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.     

A Note on Property(ω)

In this note we study the property(ω),a variant of Weyl's theorem introduced by Rakoevic,by means of the new spectrum.We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property(ω) and approximate Weyl's theorem hold.As a consequence of the main result,we study the property(ω) and approximate Weyl's theorem for a class of operators which we call the λ-weak-H(p) operators.     

$(\omega')$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.     

Property (ω) and Its Perturbations

In the note,we establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for the stability of property(ω) by means of the variant of the essential approximate point spectrum and the induced spectrum of consistency in Fredholm and index.In addition,the stability of property(ω) for H(P) operators is considered.     

Wely定理与$(\omega)$性质的等价性

We call T C B（H） consistent in Fredholm and index （briefly a CFI operator） if for each B ∈ B（H）, TB 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.     

RIESZ IDEMPOTENT AND BROWDER＇S THEOREM FOR ABSOLUTE-（p,r ）-PARANORMAL OPERATORS

An operator T is said to be paranormal if ||T 2x|| ≥ ||T x||2 holds for every unit vector x.Several extensions of paranormal operators are considered until now,for example absolute-k-paranormal and p-paranormal introduced in [10],[14],respectively.Yamazaki and Yanagida [38] introduced the class of absolute-(p,r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators.An operator T ∈ B(H) is called absolute-(p,r)-paranormal operator if |||T |p|T |r x||r ≥ |||T |rx||p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0.The famous result of Browder,that self adjoint operators satisfy Browder’s theorem,is extended to several classes of operators.In this paper we show that for any absolute-(p,r)paranormal operator T,T satisfies Browder’s theorem and a-Browder’s theorem.It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p,r)-paranormal operator T,then E is self-adjoint if and only if the null space of T μ,N(T μ) N(T μ).     

性质$(\omega)$的注解

In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.     

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.     

WEYL＇S TYPE THEOREMS AND HYPERCYCLIC OPERATORS

For a bounded operator T acting on an infinite dimensional separable Hilbert space H,we prove the following assertions: (i) If T or T* ∈ SC,then generalized aBrowder's theorem holds for f(T) for every ...     

Property(ω) and Its Perturbations

A Hilbert space operator T is said to have property(ω1) if σa(T)\σaw(T) ? π00(T), where σa(T) and σaw(T) denote the approximate point spectrum and the Weyl essential approximate point spectrum of T respectively, and π00(T) = {λ∈ iso σ(T), 0 dim N(T- λI) ∞}. If σa(T)\σaw(T) = π00(T), we say T satisfies property(ω). In this note, we investigate the stability of the property(ω1) and the property(ω) under compact perturbations, and we characterize those operators for which the property(ω1) and the property(ω) are stable under compact perturbations.     

A Remark on Chen’s Theorem (Ⅱ)

Let p denote a prime and P2 denote an almost prime with at most two prime factors. The author proves that for suffciently large x,sum from p≤x p 2=P2 1>(1.13Cx)/(log~2x), where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.     

一致Fredholm指标算子及广义(ω′)性质

本文给出了一致Fredholm指标算子的定义及判定,同时定义了Weyl型定理的一种新变化:广义(ω')性质.根据一致Fredholm指标性质定义出一种新的谱集,通过该谱集给出了Hilbert空间上有界线性算子满足广义(ω')性质的充要条件,并且研究了广义(ω')性质的摄动,还研究了算子的亚循环性和广义(ω')性质之间的关系.     

k-拟-*-A(ｎ)算子的性质(英文)

An operator T is called k-quasi-*-A（n） operator, if T^*k|T¹⁺ⁿ|^2/（1+n）T^k ≥T^*k|T^* |²T^k , k ∈ Z, which is a generalization of quasi-*-A（n） operator. In this paper we prove some properties of k-quasi-*-A（n） operator, such as, if T is a k-quasi-*-A（n） operator and N（T ）■N（T^* ）, then its point spectrum and joint point spectrum are identical. Using these results, we also prove that if T is a k-quasi-*-A（n） operator and N（T ）■N（T ）, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.     

P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions of the principle and their improvements.     

Cartan’s Second Main Theorem and Mason’s Theorem for Jackson Difference Operator

Let f : C → Pⁿ be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.     

Picard’s theorem and the Rickman construction

Nevanlinna theory (value-distribution theory) has its genesis in Picard’s discovery that a function analytic in the plane which omits two values is constant. Nearly a century later, attention turned to the analogous situation in Rn, n≥3, where entire functions are necesarily replaced by entire quasiregular mappings. This expository article centers on one of Seppo Rickman’s main contributions to this issue, including an outline of his famous example showing that the omitted set in R3, while finite, can be much larger than possible in the plane.     

(ω)性质的判定

利用新定义的谱集,刻画了Hilbert空间上有界线性算子满足(ω_1)性质和(ω)性质的等价条件.另外,利用该谱集,对算子函数的(ω)性质进行了判定.     

Lω-空间的拟ω-Lindel(o)f性

在Lω-空间中引入拟ω-Lindel(o)f性的概念,讨论拟ω-Lindel(o)f性的一些基本性质,给出拟ω-Lindel(o)f性的几个等价刻画.     

Lω-空间的ω-Lindel(o)f性质

在Lω-空间中引入ω-Lindel(o)f性质和ω-Lindel(o)f空间等概念,给出了其等价刻画,并证明它保持L-拓扑空间中许多良好的性质,如闭遗传性、L-好的推广、被连续的L值Zadeh型函数所保持.此外,引入了ω-紧性的概念,研究了其若干性质.     

(ω)性质及Weyl型定理

(ω)性质是Rakocevic给出的Weyl定理的一种变化.本文通过定义新的谱集,给出了有界线性算子同时满足(ω)性质和a-Weyl定理的充要条件.同时,利用所得的主要结论,研究了H(p)算子的(ω)性质.