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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2403.01473 |
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| _version_ | 1866929263044722688 |
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| author | Arlinskii, Yury |
| author_facet | Arlinskii, Yury |
| contents | Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families $\{{\mathcal T}_z\}_{{\rm Im\,} z>0}$ of closed densely defined symmetric operators with the properties: (I) the domain of ${\mathcal T}_z^2$ is a core of ${\mathcal T}_z$, (II) the domain of ${\mathcal T}_z^2$ is dense but note a core of ${\mathcal T}_z$, (III) the domain of ${\mathcal T}_z^2$ is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case ${\rm dom\,} {\mathcal T}_z^2=\{0\}$ has been considered in [5].
Given a densely defined closed symmetric operator $S$, in terms of the intersection of the domain of $S$ with ${\rm ran\,} (S-λI)$ and the projection of the domain of the adjoint $S^*$ on ${\rm ran\,} (S-λI)$, $λ\in{\mathbb C}\setminus{\mathbb R}$, necessary and sufficient conditions for the cases (I)--(III) related to the domain of $S^2$, are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Squares of symmetric operators Arlinskii, Yury Functional Analysis Primary 47B25, 47B44, Secondary 47A20 Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families $\{{\mathcal T}_z\}_{{\rm Im\,} z>0}$ of closed densely defined symmetric operators with the properties: (I) the domain of ${\mathcal T}_z^2$ is a core of ${\mathcal T}_z$, (II) the domain of ${\mathcal T}_z^2$ is dense but note a core of ${\mathcal T}_z$, (III) the domain of ${\mathcal T}_z^2$ is nontrivial but non-dense. For this purpose a class of maximal dissipative operators is defined and studied. The case ${\rm dom\,} {\mathcal T}_z^2=\{0\}$ has been considered in [5]. Given a densely defined closed symmetric operator $S$, in terms of the intersection of the domain of $S$ with ${\rm ran\,} (S-λI)$ and the projection of the domain of the adjoint $S^*$ on ${\rm ran\,} (S-λI)$, $λ\in{\mathbb C}\setminus{\mathbb R}$, necessary and sufficient conditions for the cases (I)--(III) related to the domain of $S^2$, are obtained. |
| title | Squares of symmetric operators |
| topic | Functional Analysis Primary 47B25, 47B44, Secondary 47A20 |
| url | https://arxiv.org/abs/2403.01473 |