A non-surjective Wigner-type theorem in terms of equivalent pairs of subspaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916161399029760 |
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| author | Pankov, Mark |
| author_facet | Pankov, Mark |
| contents | Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal G}_{\infty}(H)$ be the set of all closed subspaces of $H$ whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of ${\mathcal G}_{\infty}(H)$ sending every pair of subspaces to an equivalent pair of subspaces; two pairs of subspaces are equivalent if there is a linear isometry sending one of these pairs to the other. Let $f$ be such a transformation. We show that there is a unique up to a scalar multiple linear or conjugate linear isometry $L:H\to H$ such that for every $X\in {\mathcal G}_{\infty}(H)$ the image $f(X)$ is the sum of $L(X)$ and a certain closed subspace $O(X)$ orthogonal to the range of $L$. In the case when $H$ is separable, we give the following sufficient condition to assert that $f$ is induced by a linear or conjugate linear isometry: if $O(X)=0$ for a certain $X\in {\mathcal G}_{\infty}(H)$, then the same holds for all $X\in {\mathcal G}_{\infty}(H)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10886 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A non-surjective Wigner-type theorem in terms of equivalent pairs of subspaces Pankov, Mark Mathematical Physics Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal G}_{\infty}(H)$ be the set of all closed subspaces of $H$ whose dimension and codimension both are infinite. We investigate (not necessarily surjective) transformations of ${\mathcal G}_{\infty}(H)$ sending every pair of subspaces to an equivalent pair of subspaces; two pairs of subspaces are equivalent if there is a linear isometry sending one of these pairs to the other. Let $f$ be such a transformation. We show that there is a unique up to a scalar multiple linear or conjugate linear isometry $L:H\to H$ such that for every $X\in {\mathcal G}_{\infty}(H)$ the image $f(X)$ is the sum of $L(X)$ and a certain closed subspace $O(X)$ orthogonal to the range of $L$. In the case when $H$ is separable, we give the following sufficient condition to assert that $f$ is induced by a linear or conjugate linear isometry: if $O(X)=0$ for a certain $X\in {\mathcal G}_{\infty}(H)$, then the same holds for all $X\in {\mathcal G}_{\infty}(H)$. |
| title | A non-surjective Wigner-type theorem in terms of equivalent pairs of subspaces |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2403.10886 |