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.     

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.     

倍幂等元与倍对合元的换位子的可逆性

本文研究在一个有单位元的环中两个倍幂等元的换位子与两个倍对合元的换位子的可逆性问题.利用倍幂等元和倍对合元的性质,得到了两个倍幂等元的换位子与两个倍对合元的换位子可逆的几个充要条件,揭示了倍幂等元和倍对合元与它们的换位子的内在联系.     

半群环的幂等元

讨论了半群环R[S]的幂等元问题．对[1]提出的公开问题9作了一个肯定回答，同时就一般半群环的幂等元的具体形式作了深入的研究，给出了若干情形下的幂等元刻划．     

广义幂等算子差的可逆性

设P, Q为Hilbert空间H上的幂等算子, 关于算子$P$的广义幂等算子类ω(P)定义为ω(P)={A∈B}(H): A²=αA+βP, AP=PA=A,P²=P,∨α, β∈C}. 对任意A ∈ω(P), B∈ω(Q)使得A²=αA +βP, B²=mB+nQ,βn≠ 0, 得到了如下的结论: 值域R(PQ)是闭的充要条件是值域R(AB)是闭的; 如果P-Q是可逆的, 则A-B是可逆的.     

幂等阵幂零阵多项式可逆的充要条件及一般方阵多项式可逆性的判定

1　幂等阵多项式设Ａ∈Ｆｎ×ｎ 为一幂等矩阵 ,由于Ａ2 =Ａ ,所以任取ｆ(ｘ) ∈Ｆ[ｘ].ｆ(Ａ)总可以化为ｋＡ+ｌＩｎ的形式 (ｋ,ｌ∈Ｆ ,Ｉｎ 为ｎ阶单位阵 ) .对此我们有定理 1　若Ａ为ｎ阶非零幂等矩阵 ,ｋ≠ 0 ,ｌ≠ 0 ,则ｋＡ +ｌＩｎ 可逆 ｋ≠-ｌ.证　充分性因为ｌ≠ 0 ,ｋ≠-ｌ,所以ｌ(ｋ +ｌ)Ｉｎ 可逆又　 (ｋＡ +ｌＩｎ) [-ｋＡ+(ｋ+ｌ)Ｉｎ]=-ｋ2 Ａ2 +ｋ2 Ａ+ｋｌＡ-ｋｌＡ +ｌ(ｋ+ｌ)Ｉｎ=ｌ(ｋ+ｌ)Ｉｎ所以ｋＡ+ｌＩｎ 可逆 .必要性若ｋＡ+ｌＩｎ 可逆 ,而ｋ=-ｌ,则由Ａ(ｋＡ+ｌＩｎ) =ｋＡ2 +ｌＡ =0 ｎｎ,可知Ｒ(Ａ) +Ｒ(…     

m次数量幂等矩阵线性组合的可逆性

如果有非零数λ与μ使Pm=λP,Qm=μQ,则称P,Q分别是由λ,μ确定的m次数量幂等矩阵．本文证明了,若有非零数a与b,当μam-1（-1）m-1μbm-1≠0时,使可交换的分别由λ,μ确定的m次数量幂等矩阵P,Q的线性组合aP＋bQ是可逆的,那么对任意非零数u,u,当λμm-1-（-1）m-1μvm-1≠0时,uP＋vQ也是可逆的．本文主要结果和方法的应用,可以推广已有文献的2次、3次幂等矩阵的线性组合可逆的结论．     

具有弱中间幂等元的正则半群

本文在正则半群上引入弱中间幂等元和拟中间幂等元,着重探讨了这两类幂等元的性质特征.构造了若干具有弱(拟)中间幂等元的正则半群,确定了弱中间幂等元和拟中间幂等元之间的关系,给出了弱中间幂等元和拟中间幂等元各自的等价判定,利用拟中间幂等元刻画了纯正半群.最后,得到了构造具有拟中间幂等元的正则半群的一般途径,并在此基础上进一步给出了判定正则半群是否具有乘逆断面的方法.     

全序幂等元剩余格

研究了全序幂等元剩余格,给出了全序幂等元剩余格的另一种新的构造方法,并且得到了这类剩余格的结构定理,推广了相关文献的结论．     

关于斜多项式环的左半中心幂等元的性质

讨论并证明斜多项式环的σ－左半中心幂等元的若干性质。     

On Positive Invertibility of matrices

A new criterion for the positive invertibility of matrices is established. It improves the theorem by Collatz for matrices which are near to triangular ones.     

Invertibility of symmetric random matrices

We study symmetric random matrices H, possibly discrete, with iid above‐diagonal entries. We show that H is singular with probability at most , and . Furthermore, the spectrum of H is delocalized on the optimal scale . These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdös, Schlein and Yau.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 135‐182, 2014     

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.     

On the algebraic closure in rings

The ``algebraic closure" of a subset of a ring is an algebraic analogue of topological closure.

     

输入存贮线性有限自动机的弱可逆性

应用输入存贮线性有限自动机的结构矩阵讨论了输入存贮线性有限自动机的弱可逆性,得出输入存贮线性有限自动机延迟0步弱可逆的充要条件、延迟τ步弱可逆和严格延迟τ步弱可逆的充分条件,由此条件得出延迟τ步弱可逆和严格延迟τ步弱可逆的输入存贮线性有限自动机的构造方法并且求出延迟0步弱可逆输入存贮线性有限自动机的一个弱逆.     

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.     

Invertibility and Spectrum of Hille–Tamarkin Matrices

Hille–Tamarkin matrices are investigated.Invertibility conditions and estimates for the norm of the inverse matrices are established. In addition, bounds for the spectrum are suggested. In particular, new estimates for the spectral radius are derived.     

Invertibility of linear combinations of two idempotents

Let and be two idempotents on a Hilbert space. In this note, we prove that the invertibility of the linear combination is independent of the choice of , if and

     

Invertibility and the fredholm property of difference operators

We obtain conditions for the invertibility and the Fredholm property of the difference operator (Dx)(n)=x(n) -U(n)x(n − 1),n ε ℤ, in the Banach space l _p (ℤ, X),p ε [1, ∞], of vector sequences, whereX is a Banach space andU is a bounded operator function. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 816–827, June, 2000.     

Asymptotic Invertibility and the Collective Asymptotic Spectral Behavior of Generalized One-Dimensional Discrete Convolutions

We study the asymptotic invertibility as of matrices of the form and , where a and b are functions defined on the sets . The joint asymptotic behavior of the spectrum of these matrices is analyzed.