性等等。在研究了p-w-亚正常算子类和wF(p,r,q)算子类的基础上，进一步扩大研究成果； 4）利用正算子间的平均理论来研究保序算子不等式﹑算子方程、算子函数单调性，以及Furuta型算子函数的单调性及其最优单调区间，特别是具有负指数情形Furuta型不等式外部指数的最优性。

本项目主要开展了空间几何常数，解析函数空间上的复合算子及正算子性质等方面的研究。首先利用凸性模及经典分析方法求出了一些具体Banach空间如Day-James空间，Banas-Fraczek型空间，Z_{p,q}空间等的James常数和Jordan-Neumann常数的精确值，这对进一步研究空间的正规结构等几何性质具有重要意义。其次，对各种解析函数空间如Fock空间，Zygmund空间等中的分析性质，如复合算子的有界性和紧性，泰勒级数的收敛性等获得了较深刻的结果。最后，对算子平均不等式，Hilbert 空间C*模，若干正算子类的谱性等问题作了较深刻的研究。总之，经过四年的努力，项目按计划圆满完成，取得了预期的成果，发表标注本项目资助编号的SCI收录学术论文33篇，科学出版社出版学术专著一部。