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

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Acta Mathematica Sinica,English Series 2007年第5期23卷 951-960页

作者： xiao hong cao College of Mathematics and Information Science Shaanxi Normal University

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 ... 详细信息

关键词： a-Browder's theorem generalized a-Weyl's theorem spectrum

Property(R)for Upper Triangular Operator Matrices

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Acta Mathematica Sinica,English Series 2023年第3期39卷 523-532页

作者： Li Li YANG xiao hong cao School of Mathematics and Statistics Shaanxi Normal UniversityXi’an 710119P.R.China

Property(R)holds for an operator when the complement in the approximate point spectrum of the Browder essential approximate point spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite *** A∈B(H)and B∈B(K),where H and K are complex infinite dimensional separable Hilbert *** denote by M_(C)the operator acting on H⊕K of the form M_(C)=(AC0B).In this paper,we give a sufficient and necessary condition for M_(C)∈(R)for all C∈B(K,H).

关键词： property(R) upper triangular operator matrix spectrum

Property(ω) and Its Perturbations

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Acta Mathematica Sinica,English Series 2014年第5期30卷 797-804页

作者： Wei Juan SHI xiao hong cao College of Mathematics and Information Science Shaanxi Normal University

Abstract A Hilbert space operator T is said to have property （ω1） if σα（T）／σaw（T） π00（T）, where σα（T） andσαw（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）〈 ∞}. Ifσα（T）／σαw（T） = π00（T）, we say T satisfies property （w）. In this note, we investigate the stability of the property （wi） and the property （w） under compact perturbations, and we characterize those operators for which the property （wi） and the property （w） are stable under compact perturbations.

关键词： Property (ω1) property (ω) compact perturbations

Property(ω)and the Single-valued Extension Property

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Acta Mathematica Sinica,English Series 2021年第8期37卷 1254-1266页

作者： Lei DAI xiao hong cao Qi GUO School of Mathematics and Statistics Weinan Normal UniversityWeinan 714099P.R.China School of Computer Science Shaanxi Normal UniversityXi’an 710119P.R.China School of Mathematics and Information Science Shaanxi Normal UniversityXi’an 710119P.R.China

By the new spectrum originated from the single-valued extension property,we give the necessary and sufficient conditions for a bounded linear operator defined on a Banach space for which property(ω)***,the relationsh... 详细信息

关键词： Property(ω) the single-valued extension property hypercyclic property supercyclic property

On Weyl’s Theorem for Functions of Operators

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Acta Mathematica Sinica,English Series 2019年第8期35卷 1367-1376页

作者： Jiong DONG xiao hong cao Lei DAI School of Mathematics and Information Science Shaanxi Normal University Xi'an 710119 P. R. China College of Mathematics and Physics Weinan Normal University Weinan 714000 P. R. China

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl’s theorem, where f ∈ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl’s theorem for f(T).

关键词： Spectrum Weyl's theorem functions of operators compact perturbation

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

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Acta Mathematica Sinica,English Series 2006年第1期22卷 169-178页

作者： xiao hong cao Mao Zheng GUO Bin MENG College of Mathematics and Information Science Shanxi Normal University Xi'an 710062 P. R. China LMAM School of Mathematical Sciences Peking University Beijing 100871 P. R. China

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋（B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋（B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.

关键词： Semi-Fredholm operator Fredholm operator Spectrum Weyl's theorem

BMI Approach to the Interconnected Stability and Cooperative Control of Linear Systems

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自动化学报 2008年第4期34卷 438-444页

作者： NIAN xiao-hong cao Li School of Information Science and Engineering Central SouthUniversity Changsha 410075 P.R. China

这份报纸学习互连的稳定性和大规模线性系统的合作控制。用双线性的矩阵不平等(BMI ) 的技术，必要、足够的条件为互连的稳定性和二个分系统的合作稳定被给。就算分系统不是稳定的，系统能合作地被稳定，这被显示出就算分系统不是稳定... 详细信息

这份报纸学习互连的稳定性和大规模线性系统的合作控制。用双线性的矩阵不平等(BMI ) 的技术，必要、足够的条件为互连的稳定性和二个分系统的合作稳定被给。就算分系统不是稳定的，系统能合作地被稳定，这被显示出就算分系统不是稳定的。我们假定分系统的稳定性不是必要的。而且，设计的问题相连，合作控制器用 BMI 限制被变换成优化问题。解决这些问题，某些最佳的交替的算法被建议，并且为算法的集中的证明被介绍。最后，几个例子被给说明优化结果。

关键词：自动化系统稳定性控制系统线性控制

Property (ω′) and Its Perturbation

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Journal of Mathematical Research and Exposition 2011年第5期31卷 879-886页

作者： Yan Hua ZHANG xiao hong cao College of Mathematics and Information Science Shaanxi Normal University Shaanxi 710062 P. R. China

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.

关键词： property (ω′) spectrum Weyl’s theorem.

A Note on Property(ω)

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Journal of Mathematical Research and Exposition 2011年第1期31卷 100-108页

作者： Ji Rong WANG xiao hong cao Department of Mathematics Yuncheng University Shanxi 044000 P. R. China College of Mathematics and Information Science Shaanxi Normal University Shaanxi 710062 P. R. China

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.

关键词： approximate Weyl's theorem property (ω) Browder operator.