發(fā)布時(shí)間：2018-07-31 08:04

【摘要】：數(shù)值域是當(dāng)今數(shù)學(xué)比較熱門的話題之一,自從Toeplitz-Hausdorff定理出現(xiàn)之后,關(guān)于數(shù)值域的研究開始變得活躍起來.關(guān)于數(shù)值域的研究涉及到基礎(chǔ)數(shù)學(xué)和應(yīng)用數(shù)學(xué)的許多分支,并且在這些領(lǐng)域取得了廣泛的應(yīng)用.自從1990年,Ariyadasa Aluthge引入Aluthge變換(?)與2001年,Takeaki Yamazaki引入*-Aluthge變換(?)(*)之后,關(guān)于T,(?),(?)(*)等算子各種性質(zhì)的研究也引起大多數(shù)學(xué)者的注意,本文主要整理前人的這些結(jié)果.下面介紹本文的主要內(nèi)容:第一章是引言及相關(guān)的預(yù)備知識(shí).第二章是Aluthge變換及廣義Aluthge變換的一些結(jié)論,首先介紹(?),(?)(*)及(?)λ,(?)λ(*)的定義,其次介紹它們的一些基本性質(zhì),最后介紹W(T),W((?)),以及W((?)(*))之間的關(guān)系,總結(jié)了W((?))=W((?)(*))這一結(jié)論.對(duì)比著也有(?)λ與(?)λ(*)數(shù)值域相等的結(jié)論.第三章主要總結(jié)關(guān)于Aluthge變換的譜圖形的相關(guān)結(jié)論,首先介紹譜圖形的定義,再通過一些引理及定理,最后總結(jié)出:在大多數(shù)情況下,T的譜圖形與(?)的譜圖相一致.第四章主要總結(jié)關(guān)于復(fù)對(duì)稱算子Aluthge變換的一些結(jié)論,首先介紹共軛及復(fù)對(duì)稱的定義,再通過一些引理及定理總結(jié)本章的五個(gè)主要結(jié)論:(1)復(fù)對(duì)稱算子的Aluthge變換仍然是復(fù)對(duì)稱的.(2)若T是復(fù)對(duì)稱的算子,則((?))*與((?)*)是酉等價(jià)的.(3)若T是復(fù)對(duì)稱算子,則(?)=T(?)T是正規(guī)的.(4)(?)=0(?)T 2=0.(5)滿足T 2=0的算子一定是復(fù)對(duì)稱算子.第五章主要總結(jié)關(guān)于Aluthge變換極分解的一些結(jié)論,介紹了Aluthge變換極分解的形式以及雙正規(guī)算子的一些結(jié)論.
[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.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O177

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