On Positive Vectors in Indefinite Inner Product Spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916470966976512 |
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| author | Bagarello, Fabio Kuzel, Sergiusz |
| author_facet | Bagarello, Fabio Kuzel, Sergiusz |
| contents | Let $\mathcal{H}$ be a linear space equipped with an indefinite inner product $[\cdot, \cdot]$. Denote by $\mathcal{F}_{++}=\{f\in\mathcal{H} \ : \ [f,f]>0\}$ the nonlinear set of positive vectors in $\mathcal{H}$. We demonstrate that the properties of a linear operator $W$ in $\mathcal{H}$ can be uniquely determined by its restriction to $\mathcal{F}_{++}$. In particular, we prove that the bijectivity of $W$ on $\mathcal{F}_{++}$ is equivalent to $W$ being {\em close} to a unitary operator with respect to $[\cdot, \cdot]$. Furthermore, we consider a one-parameter semi-group of operators $W_+ = \{W(t) : t \geq 0\}$, where each $W(t)$ maps $\mathcal{F}_{++}$ onto itself in a one-to-one manner. We show that, under this natural restriction, the semi-group $W_+$ can be transformed into a one-parameter group $U = \{U(t) : t\in\mathbb{R}\}$ of operators that are unitary with respect to $[\cdot, \cdot]$. By imposing additional conditions, we show how to construct a suitable definite inner product $\langle\cdot, \cdot\rangle$, based on $[\cdot, \cdot]$, which guarantees the unitarity of the operators $U(t)$ in the Hilbert space obtained by completing $\mathcal{H}$
with respect to $\langle\cdot, \cdot\rangle$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04247 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Positive Vectors in Indefinite Inner Product Spaces Bagarello, Fabio Kuzel, Sergiusz Functional Analysis Mathematical Physics 46C20, 47B50, 47D03 Let $\mathcal{H}$ be a linear space equipped with an indefinite inner product $[\cdot, \cdot]$. Denote by $\mathcal{F}_{++}=\{f\in\mathcal{H} \ : \ [f,f]>0\}$ the nonlinear set of positive vectors in $\mathcal{H}$. We demonstrate that the properties of a linear operator $W$ in $\mathcal{H}$ can be uniquely determined by its restriction to $\mathcal{F}_{++}$. In particular, we prove that the bijectivity of $W$ on $\mathcal{F}_{++}$ is equivalent to $W$ being {\em close} to a unitary operator with respect to $[\cdot, \cdot]$. Furthermore, we consider a one-parameter semi-group of operators $W_+ = \{W(t) : t \geq 0\}$, where each $W(t)$ maps $\mathcal{F}_{++}$ onto itself in a one-to-one manner. We show that, under this natural restriction, the semi-group $W_+$ can be transformed into a one-parameter group $U = \{U(t) : t\in\mathbb{R}\}$ of operators that are unitary with respect to $[\cdot, \cdot]$. By imposing additional conditions, we show how to construct a suitable definite inner product $\langle\cdot, \cdot\rangle$, based on $[\cdot, \cdot]$, which guarantees the unitarity of the operators $U(t)$ in the Hilbert space obtained by completing $\mathcal{H}$ with respect to $\langle\cdot, \cdot\rangle$. |
| title | On Positive Vectors in Indefinite Inner Product Spaces |
| topic | Functional Analysis Mathematical Physics 46C20, 47B50, 47D03 |
| url | https://arxiv.org/abs/2411.04247 |