Spectra and norms of algebraic combinations of projection pairs

Zhu Tong; Song Xianhua

doi:10.11714/acta.snus.ZR20260012

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Spectra and norms of algebraic combinations of projection pairs

更新时间：2026-07-13

- Spectra and norms of algebraic combinations of projection pairs
- Acta Scientiarum Naturalium Universitatis Sunyatseni Pages: 1-10(2026)
- 作者机构：
  
  青海师范大学数学与统计学院，青海西宁 810008
- 作者简介：
- 基金信息：
- DOI：10.11714/acta.snus.ZR20260012
  CLC： O177.2
- Received：07 January 2026，
  
  Revised：2026-05-24，
  
  Accepted：05 June 2026，
  
  Online First：13 July 2026，
- 稿件说明：
移动端阅览
Zhu Tong, Song Xianhua. Spectra and norms of algebraic combinations of projection pairs[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-10.
DOI：

Zhu Tong, Song Xianhua. Spectra and norms of algebraic combinations of projection pairs[J/OL]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2026, 1-10. DOI： 10.11714/acta.snus.ZR20260012.

摘要

首先研究Hilbert空间

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823328&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823315&type=

2.28600001

上投影对

</mo><mi>Q</mi></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823353&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823340&type=

5.67266655

2.96333337

的积

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823329&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823316&type=

4.06400013

2.96333337

，Jordan积

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823354&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823341&type=

12.02266598

2.96333337

，三乘Jordan积

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823330&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823317&type=

6.01133299

2.96333337

，拟积

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823355&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823342&type=

15.83266735

2.96333337

和Lie积

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823331&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823318&type=

12.02266598

2.96333337

的谱. 其次，研究了投影对线性组合

</mo><mi>β</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>\</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823356&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823343&type=

31.83466530

3.80999994

的谱和范数. 同时，也得到了非零投影对的和、差、积，Jordan积及Lie积的范数之间的关系.

Abstract

This paper first investigates the spectra of the product

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823332&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823319&type=

4.74133301

3.38666677

， Jordan product

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823357&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823344&type=

14.05466747

3.47133350

， triple Jordan product

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823333&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823320&type=

7.02733374

3.38666677

， quasi-product

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823358&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823345&type=

18.54199982

3.47133350

and Lie product

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823334&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823321&type=

14.05466747

3.47133350

for pairs of projections

</mo><mi>Q</mi></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823359&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823346&type=

6.60400009

3.47133350

on Hilbert space

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823335&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823322&type=

2.70933342

2.62466669

. Secondly， it investigates the spectra and norms of the linear combination

</mo><mi>β</mi><mo>∈</mo><mi mathvariant="double-struck">C</mi><mo>\</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></mrow></mfenced></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823360&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=112823347&type=

37.16867065

4.48733330

of projection pairs. Meanwhile， the relationships between the norms of the sum， difference， product， Jordan product and Lie product of non-zero projection pairs are also derived.

关键词

Keywords

references

蒋万林，左可正，李昱， 2021 . 两个正交投影算子组合的性质［J］. 华中师范大学学报（自然科学版）， 55 （ 3 ）： 356 - 364 .

马楠，李愿， 2024 . 两个正交投影组合的谱和范数［J］. 山东大学学报（理学版）， 59 （ 12 ）： 102 - 108 .

吴文明，蒋叶聪，阮颖彬，等， 2021 . 投影算子组的联合谱［J］. 中国科学：数学， 51 （ 5 ）： 711 - 722 .

Altwaijry N ， Conde C ， Feki K ， et al ， 2025 . Operator norm of certain sums of orthogonal projections in Hilbert spaces ［J］. Bull Malays Math Sci Soc ， 48 （ 5 ）： 157 .

Bikchentaev A ， 2022 . Differences and commutators of projections on a Hilbert space ［J］. Int J Theor Phys ， 61 （ 1 ）： 2 .

Böttcher A ， Spitkovsky I M ， 2010 . A gentle guide to the basics of two projections theory ［J］. Linear Algebra Appl ， 432 （ 6 ）： 1412 - 1459 .

Conde C ， 2022 . A note about the norm of the sum and the anticommutator of two orthogonal projections ［J］. J Math Anal Appl ， 505 （ 2 ）： 125650 .

Cui J L ， Li C K ， 2012 . Maps preserving peripheral spectrum of Jordan products of operators ［J］. Oper Matrices ， 6 （ 1 ）： 129 - 146 .

Du H K ， Deng C Y ， 2005 . A new characterization of gaps between two subspaces ［J］. Proc Amer Math Soc ， 133 （ 10 ）： 3065 - 3070 .

Halmos P R ， 1969 . Two subspaces ［J］. Trans Amer Math Soc ， 144 ： 381 - 389 .

Klaja H ， 2014 . The numerical range and the spectrum of a product of two orthogonal projections ［J］. J Math Anal Appl ， 411 （ 1 ）： 177 - 195 .

Li Q H ， 2004 . Commutators of orthogonal projections ［J］. Nihonkai Math J ， 15 （ 1 ）： 93 - 99 .

Wang Y Q ， Du H K ， 2008 . Norms of commutators of self-adjoint operators ［J］. J Math Anal Appl ， 342 （ 1 ）： 747 - 751 .

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