A note on Weyl's theorem

The Kato spectrum of an operator is deployed to give necessary and sufficient conditions for Browder's theorem to hold.

     

解析hyponormal算子的α-Weyl定理

本文通过定义新的谱集,给出了算子满足a-Browder定理和a-Weyl定理的充要条件,运用了文章中新定义的谱集,研究了解析hyponormal算子的a-Weyl定理．     

解析hyponormal算子的α-Weyl定理

本文通过定义新的谱集,给出了算子满足a-Browder定理和a-Weyl定理的充要条件,运用了文章中新定义的谱集,研究了解析hyponormal算子的a-Weyl定理．     

Another note on Weyl's theorem

``Weyl's theorem holds" for an operator on a Banach space when the complement in the spectrum of the ``Weyl spectrum" coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the ``Browder spectrum", which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in . In this note we try to explore these distinctions.

     

Analytically Class A operators and Weyl's theorem

Using the new spectrum set defined in this note, we give the necessary and sufficient condition for T which the Weyl's theorem holds. We also consider how the Weyl's theorem survives for analytically Class A operators.     

Weyl type theorems and hypercyclic operators II

In this note, the relation between hypercyclic operator matrices (or supercyclic operator matrices) and the operator matrices which satisfy Weyl type theorems is discussed. Also, using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for for which a-Browder's theorem or a-Weyl's theorem holds.

     

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．     

Essential point spectra of operator matrices trough local spectral theory

For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper, we study defect set (Σ(A)∪Σ(B))?Σ(MC), where Σ is the Browder spectrum, the essential approximate point spectrum and Browder essential approximate point spectrum. We then give application for Weyl's and Browder's theorems.     

A-Browder＇s Theorem and Generalized a-Weyl＇s Theorem

Two variants of the essential approximate point spectrum are discussed. We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl＇s theorem holds, and conversely for a-isoloid operators. We also study the generalized a-Weyl＇s theorem for Class A operators.     

Weyl type theorem and hypercyclic operators

Using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for T for which the a-Browder's theorem or the a-Weyl's theorem holds. Also, the relation between hypercyclic operators (or supercyclic operators) and the operators which satisfy Weyl type theorem is discussed.     

A-Weyl定理及其摄动

本文讨论了本质逼近点谱的一种变形,并利用该变形定义的新的谱集,研究了α-Weyl定理在紧摄动下的稳定性.同时,给出了对任意的正整数n∈N,算子T~n∈B(H)不满足α-Weyl定理的稳定性的充分条件,其中H表示无限维的复可分Hilbert空间,B(H)表示H上的有界线性算子的全体.     

Property (ωˊ) and Its Perturbation

In this note we define the property (ωˊ),a variant of Weyl's theorem,and establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for which property (ωˊ) holds by means of the variant of the essential approximate point spectrum σ1(.) and the spectrum defined in view of the property of consistency in Fredholm and index.In addition,the perturbation of property (ωˊ) is discussed.     

Consistent invertibility and Weyl's theorem

A Banach space operator T∈B(X) may be said to be “consistent in invertibility” provided that for each S∈B(X), TS and ST are either both or neither invertible. The induced spectrum contributes the conditions equivalent to various forms of “Weyl's theorem”.     

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.     

一般边界条件下Sturm-Liouville算子的Ambarzumyan型定理

该文研究有限区间上一般自伴边界条件下的Sturm-Liouville方程的逆特征值问题.将Neumann边界条件下Sturm-Liouville方程的Ambarzumyan型定理推广到一般自伴边界条件下情形,证明了如果它的特征值与零势的特征值一样,则Sturm-Liouville方程的势为零.     

A characterization of some subsets of Schechter's essential spectrum and application to singular transport equation

The purpose of this paper is to provide a detailed treatment of some subsets of Schechter's essential spectrum of closed, densely defined linear operators subjected to additive perturbations. Our results are used to describe the essential approximate point spectrum and the essential defect spectrum of singular neutron transport operators in bounded geometries.     

The generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces

In this paper, we introduce the concepts of generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces. The concept of generalized regular points is an extension of the concept regular points, and so, the set of all spectrum points is reduced to the narrow spectrum. We present not only the same and different properties of spectrum and of narrow spectrum but also show the relationship between them. Finally, the well known problem about the invariant subspaces of bounded linear operators on separable Hilbert spaces is simplified to the problem of the operator with narrow spectrum only.     

$(\omega')$性质及其摄动

In this note we define the property （ω′）, a variant of Weyl’s theorem, and establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for which property （ω′） holds by means of the variant of the essential approximate point spectrum σ1（·） and the spectrum defined in view of the property of consistency in Fredholm and index. In addition, the perturbation of property （ω′） is discussed.     

k-拟-*-A算子的谱性质及其应用

主要给出k-拟-*-A算子的谱性质及其应用,若T是k-拟-*-A算子且N(T)■N(T~*),则Weyl谱的谱映射定理及本质近似点谱的谱映射定理成立;若T是k-拟-*-A算子,N(T)■N(T~*)且S～T,则a-Browder's定理对f(S)成立,其中f∈H(σ(S)).     

The Equivalence between Property (ω) and Weyl’s Theorem

We call T ∈ B(H) consistent in Fredholm and index (briefly a CFI operator) if for each B ∈ B(H),T B and BT are Fredholm together and the same index of B,or not Fredholm together.Using a new spectrum defined in view of the CFI operator,we give the equivalence of Weyl’s theorem and property (ω) for T and its conjugate operator T* .In addition,the property (ω) for operator matrices is considered.