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Linear preservers of rank k projections.

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Title: Linear preservers of rank k projections.
Authors: Plevnik, Lucijan1 (AUTHOR) lucijan.plevnik@fmf.uni-lj.si
Source: Linear Algebra & its Applications. Aug2026, Vol. 742, p101-130. 30p.
Subjects: Hilbert space, Selfadjoint operators, Linear operators, Unitary operators, Matrices (Mathematics)
Abstract: Let H be a complex Hilbert space and F s (H) the real vector space of all self-adjoint finite rank bounded operators on H. We generalize the famous Wigner's theorem by characterizing linear maps on F s (H) which preserve the set of all rank k projections. In order to do this, we first characterize linear maps on the real vector space H 0 , 2 k of trace zero (2 k) × (2 k) hermitian matrices which preserve the subset of unitary matrices in H 0 , 2 k. We also study linear maps from F s (H) to F s (K) sending projections of rank k to finite rank projections. We prove some properties of such maps, e.g. that they send rank k projections to projections of a fixed rank. We give the complete description of such maps in the case dim ⁡ H = 2. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated. [ABSTRACT FROM AUTHOR]
Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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An: 193369338
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  Data: Linear preservers of rank k projections.
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  Data: <searchLink fieldCode="DE" term="%22Hilbert+space%22">Hilbert space</searchLink><br /><searchLink fieldCode="DE" term="%22Selfadjoint+operators%22">Selfadjoint operators</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+operators%22">Linear operators</searchLink><br /><searchLink fieldCode="DE" term="%22Unitary+operators%22">Unitary operators</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink>
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  Data: Let H be a complex Hilbert space and F s (H) the real vector space of all self-adjoint finite rank bounded operators on H. We generalize the famous Wigner's theorem by characterizing linear maps on F s (H) which preserve the set of all rank k projections. In order to do this, we first characterize linear maps on the real vector space H 0 , 2 k of trace zero (2 k) × (2 k) hermitian matrices which preserve the subset of unitary matrices in H 0 , 2 k. We also study linear maps from F s (H) to F s (K) sending projections of rank k to finite rank projections. We prove some properties of such maps, e.g. that they send rank k projections to projections of a fixed rank. We give the complete description of such maps in the case dim ⁡ H = 2. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1016/j.laa.2026.04.007
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      – Code: eng
        Text: English
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        PageCount: 30
        StartPage: 101
    Subjects:
      – SubjectFull: Hilbert space
        Type: general
      – SubjectFull: Selfadjoint operators
        Type: general
      – SubjectFull: Linear operators
        Type: general
      – SubjectFull: Unitary operators
        Type: general
      – SubjectFull: Matrices (Mathematics)
        Type: general
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      – TitleFull: Linear preservers of rank k projections.
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            NameFull: Plevnik, Lucijan
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          Dates:
            – D: 01
              M: 08
              Text: Aug2026
              Type: published
              Y: 2026
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              Value: 742
          Titles:
            – TitleFull: Linear Algebra & its Applications
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