Linear maps preserving equivalence or asymptotic equivalence on Banach space
Let XX be a complex Banach space with dimension at least two and B(X)B\left(X) the algebra of all bounded linear operators on XX. We show that a bijective linear map Φ\Phi preserves asymptotic equivalence if and only if it preserves equivalence, and in turn, if and only if there exist invertible bo...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-10-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2023-0125 |
| _version_ | 1797659605171437568 |
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| author | Qin Zijie Chen Lin |
| author_facet | Qin Zijie Chen Lin |
| author_sort | Qin Zijie |
| collection | DOAJ |
| description | Let XX be a complex Banach space with dimension at least two and B(X)B\left(X) the algebra of all bounded linear operators on XX. We show that a bijective linear map Φ\Phi preserves asymptotic equivalence if and only if it preserves equivalence, and in turn, if and only if there exist invertible bounded linear operators TT and SS such that either Φ(A)=TAS\Phi \left(A)=TAS or Φ(A)=TA*S\Phi \left(A)=T{A}^{* }S for all A∈B(X)A\in B\left(X). |
| first_indexed | 2024-03-11T18:17:35Z |
| format | Article |
| id | doaj.art-d2dd5a626d954ea99ceaa77335f54882 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-03-11T18:17:35Z |
| publishDate | 2023-10-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-d2dd5a626d954ea99ceaa77335f548822023-10-16T06:06:24ZengDe GruyterOpen Mathematics2391-54552023-10-012112257226810.1515/math-2023-0125Linear maps preserving equivalence or asymptotic equivalence on Banach spaceQin Zijie0Chen Lin1School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, P. R. ChinaSchool of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, P. R. ChinaLet XX be a complex Banach space with dimension at least two and B(X)B\left(X) the algebra of all bounded linear operators on XX. We show that a bijective linear map Φ\Phi preserves asymptotic equivalence if and only if it preserves equivalence, and in turn, if and only if there exist invertible bounded linear operators TT and SS such that either Φ(A)=TAS\Phi \left(A)=TAS or Φ(A)=TA*S\Phi \left(A)=T{A}^{* }S for all A∈B(X)A\in B\left(X).https://doi.org/10.1515/math-2023-0125linear preserversasymptotic equivalenceequivalence relationsbanach spaces47b49 |
| spellingShingle | Qin Zijie Chen Lin Linear maps preserving equivalence or asymptotic equivalence on Banach space Open Mathematics linear preservers asymptotic equivalence equivalence relations banach spaces 47b49 |
| title | Linear maps preserving equivalence or asymptotic equivalence on Banach space |
| title_full | Linear maps preserving equivalence or asymptotic equivalence on Banach space |
| title_fullStr | Linear maps preserving equivalence or asymptotic equivalence on Banach space |
| title_full_unstemmed | Linear maps preserving equivalence or asymptotic equivalence on Banach space |
| title_short | Linear maps preserving equivalence or asymptotic equivalence on Banach space |
| title_sort | linear maps preserving equivalence or asymptotic equivalence on banach space |
| topic | linear preservers asymptotic equivalence equivalence relations banach spaces 47b49 |
| url | https://doi.org/10.1515/math-2023-0125 |
| work_keys_str_mv | AT qinzijie linearmapspreservingequivalenceorasymptoticequivalenceonbanachspace AT chenlin linearmapspreservingequivalenceorasymptoticequivalenceonbanachspace |