几类新型广义逆的研究

发布时间：2018-03-17 12:56

本文选题：广义逆　切入点：外逆　出处：《东南大学》2016年博士论文　论文类型：学位论文

【摘要】：矩阵的广义逆是矩阵理论研究的一个重要课题.1955年R.Penrose利用四个矩阵方程给出矩阵广义逆的定义(现称为Moore-Penrose逆),以及1958年M.P.Drazin在半群和环上给出Drazin逆的定义.自此之后,广义逆理论得到迅速发展,并在许多学科领域有着重要应用.人们分别从复矩阵、Banach空间(Hilbert空间)上的有界线性算子、Banach代数(C*-代数)及环和半群等方向对广义逆展开研究.随着广义逆理论的不断发展,又产生了几类新型广义逆,如Bott-Duffin(e,f)-逆、核逆和对偶核逆及(b,c)-逆.本文主要围绕这些新型广义逆,从环的角度展开研究,得到一些有意义的结果.主要内容如下:第一部分主要研究了环上Bott-Duffin(e,f)-逆.首先利用环上可逆元素给出元素的Bott-Duffin(e,f)-逆存在的充要条件.其次在一定条件下讨论了三个元素乘积的Bott-Duffin(e,f)-逆,建立了乘积paq 的Bott-Duffin(e,f)-逆与 pa 的 Bott-Duffin(e1,f1)-逆和aq的Bott-Duffin(e2,f2)-逆之间的关系.最后作为应用给出了环上2 × 2矩阵的Bott-Duffin(E,F)-逆的存在性和表达式.第二部分考虑了*-环上的核逆和对偶核逆,研究了一定条件下三个元素乘积的核逆和对偶核逆的存在性.作为应用,对两种分块矩阵T=(?)和M=(?),给出了当a是核可逆(或d是对偶核可逆)时和当a可逆时,矩阵T和M的核逆和对偶核逆存在的充要条件和表达式.第三部分主要在半群和环上研究(b,c)-逆.首先,在*-环上给出了(b,c)-逆的刻画和表示,推广了 D.Mosic有关像-核(p,q)-逆的相关结果.其次,在半群上建立了(b,c)-逆和Bott-Duffin(e,f)-逆之间的新关系,即当b,c均为正则元时元素a是(b,c.)-可逆的充要条件是它是Bo(bb-Duffin(bb-,c-c)-可逆的,这里b,c分别为b,c的内逆.然后在一定条件下讨论了三个元素乘积的(b,c.)-逆的存在性,给出paq的(b,c)-逆与a的(b',c')-逆之间的关系.最后,作为应用考虑了环上分块下三角矩阵的(B,C)-逆的存性和表达式.特别地,给出了任意环中分块下三角矩阵A的Mary逆的存在性和表达式,此表达式简化了 X.Mary和P.Patricio在Dedekind有限环时的结果.第四部分在环和半群中考虑(b,c)-逆的反序律.首先,在环中给出了(b,c)-逆存在的一些充要条件,并在一定条件下利用群逆给出(b,c)-逆存在的一些表示.其次,考虑了半群上(b,c)-逆的反序律(α1α2)(b,c =α2(b,s)α1(t,c)和多种混合反序律成立的充要条件,推广了H.H.Zhu等人关于Maty逆的相关结果.同时还考虑了一般情况下的反序律(a1a2)(b3,c3)=α2(b2,c2)α1(b1,c1)成立的条件.第五部分在环中引入了一类新型广义逆-单边(b,c)-逆,单边零化(b,c)-逆.这类广义逆可以看作是M.P.Drazin定义的(b,c)-逆和H.H.Zhu等人所定义的单边Mary逆的推广.研究了这类新型广义逆的存在性,双重交换性及广义Cline公式.
[Abstract]:In 1955, R. Penrose gave the definition of generalized inverse of matrix (now called Moore-Penrose inverse) by using four matrix equations, and M. P. Drazin gave the definition of Drazin inverse on semigroup and ring in 1958. The generalized inverse theory has developed rapidly. And it has important applications in many disciplines. The generalized inverses are studied from the bounded linear operators on the complex matrix Banach space Hilbert space) and the rings and Semigroups, respectively. With the development of the generalized inverse theory, Several new types of generalized inverses are produced, such as Bott-Duffinne fan-inverse, nuclear inverse and dual nuclear inverse, and bbbcng-inverse. In this paper, we mainly study these new generalized inverses from the point of view of rings. Some meaningful results are obtained. The main contents are as follows: in the first part, we study the Bott-Duffinessen fan-inverse over rings. Firstly, we give the necessary and sufficient conditions for the existence of Bott-Duffininefni-inverse of elements by using reversible elements over rings. Secondly, we discuss three necessary and sufficient conditions under certain conditions. The Bott-Duffinnian fan-inverse of the product of elements, The relations between Bott-Duffinine FT-inverse of the product paq and Bott-Duffininine f1k-inverse of pa and Bott-Duffinni-e2f2f2-inverses of AQ are established. Finally, as applications, the existence and expression of Bott-DuffinEffinEOFI-inverse of 2 脳 2 matrices over rings are given. In the second part, the nuclear inverses and dual nuclear inverses are considered. The existence of kernel inverse and dual kernel inverse of product of three elements under certain conditions are studied. ) and Mavin? In this paper, we give the necessary and sufficient conditions for the existence of kernel inverse and dual nuclear inverse of matrix T and M when a is nuclear reversible (or d is invertible to dual nucleus). In this paper, we give the characterizations and representations of the ~ ~ ~ inverse, and generalize the relevant results of D. Mosic about the image-kernel ~ ~ ~. That is, the element a is invertible if and only if it is Bobb-Duffinbb-bb-nc-ca-invertible, where BJ _ c is the internal inverse of BJ _ c respectively. Then, under certain conditions, we discuss the existence of the product of three elements, bbc.-inverses. In this paper, we give the relation between paq's inverses and a's. Finally, as an application, we consider the existence and expression of paq's inverses of the lower triangular matrices in blocks. In particular, the existence and expression of the inverses are considered. In this paper, the existence and expression of Mary inverse of block lower triangular matrix A in arbitrary rings are given. This expression simplifies the results of X. Mary and P. Patricio in Dedekind finite ring. Part 4th considers the inverse order-law of Mary inverse in rings and Semigroups. In this paper, we give some necessary and sufficient conditions for the existence of bbcng-inverses in rings, and give some representations of the existence of inverses by using group inverses under certain conditions. Secondly, we consider the inverse ordering law (伪 _ 1 伪 _ 2n ~ (2) B ~ (+) ~ ()) 伪 _ (1) and the necessary and sufficient conditions for the existence of some mixed inverse order laws on Semigroups, and the necessary and sufficient conditions for the existence of some mixed inverse order laws are also given, and the necessary and sufficient conditions for the existence of the inverse inverses are given. In this paper, we generalize the relevant results of H. H. Zhu et al on Maty inverse. We also consider the condition that the inverse order law a1a2nb ~ (3) C ~ (3 +) = 伪 ~ (2) B ~ (2 +) C ~ (2)) 伪 ~ (1) B _ (1) C _ (1)) holds. In part 5th, we introduce a new kind of generalized inverse _ one-sided bbc- ~ ~ ~ (1) in the ring. This kind of generalized inverses can be regarded as a generalization of the one-sided Mary inverses defined by M. P. Drazin and H. H. Zhu et al. The existence, double commutativity and generalized Cline formulas of these new generalized inverses are studied.
【学位授予单位】：东南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O151.21

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