上三角算子矩阵Browder定理稳定性的判定

用σ(T)和σ_w)分别表示算子T的谱与weyl谱,π_(00)(T)={λ∈isoσ(T),0dimN(T-λI)∞},若σ(T)\σ_w(T)■π_(00)(T)成立,则就认为T满足Browder定理.主要研究了2×2上三角算子矩阵的Browder定理在紧摄动下的稳定性,并给出了判定稳定性的等价条件.     

广义Kato分解与Weyl型定理的摄动

称T∈B(H)有广义Kato分解,若存在一对T的闭的不变子空间(M,N)使得H=M⊕Ⅳ,其中T|_M为上半Fredholm算子且具有非负的指标,T|_N是幂零的本文利用算子的广义Kato分解性质,研究了Weyl型定理在紧摄动下的稳定性.此外还研究了2×2上三角算子矩阵的Weyl型定理在紧摄动下的稳定性.     

Browder定理和Weyl定理

本文通过定义两个新的谱集,给出了Browder定理和Weyl定理对算子T以及f(T)成立的充要条件,其中f∈H(σ(T)),H(σ(T))表示在谱集σ(T)的开邻域上解析的函数的全体。     

广义Kato分解与拓扑一致降指数

讨论了算子的广义Kato分解与拓扑一致降指数,给出算子既有广义Kato分解又有拓扑一致降指数的等价刻画,并由此对算子谱精细结构进行进一步的细化.     

算子序列的升标和降标

本文研究了与算子紧序列有关的一类算子序列和算子上半Fredholm序列的升标和降标的一些性质.利用同伦的思想证明了具有有限升标(降标)的算子上半Fredholm序列在其摄动类中交换元的摄动下具有同样性质,从而把单个半Fredholm算子的有关结果推广到算子序列.     

Fredholm映射的集值紧摄动的拓扑度

Suppose that X is a paracompact oriented C` -Banach-Fredholm manifold, Ω is an open subset of X, E is a Banach space, f:Ω→E is a proper continuous map, f:Ω→E is an admissible (nonlinear) C′-Fredholm map of index zero,G:Ω→2^E is an upper semicontinuous compact set-valued map with compact convex values, p∈ E(f-G)(?Ω). In this paper the degree deg(f-G,Ω,p) is defined.     

3×3上三角算子矩阵的Weyl型定理

设A∈B(H1),B∈B(H2),C∈B(H3)为给定的三个算子,用M(D,E,F)= 表示一个作用在H1(?)H2(?)H3上的3×3算子矩阵．本文首先给出存在算子D∈B(H2,H1),E∈B(H3,H1),F∈B(H3,H2),使得M(D,E,F)为上半Fredholm算子(下半Fredholm算子)的充要条件．同时研究了3×3算子矩阵 M(D,E,F)的Weyl定理,α-Weyl定理,Browder定理和α-Browder定理．     

Drazin谱和算子矩阵的Weyl定理

A∈B(H)称为是一个Drazin可逆的算子,若A有有限的升标和降标．用σ_D(A)={λ∈C：A-λI不是Drazin可逆的)表示Drazin谱集．本文证明了对于Hilbert空间上的一个2×2上三角算子矩阵M_C=■,从σ_D(A)∪σ_D(G)到σ_D(M_C)的道路需要从前面子集中移动σ_D(A)∩σ_D(B)中一定的开子集,即有等式：σ_D(A)∪σ_D(B)=σ_D(M_C)∪G,其中G为σ_D(M_C)中一定空洞的并,并且为σ_D(A)∪σ_D(B)的子集．2×2算子矩阵不一定满足Weyl定理,利用Drazin谱,我们研究了2×2上三角算子矩阵的Weyl定理,Browder定理,a-Weyl定理和a-Browder定理．     

Weyl型定理的稳定性

称Hilbert空间算子T∈B(H)满足a-Browder定理,如果σ_a(T)\σ_(aw)(T)=π_(00)~a(T),其中σ_a(T)和σ_(aw)(T)分别表示逼近点谱和Weyl本性逼近点谱,π_(00)~a(T)={λ∈isoσ_a(T),0dim N(T-λI)∞}.如果σ_a(T)\σ_(aw)(T)=π_(00)~A(T),称T满足a-Weyl定理.如果对所有的紧算子K,T+K都满足a-Browder定理(a-Weyl定理),则称T关于a-Browder定理(a-Weyl定理)是稳定性的.该文研究了a-Browder定理和a-Weyl定理的稳定性,给出了算子满足a-Browder定理和a-Weyl定理紧扰动的等价刻画.     

有一致降指数的算子

本文讨论了有拓扑一致降指数的算子的可交换拟幂零摄动:举出反例说明有拓扑一致降指数的算子在可交换拟幂零摄动下是不稳定的;得到有拓扑一致降指数的算子和它的可交换拟幂零摄动的超值域与超核之间的关系;利用这些关系,证明了左(右)Drazin可逆算子在可交换幂零摄动下是稳定的.最后还讨论了Banach空间上B-正则类BR_i(1≤i≤13)中算子的R_i型Kato分解与R_i型超Kato分解.     

A Descent Theorem in Topological K-Theory

We prove the Lichtenbaum–Quillen conjecture in the topological context: in other words, real K-theory can be deduced from complex K-theory via the usual descent spectral sequence. More precise results are proved, however, and new applications are stated. The main ingredients in the proof are Atiyah's KR-theory and the definition of higher K-groups via Clifford algebras.     

Weyl's law for the cuspidal spectrum of SLn

Let Γ be a principal congruence subgroup of $\text{SL}{}_{\mathrm{n}}\text{(}\text{Z}\text{)}$ and let σ be an irreducible unitary representation of SO(n). Let N_cus^Γ(λ,σ) be the counting function of the eigenvalues of the Casimir operator acting in the space of cusp forms for Γ which transform under SO(n) according to σ. In this Note we prove that the counting function N_cus^Γ(λ,σ) satisfies Weyl's law. In particular, this implies that there exist infinitely many cusp forms for the full modular group $\text{SL}{}_{\mathrm{n}}\text{(}\text{Z}\text{)}$. To cite this article: W. Müller, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A Tauberian theorem related to Weyl's criterion

Let {x_n} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σ_n=1^Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t₀ for which A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞. In case f(s) = Σ_n=1^∞a(n)e^?sx_n, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)^?1f(σ + it₀) → 0 as σ → 0 + if and only if A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞.     

Weyl's theorem for closed operators on Fréchet spaces

A celebrated theorem of H. Weyl asserts that if A is a normal operator on a Hilbert space X, then the points in the spectrum of A which can be removed by perturbing A with a compact operator are precisely the eigenvalues of finite multiplicity which are isolated points of the spectrum of A. In these notes we develop appropriate Fredholm theory that enables us to provide interesting sufficiency and necessary conditions for a closed operator on a Fréchet space to satisfy Weyl's theorem. We complement and extend results of L. A. Coburn, V. Istr??escu, S. K. Berberian, and M. Schechter, respectively.     

Legendre's Theorem and Quadratic Reciprocity

As Gauss noted already, his Quadratic Reciprocity Law cannot be deduced from Legendre's Theorem without the existence of primes in arithmetic progressions. Here the deduction is made, with Dirichlet's Theorem replaced by the more elementary result of Selberg, which states that every non-square is a quadratic residue to half the primes.     

Manifestations of Polya's Counting Theorem

The cycle index polynomial of combinatorial analysis is discussed in various contexts.     

A short proof of Chvátal's Watchman Theorem

This note contains a short proof of Chvátal's Watchman Theorem using the existence of a three-coloring of a triangulated polygon.     

子流形与拓扑球面定理

本文建立了球面Sn p(c) (c>0 )中的完备子流形的一个拓扑球面定理 ,结果表明完备子流形的拓扑是受其内在和外在曲率不变量满足的某些条件所影响的     

Cauchy积分中值定理逆问题的注记

使用函数严格单调性,获得了定积分的若干性质.应用这些性质,解决了Cauchy积分中值定理中值点的唯一性,并得到了该定理及其逆定理的较弱表述,进一步解决了其逆定理中积分区间端点的唯一性.