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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.10645 |
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| _version_ | 1866914478232174592 |
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| author | Plevnik, Lucijan |
| author_facet | Plevnik, Lucijan |
| contents | Let $\mathcal H$ be a complex Hilbert space and $\mathcal F_s (\mathcal H)$ the real vector space of all self-adjoint finite rank bounded operators on $\mathcal H$. We generalize the famous Wigner's theorem by characterizing linear maps on $\mathcal F_s (\mathcal H)$ which preserve the set of all rank $k$ projections. In order to do this, we first characterize linear maps on the real vector space $\mathcal H_{0, 2k}$ of trace zero $(2k) \times (2k)$ hermitian matrices which preserve the subset of unitary matrices in $\mathcal H_{0, 2k}$.
We also study linear maps from $\mathcal F_s (\mathcal H)$ to $\mathcal F_s (\mathcal K)$ 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 $\dim \mathcal H = 2$. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10645 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear preservers of rank k projections Plevnik, Lucijan Functional Analysis Let $\mathcal H$ be a complex Hilbert space and $\mathcal F_s (\mathcal H)$ the real vector space of all self-adjoint finite rank bounded operators on $\mathcal H$. We generalize the famous Wigner's theorem by characterizing linear maps on $\mathcal F_s (\mathcal H)$ which preserve the set of all rank $k$ projections. In order to do this, we first characterize linear maps on the real vector space $\mathcal H_{0, 2k}$ of trace zero $(2k) \times (2k)$ hermitian matrices which preserve the subset of unitary matrices in $\mathcal H_{0, 2k}$. We also study linear maps from $\mathcal F_s (\mathcal H)$ to $\mathcal F_s (\mathcal K)$ 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 $\dim \mathcal H = 2$. We give several examples which show that in the general case the problem to describe all such maps seems to be complicated. |
| title | Linear preservers of rank k projections |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2512.10645 |