几类经典算子不等式及其应用研究

发布时间：2018-03-22 15:02

本文选题：Young不等式　切入点：Hermite-Hadamard不等式　出处：《重庆大学》2016年博士论文　论文类型：学位论文

【摘要】：算子理论产生于二十世纪初,是泛函分析理论重要的组成部分,不仅深入到矩阵理论、运筹学与控制理论、统计学等众多理论研究学科,而且在量子力学、微分动力系统等许多应用学科领域中都有着广泛的实际应用,是一个十分广阔的研究领域。作为算子理论中重要的一分支,算子不等式的研究就显得尤为重要,尤其是一些经典算子不等式的研究。近年来,越来越多新型的算子不等式版本展示出来,在方法与技巧上也体现出多元化,以及这些算子不等式在交叉应用学科中的深入或拓展应用。因此对算子不等式做更深入的研究是非常必要的。本文借助于连续函数的凹凸性,矩阵的谱分解,Hermite范数的酉不变性以及函数演算等工具,将几类经典算子不等式加以推广并给出一系列重要的算子不等式。主要研究工作有:1.推广算术、几何平均算子不等式的适用范围,并得到相应的一系列准算术几何平均算子不等关系式。2.基于算子函数的单调性原理及细化的标量形式的Young及其逆不等式,给出相应Young及其逆的算子版本的不等式。3.运用Hilbert-Schmidt范数的酉不变性及改进的标量形式的Young及其逆不等式,建立新的Young及其逆的矩阵版本的不等式。4.引入Kantorovich常数,得到多参数具有Kantorovich常数的Young及其逆的标量形式以及相应的算子形式的不等式,其中也包括一些Heinz型的均值算子不等式。5.借助连续凸函数的性质,给出酉不变范数下Heinz均值算子与Heron均值算子之间的不等关系式。6.给出直角坐标系下多参数连续s-凸函数的一个相关性质,并得到相应的多参数Hermite-Hadamard型积分算子不等式。7.利用统计学中的绝对中心距推广含参形式的Samuelson型算子不等式,并应用研究复数域矩阵数值特征值以及复系数多项式特征根的估计与定位问题,进一步得到相关的估计区域。
[Abstract]:Operator theory, which originated in the early 20th century, is an important part of functional analysis theory. It not only goes deep into matrix theory, operational research and control theory, statistics and many other theoretical research disciplines, but also in quantum mechanics. As an important branch of operator theory, the study of operator inequality is especially important. In recent years, more and more new versions of operator inequalities have been shown, and the methods and techniques have been diversified. And the application of these operator inequalities in cross-applied disciplines. Therefore, it is very necessary to do more in-depth research on operator inequalities. In this paper, by means of concave convexity of continuous functions, The spectral decomposition of matrices and the unitary invariance of Hermite norm and function calculus are used to generalize several classical operator inequalities and give a series of important operator inequalities. The range of application of geometric mean operator inequality, and a series of unequal relations of quasi-arithmetic and geometric mean operator are obtained. 2. Based on the monotonicity principle of operator function and the Young and its inverse inequality in the form of refined scalar, The inequality of the operator version of the corresponding Young and its inverse is given. 3.Using the unitary invariance of the Hilbert-Schmidt norm and the Young and its inverse inequality of the improved scalar form, the inequality of the new version of Young and its inverse matrix is established. 4. The Kantorovich constant is introduced. The scalar form of Young with Kantorovich constant and its inverse and the inequalities of corresponding operator forms are obtained, which also include some mean operator inequalities of Heinz type .5. with the help of the properties of continuous convex functions, The unequal relation between Heinz mean operator and Heron mean operator under unitary invariant norm is given. 6. A correlation property of continuous sconvex function with multiple parameters in rectangular coordinate system is given. The corresponding inequality of Hermite-Hadamard type integral operator with multiple parameters .7.Using the absolute center distance in statistics to generalize the inequality of Samuelson type operator with parameter form, The problem of estimating and locating the eigenvalues of complex matrix and polynomial with complex coefficients is studied, and the related estimation regions are obtained.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177

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