有界线性算子乘积以及和的广义Drazin可逆性

本文讨论了两个有界线性算子的乘积以及和的广义Drazin可逆性及其广义Drazin逆的表达式.在新条件下,采用空间分解的方法证明了算子乘积PQ以及算子和P+Q是广义Drazin可逆的,并给出(PQ)^d和(P+Q)^d的具体表达式.     

2×2上三角算子矩阵的Drazin谱

设MC=[A C 0 B]是从Hilbert空间H⊕K到H⊕K中的2×2上三角算子矩阵.该文主要研究MC的Drazin可逆性和MC的Drazin谱.此外,对给定算子A∈B(H)和B∈B(K),将给出在一定条件下所有上三角算子矩阵Mc的Drazin谱的交∩C∈B(K,K)σD(MC)的具体表达式.     

反三角算子矩阵的Drazin可逆性及其表示

本文讨论了反三角算子矩阵■的Drazin可逆性及其Drazin逆的表达式.在CB=CAB=CA~2B,A~3=A~2条件下,采用预解式的Laurent展开方法证明了反三角算子矩阵M是Drazin可逆的,并给出M的含有A~D和(CB)~D的Drazin逆的表达式.最后给出算例,说明了结果的有效性.     

2×2上三角算子矩阵的Drazin谱

设MG=[ O B^A C]是从Hilbert空间H＋K到H＋K中的2×2上三角算子矩阵．该文主要研究MC的Drazin可逆性和Mc的Drazin谱．此外,对给定算子A∈B（H）和B∈B（K）,将给出在一定条件下所有上三角算子矩阵Mc的Drazin谱的交∩C∈B（K,H）σD（Mc）的具体表达式。     

2×2 上三角算子矩阵的 Drazin 谱

设M_C= [ AC ; 0 B ]是从Hilbert空间H K 到HK 中的 2×2 上三角算子矩阵. 该文主要研究 M_C的Drazin可逆性和M_C 的 Drazin谱.此外, 对给定算子A∈B}(H) 和 B∈B}(K), 将给出在一定条件下所有上三角算子矩阵M_C的Drazin谱的交∩σ_D (M_C) 的具体表达式.     

算子AB和BA的Drazin可逆性

给定Hilbert空间${\cal H}$上的有界线性算子$A$和$B$, 本文证明了$AB$和$BA$的Drazin可逆性是等价的. 作为应用, 我们证明了$\sigma_D(AB)=\sigma_D(BA)$和$\sigma_D(A)=\sigma_D(\widetilde{A})$,这里$\sigma_D(M)$和$\widetilde{M}$分别表示算子$M$的Drazin谱和Aluthge变换.     

有界线性算子的Drazin逆的逆序律

该文讨论了两个有界线性算子乘积的Drazin可逆性及其逆序律,分别在P与PQP可交换(即P~2QP=PQP~2)和Q与QPQ可交换(即Q~2PQ=QPQ~2)等条件下,采用空间分解的方法得到了PQ的Drazin可逆性及其逆序律(PQ)~D=Q~DP~D成立的等价条件.     

有界线性算子和的Drazin逆表示

本文讨论了两个有界线性算子和的Drazin可逆性及其表达式.在PQ~3=0,P~2Q=0,QPQ~2=0的条件下,采用预解式的Laurent展开方法,证明了P+Q是Drazin可逆的,并得到了P+Q的Drazin逆的表达式.同时,还确定出P+Q的指标的范围ind(P+Q)≤2t+r+s—1,给出数值算例说明结论的有效性.     

Hilbert空间上算子W-加权Drazin逆的刻画及表示

该文研究了Hilbert空间上线性算子的W-加权Drazin逆,利用算子的分块矩阵表示,给出了W-加权Drazin逆的刻画及表示,所获结果推广了魏益民等的相关结果.     

两个有界线性算子和的Drazin逆

本文研究了两个有界线性算子和的Drazin逆的问题.利用算子的预解式展开的方法,得到了(P+Q)~D的具体表达式,并将其应用到四分块算子矩阵M=[A B C D]的Drazin逆上,推广了文献[14,15]的结果.     

Characterizations and representations of the Drazin inverse involving idempotents

This paper is to present some results on the Drazin invertibility of products and differences of idempotents. In addition, some formulae for the Drazin inverse of sums, differences and products of idempotents are also established.     

On Jacobson’s lemma and Drazin invertibility

In this note we give an answer to a question of Patricio and da Costa, to note that Jacobson’s lemma extends to Drazin invertibility, preserving also the “Drazin index”. Also, we consider the interaction between abstract versions of Drazin invertibility and Jacobson’s lemma.     

The Drazin inverse in an arbitrary semiring

In this article, we investigate the Drazin invertibility for the elements of an arbitrary semiring. We give necessary and sufficient conditions for the existence and expressions of the Drazin inverse of an element in an arbitrary semiring. Moreover, we consider the product paq under some additional necessary conditions for which the Drazin inverse of the product paq exists.     

The Pseudo Drazin inverses in Banach Algebras

Let (A) be a complex Banach algebra and J be the Jacobson radical of(A).(1) We firstly show that a is generalized Drazin invertible in (A) if and only if a+J is generalized Drazin invertible in (A)/J.Then we prove that a is pseudo Drazin invertible in (A) if and only if a + J is Drazin invertible in (A)/J.As its application,the pseudo Drazin invertibility of elements in a Banach algebra is explored.(2) The pseudo Drazin order is introduced in (A).We give the necessary and sufficient conditions under which elements in (A) have pseudo Drazin order,then we prove that the pseudo Drazin order is a pre-order.     

The Drazin invertibility of the difference and the sum of two idempotent operators

In this note, some equivalents are established of the Drazin invertibility of differences and sums of idempotent operators on a Hilbert space.     

Additive property of Drazin invertibility of elements in a ring

In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab?=?ba, we show that a?+?b is Drazin invertible if and only if 1?+?a ^D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a?+?b) ^D in terms of a, a ^D , b and b ^D , but also (1?+?a ^D b) ^D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.     

On the Drazin inverses involving power commutativity

We explore the Drazin inverses of bounded linear operators with power commutativity (PQ=QmP) in a Hilbert space. Conditions on Drazin invertibility are formulated and shown to depend on spectral properties of the operators involved. Moreover, we prove that P±Q is Drazin invertible if P and Q are dual power commutative (PQ=QmP and QP=PnQ) and show that the explicit representations of the Drazin inverse D(P±Q) depend on the positive integers m,n?2.     

Strongly preserver problems in Banach algebras and C1-algebras

Let A and B be Banach algebras. Assume that A is unital. We prove that an additive map $T:A\to B$ strongly preserves Drazin (or equivalently group) invertibility, if and only if T is a Jordan triple homomorphism. When A and B are C¹-algebras, we characterize the linear maps strongly preserving generalized invertibility (in the Jordan systems’ sense), and as consequence we determine the structure of selfadjoint linear maps strongly preserving Moore–Penrose invertibility.