幂等算子线性组合的幂等性

设P与Q是Hilbert空间中的两个不同的幂等算子.本文主要刻画了幂等算子P与Q的线性组合仍是幂等算子的充要条件,从而推广了Baksalary与Baksalary (2000)的结论.值得指出的是,我们通过严密的推理发现,其定理的条件P_1P_2≠P_2P_1是非必要的.     

缺项算子矩阵的幂等补

本文获得各类二阶缺项算子矩阵存在幂等补的充分必要条件,并且给出它们各自所有幂等补的参数表示形式．     

幂等矩阵线性组合的可逆性

设T1,T2,T3是三个不同的两两相互可交换的n×n非零的三次幂等矩阵,并且c1,c2,c3是非零数.本文主要给出了线性组合c1T1 c2T2 c3T3可逆性的刻画.     

广义n幂等与超广义n幂等算子

For an integer n≥2,we say that an operator A is an n-idempotent ifA~n=A;A is a generalized n-idempotent if A~n=A~*;A is a hyper-generalized n-iclempotent if A~n=A~ .The set of all n-idempotents,all generalized n-idempotentsand all hyper-generalized n-idempotents are denoted by I_n(H),GI_n(H)andHGI_n(H),respectively.In this note,we obtain a chain of proper inclusionsGI_n(H)HGI_n(H)I_(n 2)(H).     

关于幂等元之差的可逆性

本文研究在一个有单位元的环中两个幂等元之差的可逆性问题,利用幂等元的性质,得到了两个幂等元之差可逆的几个充分必要条件,并给出了在矩阵环中的几个应用.     

正反幂等元和差的可逆性

研究在一个有单位元的环中,将幂等元的和与差的可逆性的一些已知结果,推广到正反幂等元的和与差可逆性上,并在矩阵环中得到一些运用.     

$k$-次幂等矩阵线性组合群逆和超广义幂等矩阵线性组合Moore-Penrose广义逆的表

在 $T_{1}T_{2}T_{1}=T_{2}$, $T_{2}T_{1}^{k-1}=T_{1}T_{2}^{k-1}$ 和 $T_{1}T_{2}T_{1}=T_{2}T_{1}$的条件下, 得到k-次幂等矩阵线性组合群逆的表示. 另外, 在$T_{1}T_{2}T_{1}=T_{2}$ 和 $T_{1}^{2}T_{2}=T_{2}$ 的条件下, 计算超广义幂等矩阵线性组合Moore-Penrose 广义逆的表示     

非负广义幂等矩阵

非负幂等矩阵在［１］中已有详细地讨论和重要地应用。［２］给出了Ａ≥０，Ａｋ＋１＝Ａ，ｋ是正整数，Ａｋ是零对称幂等矩阵的充要条件。本文给出了一般非负广义幂等矩阵的充要条件。并讨论了广义幂等矩阵的秩与其广义幂等指数的关系，推广了［２］中的部分结论。     

分块幂等矩阵中广义Schur补的幂等性

为了讨论分块幂等矩阵中使用A(1)与A(2)的广义Schur补的幂等性问题,定义了(M/D)I=D-CA(1)B和(M/A)_O=D-CA(2)B,讨论得到了(M/D)_I=D-CA(1)B与(M/A)O=D-CA(2)B具有幂等性的充要条件,并研究了一些特殊情况,推广了J K Baksalary和Zhou J H的结论.     

幂等矩阵的组合的可逆性

本文研究了两个幂等矩阵P与Q的组合aP+bQ-cPQ-dQP-ePQP (其中a,b,c,d,e∈(C),a≠0,b≠0)的可逆性. 利用P-Q的可逆性及幂等矩阵的性质,得到了aP+bQ-cPQ-dQP-ePQP可逆的一些充要条件. 推广了J. J.Koliha 和 V.RakoA(c)eviA(c)[1]及Zuo Kezheng[2]的结论.     

Idempotent completions of operator partial matrices

Necessary and sufficient conditions are obtained for operator partial 2 × 2 matrices to have an idempotent completion, and all such completions are parametrically represented. Project supported by National Natural Science Foundation of China and Provincial Natural Science Foundation of Shanxi     

Invertibility of the Sum of Idempotents

We study necessary and sufficient conditions for the invertibility of the sum f+g when f and g are idempotents in a unital ring or bounded linear operators in Hilbert or Banach spaces. We describe the relation between the invertibility of f+g and f -g.     

Invertibility of the Sum of Idempotents

We study necessary and sufficient conditions for the invertibility of the sum f+g when f and g are idempotents in a unital ring or bounded linear operators in Hilbert or Banach spaces. We describe the relation between the invertibility of f+g and f m g.     

无界形式Hamilton算子的可逆性

利用算子分块方法研究了具有一般定义域的形式Hamilton算子的可逆性和可逆补,还进一步给出了无穷维Hamilton算子的相关结论.     

Linear Maps Preserving Invertibility or Related Spectral Properties

We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various parts of spectrum, spectral radius, quasinilpotents, etc. We present some results on elementary operators and additive operators preserving invertibility or related properties. In particular, we give a negative answer to a problem posed by Gao and Hou on characterizing spectrumpreserving elementary operators. Several open problems are also mentioned.     

Invertibility of the Difference of Idempotents

We study conditions equivalent to the invertibility of f -g when f and g are idempotents in a unital ring, and give applications to bounded linear operators in Banach and Hilbert spaces. In the setting of rings we are able to show that many conditions previously linked to finite dimensionality, rank equalities, norm topology of bounded linear operators or to properties of C *-algebras can be in fact proved by simple algebraic arguments.     

非线性 Lipschitz 连续算子的定量性质(III)------glb-Lipschitz数

本文引进非线性Lipschitz算子T的glb-Lipschitz数l(T),并证明l(T)定量刻画非线性Lipschitz连续算子全体所构成的赋半范算子空间中可逆算子T保持可逆的最大扰动半径,因而具有特别重要意义.所获结果被应用来建立``非线性扰动引理'、非线性算子条件数、推广线性算子逼近理论和建立与矩阵理论中Gerschgorin圆盘定理对应的非线性Lipschitz连续算子谱集的包含域.     

任意域上的代数中两个幂等元线性组合的Drazin可逆性（英文）

对域上的代数中两个幂等元P和q,满足一定的条件PqP=s,其中s∈{O,P,q,Pq,qP,qPq},本文得到了它们的线性组合ip＋jq均是Drazin可逆的,并给出了其Drazin逆的表示.