有界线性算子的
Property
- 中山大学学报(自然科学版)(中英文) 2025年64卷第5期 页码：137-145
- 作者机构：
1.渭南师范学院数学与统计学院， 陕西 渭南 714099
2.陕西师范大学数学与统计学院， 陕西 西安 710119
- 作者简介：
孙晨辉（1986年生），女；研究方向：算子理论与算子代数；E-mail：sunchenhui1986@163.com
曹小红（1972年生），女；研究方向：算子理论与算子代数；E-mail：xiaohongcao@snnu.edu.cn
- 基金信息：2022年度渭南市科技局科技计划(2022ZDYFJH-11);渭南师范学院2021年人才项目(2021RC16);2024年渭南师范学院服务渭南市基础教育高质量发展教育科学研究专项(SWNZ2420)
DOI：10.13471/j.cnki.acta.snus.ZR20250027
中图分类号： O177.2收稿：2025-02-11，
修回：2025-03-18，
录用：2025-03-31，
网络出版：2025-06-16，
纸质出版：2025-09-25
- 稿件说明：
移动端阅览
孙晨辉,曹小红.有界线性算子的R性质及其稳定性[J].中山大学学报(自然科学版)(中英文),2025,64(05):137-145.
SUN Chenhui ,CAO Xiaohong .Property (R) and its stability for bounded linear operator[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(05):137-145.
孙晨辉,曹小红.有界线性算子的R性质及其稳定性[J].中山大学学报(自然科学版)(中英文),2025,64(05):137-145. DOI： 10.13471/j.cnki.acta.snus.ZR20250027.
SUN Chenhui ,CAO Xiaohong .Property (R) and its stability for bounded linear operator[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2025,64(05):137-145. DOI： 10.13471/j.cnki.acta.snus.ZR20250027.
![PDF下载](/rc-pub/front/front-article/download/109628125/lowqualitypdf/有界线性算子的<inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mfenced separators="|"><mrow><mi mathvariant="bold-italic">R</mi></mrow></mfenced></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/74BC227E-D204-490e-80CF-A06ADF71394D-M001.jpg"><?fx-imagestate width="10.83733368" height="7.45066643"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/74BC227E-D204-490e-80CF-A06ADF71394D-M001c.jpg"><?fx-imagestate width="10.83733368" height="7.45066643"?></graphic></alternatives></inline-formula>性质及其稳定性.pdf){kind=link}
摘要
设
<math id="M3"><mi mathvariant="normal">H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003365&type=
2.28600001
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003358&type=
2.28600001
为无限维复Hilbert空间，
<math id="M4"><mi>B</mi><mfenced separators="|"><mrow><mi mathvariant="normal">H</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003368&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003383&type=
7.19666624
为
<math id="M5"><mi mathvariant="normal">H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003365&type=
2.28600001
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003358&type=
2.28600001
上的有界线性算子的全体. 算子
<math id="M6"><mi>T</mi><mo>∈</mo><mi>B</mi><mfenced separators="|"><mrow><mi mathvariant="normal">H</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003385&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003395&type=
12.78466606
称为具有
<math id="M7"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003376&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003399&type=
4.57200003
性质，若
<math id="M8"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced><mo>∖</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn mathvariant="normal">00</mn></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003400&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003413&type=
31.75000000
，其中
<math id="M9"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003428&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003417&type=
7.78933382
和
<math id="M10"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003439&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003422&type=
8.55133343
分别表示算子
<math id="M11"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003444&type=
2.28600001
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003456&type=
1.94733346
的逼近点谱和Browder本质逼近点谱，
<math id="M12"><msub><mrow><mi>π</mi></mrow><mrow><mn mathvariant="normal">00</mn></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003458&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003475&type=
8.72066593
表示谱集中孤立的有限重特征值的全体. 利用算子的一致Fredholm 非负指标性质，首先给出了有界线性算子具有
<math id="M13"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003376&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003399&type=
4.57200003
性质的充要条件；其次讨论了
<math id="M14"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003376&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003399&type=
4.57200003
性质的摄动. 结合算子的一致Fredholm 非负指标的特征，得到了判定算子函数具有
<math id="M15"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003376&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003399&type=
4.57200003
性质的新途径，并揭示了
<math id="M16"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003376&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003399&type=
4.57200003
性质的稳定性与算子函数满足
<math id="M17"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003376&type=
3.21733332
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003399&type=
4.57200003
性质的内在联系.
Abstract
Let
<math id="M18"><mi mathvariant="normal">H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003460&type=
2.62466669
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003483&type=
2.70933342
be a complex infinite dimensional Hilbert space and
<math id="M19"><mi>B</mi><mfenced separators="|"><mrow><mi mathvariant="normal">H</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003481&type=
3.80999994
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003488&type=
8.38199997
be the algebra of all bounded linear operators from
<math id="M20"><mi mathvariant="normal">H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003460&type=
2.62466669
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003483&type=
2.70933342
to
<math id="M21"><mi mathvariant="normal">H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003460&type=
2.62466669
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003483&type=
2.70933342
. For
<math id="M22"><mi>T</mi><mo>∈</mo><mi>B</mi><mfenced separators="|"><mrow><mi mathvariant="normal">H</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003490&type=
3.80999994
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003492&type=
14.90133286
，
<math id="M23"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003504&type=
2.62466669
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003518&type=
2.28600001
is said to satisfy property
<math id="M24"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003520&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003528&type=
5.24933338
， if
<math id="M25"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced><mo>∖</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn mathvariant="normal">00</mn></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003511&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003512&type=
36.83000183
， where
<math id="M26"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003544&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003546&type=
9.05933285
is the approximate point spectrum of
<math id="M27"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003504&type=
2.62466669
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003518&type=
2.28600001
，
<math id="M28"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003535&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003537&type=
9.90600014
is the Browder essential approximate point spectrum of
<math id="M29"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003504&type=
2.62466669
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003518&type=
2.28600001
and
<math id="M30"><msub><mrow><mi>π</mi></mrow><mrow><mn mathvariant="normal">00</mn></mrow></msub><mfenced separators="|"><mrow><mi>T</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003564&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003566&type=
10.07533360
denotes the isolates points of the spectrum which are eigenvalues of finite multiplicity. By using the consistent Fredholm non negative index property of operators， the necessary and sufficient conditions of property
<math id="M31"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003520&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003528&type=
5.24933338
for bounded linear operators are given. Then the stability of property
<math id="M32"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003520&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003528&type=
5.24933338
are studied. Based on the consistent Fredholm non negative index property of operators， the new judgement method of property
<math id="M33"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003520&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003528&type=
5.24933338
for operator function is obtained. In addition， the relationship between the stability of property
<math id="M34"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003520&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003528&type=
5.24933338
a
nd property
<math id="M35"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003520&type=
3.72533321
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91003528&type=
5.24933338
for operator function is revealed.
关键词
Keywords
references
AIENA P ， TRIOLO S ， 2018 . Weyl-type theorems on Banach spaces under compact perturbations ［J］. Mediterr J Math ， 15 （ 3 ）： 126 .
BERKANI M ， KACHAD M ， 2015 . New Browder and Weyl type theorems ［J］. Bull Korean Math Soc ， 52 （ 2 ）： 439 - 452 .
CAO X H ， 2008 . Weyl spectrum of the products of operators ［J］. J Korean Math Soc ， 45 （ 3 ）： 771 - 780 .
CAO X H ， GUO M Z ， MENG B ， 2003 . Weyl spectra and Weyl's theorem ［J］. J Math Anal Appl ， 288 （ 2 ）： 758 - 767 .
DJORDJEVIĆ D S ， 1999 . Operators obeying a -Weyl's theorem ［J］. Publ Math Debrecen ， 55 （ 3/4 ）： 283 - 298 .
DONG J ， CAO X H ， 2020 . Compact perturbations of both SVEP and Weyl's theorem for <math id="M729"><mn mathvariant="normal">3</mn><mo>×</mo><mn mathvariant="normal">3</mn></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91006157&type= 2.28600001 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91006132&type= 7.11199999 upper triangular operator matrices ［J］. Linear Multilinear A ， 68 （ 10 ）： 2020 - 2033 .
GUPTA A ， KUMAR A ， 2020 . Properties （BR） and （BgR） for bounded linear operators ［J］. Rend Del Circolo Matematico Di Palermo Ser 2 ， 69 （ 2 ）： 601 - 611 .
JIA B T ， FENG Y L ， 2020 . Property <math id="M730"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91006158&type= 3.21733332 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91006134&type= 4.57200003 under compact perturbations ［J］. Mediterr J Math ， 17 （ 2 ）： 73 .
SUN C H ， CAO X H ， 2022 . Criteria for the property （UWE） and the a-Weyl theorem ［J］. Funct Anal Appl ， 56 （ 3 ）： 216 - 224 .
WU X F ， HUANG J J ， CHEN A ， 2018 . Weylness of 2×2 operator matrices ［J］. Math Nachr ， 291 （ 1 ）： 187 - 203 .
YANG L L ， CAO X H ， 2023 . Perturbations of spectra and property <math id="M731"><mfenced separators="|"><mrow><mi>R</mi></mrow></mfenced></math> https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91006158&type= 3.21733332 https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=91006134&type= 4.57200003 for upper triangular operator matrices ［J］. J Math Anal Appl ， 519 （ 1 ）： 126797 .
ZHU S ， LI C G ， ZHOU T T ， 2012 . Weyl type theorems for functions of operators ［J］. Glasgow Math J ， 54 （ 3 ）： 493 - 505 .
0
浏览量
94
下载量
0
CSCD
- 网络PDF下载
- Meta-XML下载
- BibTeX下载
- Endnote下载
- RefMan 下载
- NoteExpress下载
- NoteFirst下载
关联资源
相关文章
相关作者
相关机构
- 地址：广州市海珠区新港西路135号 中山大学南校园东北区312栋 邮编：510275
- 联系电话：020-84112585，84113223 Email：xuebaozr@mail.sysu.edu.cn
- 技术支持由北京北大方正电子有限公司提供 京ICP备09064830号-19
京公网安备11010802024621 - 本系统建议在Chrome、 IE9+ 以上版本浏览器阅读本站内容，360浏览器请切换至极速模式
- Cookies帮助我们提供服务并提供个性化体验。使用本网站，即表示您同意我们使用Cookies
