关于Aluthge变换的相关结论

发布时间：2018-07-31 08:04

【摘要】：数值域是当今数学比较热门的话题之一,自从Toeplitz-Hausdorff定理出现之后,关于数值域的研究开始变得活跃起来.关于数值域的研究涉及到基础数学和应用数学的许多分支,并且在这些领域取得了广泛的应用.自从1990年,Ariyadasa Aluthge引入Aluthge变换(?)与2001年,Takeaki Yamazaki引入*-Aluthge变换(?)(*)之后,关于T,(?),(?)(*)等算子各种性质的研究也引起大多数学者的注意,本文主要整理前人的这些结果.下面介绍本文的主要内容:第一章是引言及相关的预备知识.第二章是Aluthge变换及广义Aluthge变换的一些结论,首先介绍(?),(?)(*)及(?)λ,(?)λ(*)的定义,其次介绍它们的一些基本性质,最后介绍W(T),W((?)),以及W((?)(*))之间的关系,总结了W((?))=W((?)(*))这一结论.对比着也有(?)λ与(?)λ(*)数值域相等的结论.第三章主要总结关于Aluthge变换的谱图形的相关结论,首先介绍谱图形的定义,再通过一些引理及定理,最后总结出:在大多数情况下,T的谱图形与(?)的谱图相一致.第四章主要总结关于复对称算子Aluthge变换的一些结论,首先介绍共轭及复对称的定义,再通过一些引理及定理总结本章的五个主要结论:(1)复对称算子的Aluthge变换仍然是复对称的.(2)若T是复对称的算子,则((?))*与((?)*)是酉等价的.(3)若T是复对称算子,则(?)=T(?)T是正规的.(4)(?)=0(?)T 2=0.(5)满足T 2=0的算子一定是复对称算子.第五章主要总结关于Aluthge变换极分解的一些结论,介绍了Aluthge变换极分解的形式以及双正规算子的一些结论.
[Abstract]:Numerical range is one of the most popular topics in mathematics nowadays. Since the emergence of Toeplitz-Hausdorff theorem, the research on numerical range has become more and more active. The research on numerical range involves many branches of basic mathematics and applied mathematics, and has been widely used in these fields. Since 1990, Ariyadasa Aluthge has introduced Aluthge transform (?) After the introduction of Takeaki Yamazaki in 2001, the study on the properties of operators such as T, (?) (*) has also attracted the attention of most scholars. In this paper, these results are mainly summarized. The following is the main content of this paper: the first chapter is the introduction and related preparatory knowledge. In the second chapter, some conclusions of Aluthge transform and generalized Aluthge transform are given. Firstly, the definitions of (?), (?) (*) and (?) 位, (?) 位 (*) are introduced, and then some basic properties of W (T) W (?), and W (?) (*) are introduced, and the conclusion of W (?) W (?) (*) is summarized. In contrast, we also have the conclusion that (?) 位 and (?) 位 (*) are equal to each other. The third chapter summarizes the related conclusions about the spectral graph of Aluthge transform, first introduces the definition of spectral graph, then through some Lemma and theorem, finally concludes: in most cases, the spectral graph of T and (?) The spectral patterns are consistent with each other. In chapter 4, some conclusions about Aluthge transformation of complex symmetric operators are summarized. Firstly, the definitions of conjugate and complex symmetry are introduced. The five main conclusions of this chapter are summarized by some Lemma and theorems: (1) the Aluthge transformation of complex symmetric operators is still complex symmetric. (2) if T is a complex symmetric operator, then (?) * and (?) *) are unitary equivalent. (3) if T is a complex symmetric operator, Then T (?) T (?) T is normal. (4) 0 (?) T 2 0. (5) the operator satisfying T2G 0 must be a complex symmetric operator. In chapter 5, we summarize some conclusions about pole decomposition of Aluthge transform, and introduce the form of pole decomposition of Aluthge transform and some conclusions of bimormal operator.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177

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