On products of symmetries acting on Hilbert spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918202494156800 |
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| author | Marcoux, Laurent W. Radjavi, Heydar Zhang, Yuanhang |
| author_facet | Marcoux, Laurent W. Radjavi, Heydar Zhang, Yuanhang |
| contents | Let $\mathcal{H}$ be a complex, separable Hilbert space (of finite or infinite dimension), and let $\mathcal{U}(\mathcal{H})$ denote the group of unitary operators on $\mathcal{H}$. A symmetry is, by definition, a unitary operator $J$ with $J^2 =I$. Denote by $\text{Sym}_k(\mathcal{H})$ the subset of $\mathcal{U}(\mathcal{H})$ consisting of those operators expressible as a product of $k$ symmetries. It is known that $\mathcal{U}(\mathcal{H}) = \text{Sym}_4(\mathcal{H})$ if $\dim \, \mathcal{H} = \infty$, while the only additional condition in finite dimensions is that the determinant be $\pm 1$. Of all the sets $\text{Sym}_k(\mathcal{H})$ with $k \in \{ 1, 2, 3, 4\}$, the case $k =3$ has been the most stubborn to characterise. Among other things, we investigate which elements of $\text{Sym}_3(\mathcal{H})$ possess exactly two eigenvalues in the setting where $\mathcal{H}$ is finite-dimensional. We also consider the problem: when is the unitary orbit of an operator $T$, i.e., the set \[ \{ U^* T U : U \in \mathcal{U}(\mathcal{H}) \} \] the same as its $\text{Sym}_k$-orbit, i.e., the set \[ \{ U^* T U: U \in \text{Sym}_k(\mathcal{H})\} ? \] Clearly, the cases of interest are when $k \le 3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_12028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On products of symmetries acting on Hilbert spaces Marcoux, Laurent W. Radjavi, Heydar Zhang, Yuanhang Functional Analysis Primary: 15A23, 47A65. Secondary: 15A18, 47A08 Let $\mathcal{H}$ be a complex, separable Hilbert space (of finite or infinite dimension), and let $\mathcal{U}(\mathcal{H})$ denote the group of unitary operators on $\mathcal{H}$. A symmetry is, by definition, a unitary operator $J$ with $J^2 =I$. Denote by $\text{Sym}_k(\mathcal{H})$ the subset of $\mathcal{U}(\mathcal{H})$ consisting of those operators expressible as a product of $k$ symmetries. It is known that $\mathcal{U}(\mathcal{H}) = \text{Sym}_4(\mathcal{H})$ if $\dim \, \mathcal{H} = \infty$, while the only additional condition in finite dimensions is that the determinant be $\pm 1$. Of all the sets $\text{Sym}_k(\mathcal{H})$ with $k \in \{ 1, 2, 3, 4\}$, the case $k =3$ has been the most stubborn to characterise. Among other things, we investigate which elements of $\text{Sym}_3(\mathcal{H})$ possess exactly two eigenvalues in the setting where $\mathcal{H}$ is finite-dimensional. We also consider the problem: when is the unitary orbit of an operator $T$, i.e., the set \[ \{ U^* T U : U \in \mathcal{U}(\mathcal{H}) \} \] the same as its $\text{Sym}_k$-orbit, i.e., the set \[ \{ U^* T U: U \in \text{Sym}_k(\mathcal{H})\} ? \] Clearly, the cases of interest are when $k \le 3$. |
| title | On products of symmetries acting on Hilbert spaces |
| topic | Functional Analysis Primary: 15A23, 47A65. Secondary: 15A18, 47A08 |
| url | https://arxiv.org/abs/2511.12028 |