有界线性算子的Weyl定理的判定

doi:10.6040/j.issn.1671-9352.0.2018.134

山东大学学报（理学版） ›› 2018, Vol. 53 ›› Issue (10): 82-87.doi: 10.6040/j.issn.1671-9352.0.2018.134

有界线性算子的Weyl定理的判定

张莹¹,曹小红^1*,戴磊²

1.陕西师范大学数学与信息科学学院, 陕西西安 710119;2.渭南师范学院数理学院, 陕西渭南 714000

收稿日期:2018-03-20 出版日期:2018-10-20 发布日期:2018-10-09
作者简介:张莹(1993— ), 女, 硕士研究生, 研究方向为算子理论. E-mail:zhangying12240705@snnu.edu.cn*通信作者简介:曹小红(1972— ), 女, 教授, 博士生导师, 研究方向为算子理论. E-mail:xiaohongcao@snnu.edu.cn
基金资助:
国家自然科学基金资助项目(11471200;11501419);陕西师范大学中央高校基本科研业务费专项资金资助项目(GK201601004);渭南市科技计划资助项目(2016KYJ-3-3);渭南师范学院自然科学人才资助项目(15ZRRC10)

Judgement of Weyls theorem for bounded linear operators

ZHANG Ying¹, CAO Xiao-hong^1*, DAI Lei²

1. College of Mathematics and Information Science, Shaanxi Normal University, Xian 710119, Shaanxi, China;
2. College of Mathematics and Physics, Weinan Normal University, Weinan 714000, Shaanxi, China

Received:2018-03-20 Online:2018-10-20 Published:2018-10-09

摘要/Abstract

摘要： 令H为复的无限维可分的Hilbert空间, B(H)为H上有界线性算子的全体。称算子T∈B(H)满足Weyl定理, 若σ(T)\σw(T)=π₀₀(T), 其中σ(T)和σw(T)分别表示算子T的谱集与Weyl谱, π₀₀(T)={λ∈iso σ(T):0

关键词: Weyl定理, 谱, 单值延拓性质

Abstract: Let H be an infinite dimensional separable complex Hilbert space and B(H) be the algebra of all bounded linear operators on H. T∈B(H) satisfies Weyls theorem if σ(T)\σw(T)=π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum respectively, π00(T)={λ∈iso σ(T):0An operator T∈B(H) is said to have the single-valued extension property, if for every open set U⊆C, the only analytic solution of the equation (T-λI)f(λ)=0(for all λ∈U)is zero function on U. Using the single-valued extension property, we give a new judgement for Weyls theorem.

Key words: spectrum, single-valued extension property, Weyls theorem

中图分类号:

O177.2

引用本文

张莹,曹小红,戴磊. 有界线性算子的Weyl定理的判定[J]. 山东大学学报（理学版）, 2018, 53(10): 82-87.

ZHANG Ying, CAO Xiao-hong, DAI Lei. Judgement of Weyls theorem for bounded linear operators[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 82-87.

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链接本文: https://lxbwk.njournal.sdu.edu.cn/CN/10.6040/j.issn.1671-9352.0.2018.134

https://lxbwk.njournal.sdu.edu.cn/CN/Y2018/V53/I10/82

参考文献 17

[1] WEYL H V. Über beschränkte quadratische Formen, deren Differenz vollstetig ist[J]. Rendiconti Del Circolo Matematico Di Palermo, 1909, 27(1):373-392.

[2] BERBERIAN S K. An extension of Weyls theorem to a class of not necessarily normal operators[J]. Michigan Mathematical Journal, 1969, 16(3):273-279.

[3] LI Chunguang, ZHU Sen, FENG Youling. Weyls theorem for functions of operators and approximation[J]. Integral Equations and Operator Theory, 2010, 67(4):481-497.

[4] CURTO R E, HAN Y M. Weyls theorem for algebraically paranormal operators[J]. Integral Equations and Operator Theory, 2003, 47(3):307-341.

[5] AN I J, HAN Y M. Weyls theorem for algebraically quasi-class a operators[J]. Integral Equations and Operator Theory, 2008, 62(1):1-10.

[6] SHI Weijuan, CAO Xiaohong. Weyls theorem for the square of operator and perturbations[J]. Communications in Contemporary Mathematics, 2015, 17:1450042.

[7] COBURN L A. Weyls theorem for nonnormal operators[J]. Michigan Mathematical Journal, 1966, 13(3):285-288.

[8] DUGGAL B P. The Weyl spectrum of p-hyponormal operators[J]. Integral Equations and Operator Theory, 1997, 29(2):197-201.

[9] CAO Xiaohong. Analytically class operators and Weyls theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 320(2):795-803.

[10] DUNFORD N. Spectral operator[J]. Pacific Journal of Mathematics, 1954, 4(3):321-354.

[11] FINCH J K. The single valued extension property on a Banach space[J]. Pacific Journal of Mathematics, 1975, 58(1):61-69.

[12] ZHU Sen, LI Chunguang. SVEP and compact perturbations[J]. Journal of Mathematical Analysis and Applications, 2011, 380(1):69-75.

[13] Aiena P, Peña P. Variantions on Weyls theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 324(1):566-579.

[14] AMOUCH M. Weyl type theorem for operators satisfying the single-valued extention property[J]. Journal of Mathematical Analysis and Applications, 2007, 326(2):1476-1484.

[15] DUGGAL B P. Upper triangular operator matrices, SVEP and Browder, Weyl theorems[J]. Integral Equations and Operator Theory, 2009, 63(1):17-28.

[16] 江泽坚, 吴智泉, 纪友清. 实变函数论[M]. 3版. 北京: 高等教育出版社, 2007: 17-42. JIANG Zejian, WU Zhiquan, JI Youqing. Real variable function theory[M]. 3rd ed. Beijing: Higher Education Press, 2007: 17-42.

[17] TAYLOR A E. Theorems on ascent, descent, nullity and defect of linear operators[J]. Mathematische Annalen, 1996, 163(1):18-49.

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[1]	WEYL H V. Über beschränkte quadratische Formen, deren Differenz vollstetig ist[J]. Rendiconti Del Circolo Matematico Di Palermo, 1909, 27(1):373-392.
[2]	BERBERIAN S K. An extension of Weyls theorem to a class of not necessarily normal operators[J]. Michigan Mathematical Journal, 1969, 16(3):273-279.
[3]	LI Chunguang, ZHU Sen, FENG Youling. Weyls theorem for functions of operators and approximation[J]. Integral Equations and Operator Theory, 2010, 67(4):481-497.
[4]	CURTO R E, HAN Y M. Weyls theorem for algebraically paranormal operators[J]. Integral Equations and Operator Theory, 2003, 47(3):307-341.
[5]	AN I J, HAN Y M. Weyls theorem for algebraically quasi-class a operators[J]. Integral Equations and Operator Theory, 2008, 62(1):1-10.
[6]	SHI Weijuan, CAO Xiaohong. Weyls theorem for the square of operator and perturbations[J]. Communications in Contemporary Mathematics, 2015, 17:1450042.
[7]	COBURN L A. Weyls theorem for nonnormal operators[J]. Michigan Mathematical Journal, 1966, 13(3):285-288.
[8]	DUGGAL B P. The Weyl spectrum of p-hyponormal operators[J]. Integral Equations and Operator Theory, 1997, 29(2):197-201.
[9]	CAO Xiaohong. Analytically class operators and Weyls theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 320(2):795-803.
[10]	DUNFORD N. Spectral operator[J]. Pacific Journal of Mathematics, 1954, 4(3):321-354.
[11]	FINCH J K. The single valued extension property on a Banach space[J]. Pacific Journal of Mathematics, 1975, 58(1):61-69.
[12]	ZHU Sen, LI Chunguang. SVEP and compact perturbations[J]. Journal of Mathematical Analysis and Applications, 2011, 380(1):69-75.
[13]	Aiena P, Peña P. Variantions on Weyls theorem[J]. Journal of Mathematical Analysis and Applications, 2006, 324(1):566-579.
[14]	AMOUCH M. Weyl type theorem for operators satisfying the single-valued extention property[J]. Journal of Mathematical Analysis and Applications, 2007, 326(2):1476-1484.
[15]	DUGGAL B P. Upper triangular operator matrices, SVEP and Browder, Weyl theorems[J]. Integral Equations and Operator Theory, 2009, 63(1):17-28.
[16]	江泽坚, 吴智泉, 纪友清. 实变函数论[M]. 3版. 北京: 高等教育出版社, 2007: 17-42. JIANG Zejian, WU Zhiquan, JI Youqing. Real variable function theory[M]. 3rd ed. Beijing: Higher Education Press, 2007: 17-42.
[17]	TAYLOR A E. Theorems on ascent, descent, nullity and defect of linear operators[J]. Mathematische Annalen, 1996, 163(1):18-49.

[1]	汤步洲,胡晗. 电力安全知识图谱构建技术与应用[J]. 《山东大学学报(理学版)》, 2026, 61(5): 18-26.
[2]	庞永锋,杜亚伟,岳慧慧. 闭值域算子的Moore-Penrose逆的谱分解[J]. 《山东大学学报(理学版)》, 2026, 61(2): 37-42.
[3]	宁书怡,戴召媛,冯心池. 基于UPLC-QE-Orbitrap-MS技术的龙血竭成分定性定量分析[J]. 《山东大学学报(理学版)》, 2025, 60(9): 10-23.
[4]	钱文彬,彭嘉豪,蔡星星. 基于邻域粒度与三支决策的知识表示学习方法[J]. 《山东大学学报(理学版)》, 2025, 60(7): 94-103.
[5]	陈旭,邵荣侠,王国平. 带有割点的图的补图的距离谱半径[J]. 《山东大学学报(理学版)》, 2025, 60(2): 19-23.
[6]	齐鑫,孟蕊,王玉玉. 上同调H^1,*(A)中基元的注记[J]. 《山东大学学报(理学版)》, 2025, 60(2): 78-84.
[7]	周玉兰,魏万瑛,柳翠翠,杨青青. 离散时间正规鞅泛函空间中幂计数算子的性质[J]. 《山东大学学报(理学版)》, 2025, 60(2): 85-95.
[8]	丁瑞鹤,王才士,张丽霞. Bernoulli噪声泛函上一类位势算子的谱性质[J]. 《山东大学学报(理学版)》, 2025, 60(12): 173-177.
[9]	刘妮,焦红英,王亚利. 算子乘积AB及BA的Browder定理[J]. 《山东大学学报(理学版)》, 2025, 60(12): 178-182.
[10]	吕平涛,王才士,赵积君. 基于Hadamard游荡的一类具有相同特征值和连续谱的两态量子游荡[J]. 《山东大学学报(理学版)》, 2024, 59(8): 84-93.
[11]	桂梁,徐遥,何世柱,张元哲,刘康,赵军. 基于动态邻居选择的知识图谱事实错误检测方法[J]. 《山东大学学报(理学版)》, 2024, 59(7): 76-84.
[12]	黎超,廖薇. 基于医疗知识驱动的中文疾病文本分类模型[J]. 《山东大学学报(理学版)》, 2024, 59(7): 122-130.
[13]	王晨,许德刚,达虹鞠,唐智和,栾辉,范海浩. 基于DEM与“宽带结构”联合优化的XCH₄遥感反演算法研究[J]. 《山东大学学报(理学版)》, 2024, 59(4): 127-134.
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[15]	梁超凡,刘奋进,李玉超,柳顺义. 奇异同谱图的构造[J]. 《山东大学学报(理学版)》, 2024, 59(2): 65-70.

有界线性算子的Weyl定理的判定

Judgement of Weyls theorem for bounded linear operators

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