关于初等算子的范数及其相关问题的研究

【作者】 李江艳；

【导师】 吉国兴；

【作者基本信息】 陕西师范大学 ， 基础数学， 2008， 硕士

【摘要】 算子代数理论产生于20世纪初,由于其在数学和其它科学中的广泛应用,所以在20世纪的前三十年就得到了很大的发展.初等算子是算子代数上一类重要的线性映射.近年来,国内外诸多学者对初等算子的各种性质进行了深入研究.本文研究的主要内容为初等算子的范数,初等算子的范数可达性以及与初等算子范数有关的一些集合的稠密性.本文共分三章:第一章主要介绍了在本文中用到的符号,定义和一些比较著名的定理.首先我们介绍了一些符号的表示意义,接着引入了初等算子,数值域,正规极大数值域,算子的谱,范数可达等概念.最后,给出了一些常用的广泛熟知的定理如极分解定理、谱分解定理.第二章我们讨论了一类特殊初等算子的范数.设H是无限维可分的复Hilbert空间,B（H）表示H上有界线性算子的全体组成的Banach代数.在文献[1]中,J.Stampfli计算了B（H）上的广义导子的范数公式.在文献[2]中,M.Barraa和M.Boumazgour找到了B（H）上的初等算子△A,B（X）=AXB+X的范数为‖A‖‖B‖+1成立的一个充要条件.本章我们首先证明了初等算子U_A,B（X）=AXB+BXA的范数为2‖A‖‖B‖的充要条件是‖A～*B‖=‖A‖‖B‖且W_N（B～*A）∩W_N（A,B）≠（?）（A,B≠0）,并且给出了‖U_A,B‖=2‖A‖‖B‖的一些充分或必要条件.然后我们给出了当‖U_A,B‖=‖A‖‖B‖时,0∈W_B（A～*B）∪W_A（B～*A）的一些充分条件,并且证明了若‖B～*A‖∈W_A（B～*A）,‖AB～*‖∈W_B～*（AB～*）,‖A‖‖B‖∈W_B（A～*B）∩W_A～*（BA～*）,则‖U_A,B‖=[（‖A‖‖B‖+‖AB～*‖）（‖A‖‖B‖+‖B～*A‖）]（?）.最后给出例子说明了‖A～*B‖=‖A‖‖B‖是‖U_A,B‖=2‖A‖‖B‖成立的必要而非充分条件,这样就否定回答了A.Seddik在[3]中提出的问题.第三章我们讨论了初等算子Δ_A,B（X）=AXB+X的一些性质.首先我们研究了初等算子Δ_A,B的范数可达性,然后我们讨论了集合{（A,B）:‖Δ_A,B‖<1+‖A‖‖B‖}和集合{（A,B）:‖U_A,B‖<2‖A‖‖B‖}在B（H）×B（H）中的稠密性.最后我们证明了若A,B为paranormal算子,对于（?）λ∈isoo（ΔAB）且λ≠1,则H₀(Δ_AB-λ)=(Δ_AB-λ)～（-1）（0）.更多还原

【Abstract】 The study of operator algebra theory began in 20th century. Since it is usedwidely in mathematics and other scientific branches, it got great development atthe beginning of the 20th century. Elementary operators are important linear map-pings. In recent years, many scholars both here and abroad have focused on manycharacterization on elementary operators. In this paper we mainly and detailedlydiscuss the norm of elementary operators, norm attainability of some elementaryoperators and the density of some certain sets about the norm of some elementaryoperators.This paper contains three chapters:Chapter 1 mainly introduces some notations, definitions and some well-knowntheorems. Firstly, we give some notations. Subsequently, we introduce the defini-tions of elementary operators, numerical range, normal maximal numerical range,spectrum, norm attainability etc. Finally, we give some well-known theorems suchas polar decomposition theorem and spectral decomposition theorem.In chapter 2, we discuss the norm of some certain elementary operators. Let Hbe a separable infinite dimensional Hilbert space and B(H) be the Banach algebra ofall bounded linear operators on H. In [1], J. Stampfii compute the norm of generalderivation. In [2], M. Barraa and M. Boumazgour find the equality condition for‖△A,B‖=‖A‖‖B‖+ 1. In this chapter, firstly, we proved that‖UA，B‖=2‖A‖‖B‖if and only if‖A*B‖=‖A‖‖B‖and WN(B*A)∩WN(A，B)≠(?)(A,b≠0) and givesome other sufficient or necessary condition for‖UA，B‖= 2‖A‖‖B‖.Secondly,wegive some sufficient condition for 0∈WB(A*B)∪WA(B*A)when‖UA，B‖=‖A‖‖B‖,and we proved that if‖B*A‖∈WA(B*A)，‖AB*‖∈WB*(AB*)and‖A‖‖B‖∈WB*(A*B)∩WA*(BA*)，then‖UA，B‖= [(‖A‖‖B‖+‖AB*‖)(‖A‖‖B‖+‖B*A‖)](?)Lastly, an example is given to show that‖A*B‖=‖A‖‖B‖is necessary but notsufficient condition for‖UA，B‖= 2‖A‖‖B‖, so we negatively answer the questionsposed by A.Seddik in [3].In chapter 3, we discuss some characterization on elementary operators△A,B,where△A,B(X) = AXB+X. Firstly, we discuss the norm attainability of elementaryoperator△A,B. Secondly, we discuss the density of two sets {(A, B) :‖△A,B‖< 1+‖A‖‖B‖+‖}and{(A,B):‖UA,B‖<2‖A‖‖B‖}in B(H).Lastly,we mainlyproved that if A and B are paranormal operators,(?)λA∈isoó(△AB)andλ≠1,then Ho(△AB-λ)=(△AB-λ)～(-1)(0).更多还原