论文标题:关于算子乘积的一些不变性问题研究
Research on Some Invariance Properties of Operator Product
论文作者
论文导师 杜鸿科,论文学位 硕士,论文专业 基础数学
论文单位 陕西师范大学,点击次数 82,论文页数 50页File Size1574K
2007-05-01论文网 http://www.lw23.com/lunwen_537073387/
Operator product; Drazin inverse; generalized inverse; range; spectrum
设H，K，L，M是复可分希尔伯特空间，B(H)，B(K，H)分别表示H上的和从K到H上的有界线性算子构成的Banach空间。给定算子A∈B(H，K)，B∈B(H，L)，C∈B(M，L)，如果B的值域R(B)是闭的，则B有Moore-Penrose逆，即存在唯一的X∈B(L，H)满足下面四个方程(1)设B{i，j，…，k}表示满足上面四个方程中的(i)，(j)，…，(k)的所有算子X∈B(K，H)，且被记为B~((i，j，…，k))。当{i，j，…，k}中含有1时，则B~((i，j，…，k))叫做算子B的{i，j，…，k}-广义逆。一般情况下，算子的广义逆不唯一。 近年来，包含广义逆的矩阵乘积不变性问题吸引了一大批学者的关注，例如，J．K．Baksalary，Jürgen Grob，Yongge Tian，R．Kala，T．Pukkila等，他们从不同的角度对该问题进行了深入的研究(参见文献[9-20])。本文主要研究了包含广义逆的算子乘积不变性问题，推广了Jürgen Groβ和Yongge Tian(2006)在[12]中的和J．K．Baksalary和A．Markiewicz(1996)在[10]中的结果。 本文共分为三章，各章的主要内容如下： 第一章我们用Riesz函数演算的方法给出了算子AB和BA的Drazin可逆性等价的不同于[6]中的一个证明，其中算子A，B∈B(H)。作为应用，我们得到了σ_D(AB)=σ_D(BA)和σ_D(A)=σ_D(?)，其中σ_D(M)和(?)分别表示一个算子M∈B(H)的Drazin谱和算子M的Aluthge变换。 第二章我们主要得到了当给定三个算子A∈B(H，K)，B∈B(H，L)，C∈B(M，L)，三个算子AXC乘积和值域与X的选取无关的一些充要条件，其中算子B的值域R(B)是闭的，X是算子B的不同种类的广义逆。这将Jürgen Groβ和Yongge Tian在[12]中的主要结论推广到了无限维的情况。这里需要指出的是在证明中我们应用了算子分块矩阵技巧和解算子方程的方法，这与Jürgen Groβ和Yongge Tian所用的思想是完全不同的。 第三章我们利用算子矩阵分块技巧给出了(?)σ(AB~((1))C)=C (2)成立的充分必要条件，其中算子A∈B(H，K)，B∈B(H，L)，C∈B(K，L)给定，算子B的值域R(B)是闭的，σ(D)是算子D∈B(H)的谱，B~((1))是算子B的{1}-逆。需要指出的是我们不仅给出了(2)式对于Hilbert空间上的三个有界线性算子成立的充分必要条件，而且我们进一步指出[10]中定理B-M中给的充分条件就是式(2)成立的充分必要条件。这里我们所用的思想，方法和[10]中的是完全不同的。
Let H, K, L and M be Hilbert spaces, B(H), B(K, H) denote theset of all bounded linear operators on H and from H into K, respectively. Forgiven A∈B(H, K), B∈B(H, L), C∈B(M, L), and let the range R(B) of B beclosed. Then B has the Moore-Penrose inverse, that is, there is a unique operatorX∈B(L, H) satisfied the following equations. (3)Let B{i,j,…, k} the set of all operators X∈B(L, H) satisfied the (i), (j),…, (k)equationss of the above four equations and written by B~((i,j,…,k)). When the set{i, j,…, k} contains the number 1, then B~((i,j,…,k)) is called a (i, j,…, k) generalizedinverse of B. Generally, the generalized inverse of an operator are not unique. In the recent years, the problem of invariance properties of a triple matrixproduct involving generalized inverses has been observed by many authors, such asJ. K. Baksalary, Jürgen Grob, Yongge Tian, R. Kala, T. Pukkila and so on (see [1-9]). In this article, we mainly study the problems of invariance properties of a tripleoperator product involving generalized inverses, which generalized some results ofJürgen Groβand Yongge Tian (2006) obtained in [12] and J.K. Baksalary and A.Markiewicz (1996) obtained in [10]. There are three chapters in this article, and the main content of each chapteras follows: In the first chaper, an alternative proof of the equivalence of Drazin invertibilityof operators AB and BA is given by the Riesz functional calculus. As an application,we will prove thatσ_D(AB)=σ_D(BA) andσ_D(A)=σ_D(?), whereσ_D(M) and (?)denote the Drazin spectrum and the Aluthge transform of an operator M∈B(H),respectively. In the second chapter, we mainly study given three operators A∈B(H, K),B∈B(H, L), C∈B(M, L), and the range R(B) of B is closed, certain invariance properties of the triple operator product AXC with respect to the choice of Xare investigated by the operaior matrix technique, where X is a different type ofgeneralized inverses of B. This generalizes the results obtained by Jürgen GroβandYongge Tian in [12] to the infinite dimenional Hilbert spaces. It is worthily to pointout that our using methods are different from that by Jürgen Groβand YonggeTian. And in the chapter three, we explore for three operators A∈B(H, K), B∈B(H, L), C∈B(K, L), if the range R(B) of B is closed, then the necessary andsufficient conditions such that (?)σ(AB~((1))C)=C (4)has been obtained by using block-operator matrix technique, whereσ(D) is thespectrum of an operator D∈B(H) and B~({1}) is the {1}-inverse of B. It is worthilyto point out that not only we got the necessary and sufficient conditions such that(4) holds, but also we point out the sufficient conditions obtained in the TheoremB-M of [10] are just the necessary and sufficient conditions. Here, our using methodsare different from that used in [10].
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