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The property （
R ） and the hypercyclic property for bounded linear operators- Acta Scientiarum Naturalium Universitatis Sunyatseni Vol. 62, Issue 2, Pages: 172-180(2023)
- 作者机构：
陕西师范大学数学与统计学院，陕西 西安 710119
- 作者简介：
- 基金信息：
DOI：10.13471/j.cnki.acta.snus.2021A094
CLC： O177.2Published：25 March 2023，
Published Online：30 December 2022，
Received：22 November 2021，
Accepted：21 February 2022
扫 描 看 全 文
胡添翼,窦艳妮.有界线性算子的（R）性质及亚循环性[J].中山大学学报(自然科学版),2023,62(02):172-180.
HU Tianyi,DOU Yanni.The property （R） and the hypercyclic property for bounded linear operators[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2023,62(02):172-180.
胡添翼,窦艳妮.有界线性算子的（R）性质及亚循环性[J].中山大学学报(自然科学版),2023,62(02):172-180. DOI： 10.13471/j.cnki.acta.snus.2021A094.
HU Tianyi,DOU Yanni.The property （R） and the hypercyclic property for bounded linear operators[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2023,62(02):172-180. DOI： 10.13471/j.cnki.acta.snus.2021A094.
摘要
令
<math id="M1"><mi>H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424010&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49423996&type=
2.28600001
2.28600001
为无限维复可分的Hilbert空间，
<math id="M2"><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49423975&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49423969&type=
7.19666624
2.96333337
为
<math id="M3"><mi>H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424010&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49423996&type=
2.28600001
2.28600001
上有界线性算子的全体。
<math id="M4"><mi>σ</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424028&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424018&type=
6.77333355
2.96333337
表示算子
<math id="M5"><mi>T</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424049&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424057&type=
12.78466606
2.96333337
的谱集。称算子
<math id="M6"><mi>T</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424049&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424057&type=
12.78466606
2.96333337
满足
<math id="M7"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424229&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424238&type=
4.48733330
2.96333337
性质，若
<math id="M8"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>\</mo><mtext> </mtext><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn mathvariant="normal">00</mn></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424095&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424081&type=
31.32666588
3.89466691
，其中
<math id="M9"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424131&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424118&type=
8.04333401
3.89466691
，
<math id="M10"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424173&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424151&type=
8.80533314
3.89466691
分别表示算子
<math id="M11"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424184&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424170&type=
1.94733346
2.28600001
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><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">o</mi><mtext> </mtext><mi>σ</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>:</mo><mn mathvariant="normal">0</mn><mo><</mo><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>T</mi><mo>-</mo><mi>λ</mi><mi>I</mi><mo stretchy="false">)</mo><mo><</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424209&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424203&type=
59.85933304
3.89466691
. 给出了有界线性算子满足
<math id="M13"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424229&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424238&type=
4.48733330
2.96333337
性质的新的判定方法，并讨论了算子的
<math id="M14"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424229&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424238&type=
4.48733330
2.96333337
性质和亚循环性之间的关系。
Abstract
Let
<math id="M15"><mi>H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49427332&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49427322&type=
2.70933342
2.62466669
be an infinite dimensional separable complex Hilbert space and
<math id="M16"><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49427361&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49427359&type=
8.38199997
3.47133350
be the algebra of all bounded linear operators on
<math id="M17"><mi>H</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424301&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424285&type=
2.70933342
2.62466669
.
<math id="M18"><mi>σ</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424313&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424309&type=
7.87400007
3.47133350
denotes the spectrum of
<math id="M19"><mi>T</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424349&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424355&type=
14.30866718
3.47133350
.
<math id="M20"><mi>T</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424349&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424355&type=
14.30866718
3.47133350
is said to satisfy property
<math id="M21"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428934&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428932&type=
5.24933338
3.47133350
if
<math id="M22"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>\</mo><mtext> </mtext><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>π</mi></mrow><mrow><mn mathvariant="normal">00</mn></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424399&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424379&type=
36.40666580
4.57200003
， where
<math id="M23"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mtext> </mtext></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424418&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424393&type=
10.15999985
4.57200003
and
<math id="M24"><msub><mrow><mi>σ</mi></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424440&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424430&type=
10.24466610
4.57200003
denote the approximate point spectrum and the Browder essential approximate spectrum of
<math id="M25"><mi>T</mi></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428946&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428945&type=
2.28600001
2.62466669
respectively，
<math id="M26"><msub><mrow><mi>π</mi></mrow><mrow><mn mathvariant="normal">00</mn></mrow></msub><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">o</mi><mtext> </mtext><mi>σ</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>:</mo><mn mathvariant="normal">0</mn><mo><</mo><mi>n</mi><mo stretchy="false">(</mo><mi>T</mi><mo>-</mo><mi>λ</mi><mi>I</mi><mo stretchy="false">)</mo><mo><</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">}</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424485&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49424489&type=
69.84999847
4.57200003
. A new judgement for property
<math id="M27"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428934&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428932&type=
5.24933338
3.47133350
for bounded linear operator is given. In additional， the relations between the property
<math id="M28"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428934&type=
https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428932&type=
5.24933338
3.47133350
and the hypercyclic property are considered.
关键词
（R）性质亚循环性谱
Keywords
property （R）hypercyclic propertyspectrum
references
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AIENA P， GUILLÉN J R， PEN˜A P， 2011. Property <math id="M686"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428953&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428941&type=4.487333302.96333337 for bounded linear operators［J］. Mediterr J Math， 8（4）： 491-508.
CAO X， 2006. Weyl type theorem and hypercyclic operators［J］. J Math Anal Appl， 323（1）： 267-274.
DAI L， CAO X， GUO Q， 2021. Property <math id="M687"><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428961&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428959&type=4.572000032.96333337 and the single-valued extension property［J］. Acta Math Sin （Engl Ser）， 37（8）： 1254-1266.
HARTE R， LEE W Y， 1997. Another note on Weyl's theorem［J］. Trans Amer Math Soc， 349（5）： 2115-2124.
HERRERO D A， 1991. Limits of hypercyclic and supercyclic operators［J］. J Funct Anal， 99（1）： 179-190.
HILDEN H M， WALLEN L J， 1974. Some cyclic and non-cyclic vectors of certain operators［J］. Indiana Univ Math J， 23（7）： 557-565.
JIA B， FENG Y， 2020. Property <math id="M688"><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428965&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428963&type=4.487333302.96333337 under compact perturbations［J］. Mediterr J Math，17（2）： 73.
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RAKOČEVIĆ V， 1989. Operators obeying <math id="M689"><mi>α</mi></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428969&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428967&type=1.862666612.28600001-Weyl's theorem［J］. Rev Roumaine Math Pures Appl， 34（10）： 915-919.
TAYLOR A E， 1966. Theorems on ascent， descent， nullity and defect of linear operators［J］. Math Ann， 163（1）： 18-49.
Von WEYL H， 1909. <math id="M690"><mi mathvariant="normal">Ü</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428973&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428955&type=6.180666922.62466669 <math id="M691"><mi mathvariant="normal">b</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">ä</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi></math>https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428978&type=https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=49428975&type=15.155334472.28600001 quadratische formen， deren differenz vollstetig ist［J］. Rend Circ Mat Palermo， 27（1）： 373-392.
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