發(fā)布時間：2018-03-22 15:02

  本文選題：Young不等式　切入點：Hermite-Hadamard不等式　出處：《重慶大學》2016年博士論文　論文類型：學位論文

【摘要】：算子理論產(chǎn)生于二十世紀初,是泛函分析理論重要的組成部分,不僅深入到矩陣理論、運籌學與控制理論、統(tǒng)計學等眾多理論研究學科,而且在量子力學、微分動力系統(tǒng)等許多應用學科領(lǐng)域中都有著廣泛的實際應用,是一個十分廣闊的研究領(lǐng)域。作為算子理論中重要的一分支,算子不等式的研究就顯得尤為重要,尤其是一些經(jīng)典算子不等式的研究。近年來,越來越多新型的算子不等式版本展示出來,在方法與技巧上也體現(xiàn)出多元化,以及這些算子不等式在交叉應用學科中的深入或拓展應用。因此對算子不等式做更深入的研究是非常必要的。本文借助于連續(xù)函數(shù)的凹凸性,矩陣的譜分解,Hermite范數(shù)的酉不變性以及函數(shù)演算等工具,將幾類經(jīng)典算子不等式加以推廣并給出一系列重要的算子不等式。主要研究工作有:1.推廣算術(shù)、幾何平均算子不等式的適用范圍,并得到相應的一系列準算術(shù) 幾何平均算子不等關(guān)系式。2.基于算子函數(shù)的單調(diào)性原理及細化的標量形式的Young及其逆不等式,給出相應Young及其逆的算子版本的不等式。3.運用Hilbert-Schmidt范數(shù)的酉不變性及改進的標量形式的Young及其逆不等式,建立新的Young及其逆的矩陣版本的不等式。4.引入Kantorovich常數(shù),得到多參數(shù)具有Kantorovich常數(shù)的Young及其逆的標量形式以及相應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.借助連續(xù)凸函數(shù)的性質(zhì),給出酉不變范數(shù)下Heinz均值算子與Heron均值算子之間的不等關(guān)系式。6.給出直角坐標系下多參數(shù)連續(xù)s-凸函數(shù)的一個相關(guān)性質(zhì),并得到相應的多參數(shù)Hermite-Hadamard型積分算子不等式。7.利用統(tǒng)計學中的絕對中心距推廣含參形式的Samuelson型算子不等式,并應用研究復數(shù)域矩陣數(shù)值特征值以及復系數(shù)多項式特征根的估計與定位問題,進一步得到相關(guān)的估計區(qū)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.
【學位授予單位】：重慶大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O177

【相似文獻】

相關(guān)期刊論文 前9條

1 侯晉川;算子不等式及算子的線性組合[J];科學通報;1990年17期

2 楊長森;王亞青;;關(guān)于三個算子的算子不等式(英文)[J];數(shù)學季刊;2011年03期

3 馬麗;王芳;;有關(guān)粗糙核算子不等式的一點注記[J];贛南師范學院學報;2009年06期

4 匡繼昌;;廣義Hardy-Hilbert積分算子不等式[J];北京教育學院學報(自然科學版);2009年03期

5 金雁鳴;;Orlicz空間鞅算子不等式(英文)[J];應用數(shù)學;2012年01期

6 葉臣;;兩指標B值強鞅平削算子不等式與Banach空間的幾何性質(zhì)[J];寧波大學學報(理工版);2013年03期

7 鐘五一;;一對較精密的Hilbert型算子不等式[J];廣東教育學院學報;2009年05期

8 鐘五一;;一個較精密的Hilbert型算子不等式及其應用[J];浙江大學學報(理學版);2010年06期

9 ;[J];;年期

相關(guān)博士學位論文 前1條

1 曹海松;幾類經(jīng)典算子不等式及其應用研究[D];重慶大學;2016年

相關(guān)碩士學位論文 前5條

1 師光華;幾類算子不等式的研究[D];河南師范大學;2012年

2 段光才;幾類算子不等式的研究[D];河南師范大學;2014年

3 楊旭;關(guān)于Chaotic序下Furuta不等式及其它算子不等式相關(guān)問題的探討[D];東華大學;2012年

4 雷陽;關(guān)于Hilbert空間上的算子不等式及相關(guān)問題的研究[D];東華大學;2010年

5 向麗;三類非線性泛函微分方程解的穩(wěn)定性及其應用[D];四川大學;2006年

，

本文編號：1649162