【作者】 王玲；

【导师】 陈建龙；

【作者基本信息】 东南大学 ， 基础数学， 2021， 硕士

【摘要】 EP矩阵和EP算子是矩阵和算子理论中的重要研究对象.带对合的环上的EP元素是EP矩阵和EP算子的自然推广,吸引了众多学者对其展开研究.近些年来,人们将EP元进一步推广到指标为m的m-EP元和*-DMP元的情形,得到了许多有意义的结果.本文致力于研究带对合的环上的m-EP元,EP元以及幂是EP的元素.第二章定义并研究了 m-EP元.首先我们给出了 m-EP元和指标为m的m-EP元之间的关系.然后我们利用单边主理想给出了 m-EP元的等价刻画,推广了 EP元的相关结果.例如:设a ∈R?,则a是m-EP元当且仅当am∈a*R∩Ra*.并利用投影元给出了 m-EP元的等价条件,例如:a是m-EP元当且仅当存在投影元p,q使得Ra=Rp,a=qR,am=pamq.最后,我们研究了 m-EP元与其它广义逆之间的关系,得到了 Drazin可逆元和伪核可逆元的m-EP性的刻画,并证明了若a是(m+1)-EP元,则a是m-弱群可逆的且aD是a的一个m-弱群逆.第三章我们结合m-EP性质研究EP元(即1-EP元)及幂是EP的元素.首先我们利用交换子和双交换子给出了 EP元的等价刻画,推广了王等关于EP元的结果.然后,在am+1是Hermitian元的条件下,我们通过单边理想给出了am是EP元的刻画,推广了陈等关于EP元的相关结果.最后利用伪核可逆元给出了 am是EP元的等价条件.例如:设a∈R,m,n∈ N+,且n ≥ m.则am是EP元当且仅当a是伪核可逆的且an∈Ra(?),从而推广了许等关于核可逆元与EP元的关系的结果.更多还原

【Abstract】 While the EP matrices and EP operators play an important role in the study of matrices and operators,the EP elements in rings with involutions serve as a natural generalizations of them.In recent years,m-EP elements with index m and*-DMP elements(two main generalizations of EP elements)have been introduced and investigated by many authors.This article is devoted to studing m-EP elements,EP elements and elements whose power is EP.In Chapter 2,we define and investigate m-EP elements for some fixed m.We first give the relationship between m-EP elements and m-EP elements with index m.Then we characterize m-EP elements with one-sided principal ideals.For example,it is proved that if a ∈ R? then a is m-EP if and only if am∈a*R∩Ra*.And we characterize m-EP elements with projection:it is proved that a is m-EP if and only if there exist p,q such that Ra=Rp,aR=qR,am=pamq.Finally,the relationships between m-EP elements and some other generalized inverses are investigated.We derive the m-EP properties of Drazin invertible elements and pseudo core invertible elements,and show that if a is(m+1)-EP then a is m-weak group invertible with a m-weak group inverse aD.In Chapter 3,the m-EP properties are applied to studing EP elements(i.e.,1-EP elements)and elements whose power is EP.Firstly,we characterize EP element by the commutativity and double commutativity,which generalizes the work of Wang et al.about EP elements.Secondly,under the conditions that am+1 is Hermitian,we give the characterization of am being EP by using one-sided principal ideals,which generalizes the work of Chen et al.about EP elements.Finally,we give the equivalence conditions of am being EP if a is pseudo core invertible,which generalizes the work of Xu et al.about a being EP when a is core invertible.For example,let a∈R,m,n∈N+ and n≥m.Then am is EP if and only if a is pseudo core inverse and an∈Ra(?).更多还原