Various form closures associated with a fixed non-semibounded self-adjoint operator
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914088550924288 |
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| author | Fleige, Andreas |
| author_facet | Fleige, Andreas |
| contents | If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a Kre\uın space completion and generally, it is not unique. Here, all such closures are studied. A one-to-one correspondence between all closed symmetric forms (with ``gap point'' $0$) and all J-non-negative, J-self-adjoint and boundedly invertible Kre\uın space operators is observed. Their eigenspectral functions are investigated, in particular near the critical point infinity. An example for infinitely many closures of a fixed form $(T \cdot , \cdot)$ is discussed in detail using a non-semibounded self-adjoint multiplication operator $T$ in a model Hilbert space. These observations indicate that closed symmetric forms may carry more information than self-adjoint Hilbert space operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Various form closures associated with a fixed non-semibounded self-adjoint operator Fleige, Andreas Functional Analysis Spectral Theory 47A07 (Primary) 46C20, 47B50, 47B25 (Secondary) If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a Kre\uın space completion and generally, it is not unique. Here, all such closures are studied. A one-to-one correspondence between all closed symmetric forms (with ``gap point'' $0$) and all J-non-negative, J-self-adjoint and boundedly invertible Kre\uın space operators is observed. Their eigenspectral functions are investigated, in particular near the critical point infinity. An example for infinitely many closures of a fixed form $(T \cdot , \cdot)$ is discussed in detail using a non-semibounded self-adjoint multiplication operator $T$ in a model Hilbert space. These observations indicate that closed symmetric forms may carry more information than self-adjoint Hilbert space operators. |
| title | Various form closures associated with a fixed non-semibounded self-adjoint operator |
| topic | Functional Analysis Spectral Theory 47A07 (Primary) 46C20, 47B50, 47B25 (Secondary) |
| url | https://arxiv.org/abs/2502.09551 |