Friedrichs and Kre\uın type extensions in terms of representing maps
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910388431355904 |
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| author | Hassi, Seppo de Snoo, Henk |
| author_facet | Hassi, Seppo de Snoo, Henk |
| contents | A semibounded operator or relation $S$ in a Hilbert space with lower bound $m \in {\mathbb R}$ has a symmetric extension $S_{\rm f}=S {\, \widehat + \,} (\{0\} \times {\rm mul\,} S^*)$, the weak Friedrichs extension of $S$, and a selfadjoint extension $S_{\rm F}$, the Friedrichs extension of $S$, that satisfy $S \subset S_{\rm f} \subset S_{\rm F}$. The Friedrichs extension $S_{\rm F}$ has lower bound $γ$ and it is the largest semibounded selfadjoint extension of $S$. Likewise, for each $c \leq γ$, the relation $S$ has a weak Kre\uın type extension $S_{{\rm k},c}=S {\, \widehat + \,} (\ker (S^*-c) \times \{0\})$ and Kre\uın type extension $S_{{\rm K},c}$ of $S$, that satisfy $S \subset S_{{\rm k},c} \subset S_{{\rm K},c}$. The Kre\uın type extension $S_{{\rm K},c}$ has lower bound $c$ and it is the smallest semibounded selfadjoint extension of $S$ which is bounded below by $c$. In this paper these special extensions and, more generally, all extremal extensions of $S$ are constructed in terms of a representing map for ${\mathfrak t}(S)-c$ and their properties are being considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19041 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Friedrichs and Kre\uın type extensions in terms of representing maps Hassi, Seppo de Snoo, Henk Functional Analysis Primary 47A07, 47B25, 47A67, Secondary 47A06, 47B65 A semibounded operator or relation $S$ in a Hilbert space with lower bound $m \in {\mathbb R}$ has a symmetric extension $S_{\rm f}=S {\, \widehat + \,} (\{0\} \times {\rm mul\,} S^*)$, the weak Friedrichs extension of $S$, and a selfadjoint extension $S_{\rm F}$, the Friedrichs extension of $S$, that satisfy $S \subset S_{\rm f} \subset S_{\rm F}$. The Friedrichs extension $S_{\rm F}$ has lower bound $γ$ and it is the largest semibounded selfadjoint extension of $S$. Likewise, for each $c \leq γ$, the relation $S$ has a weak Kre\uın type extension $S_{{\rm k},c}=S {\, \widehat + \,} (\ker (S^*-c) \times \{0\})$ and Kre\uın type extension $S_{{\rm K},c}$ of $S$, that satisfy $S \subset S_{{\rm k},c} \subset S_{{\rm K},c}$. The Kre\uın type extension $S_{{\rm K},c}$ has lower bound $c$ and it is the smallest semibounded selfadjoint extension of $S$ which is bounded below by $c$. In this paper these special extensions and, more generally, all extremal extensions of $S$ are constructed in terms of a representing map for ${\mathfrak t}(S)-c$ and their properties are being considered. |
| title | Friedrichs and Kre\uın type extensions in terms of representing maps |
| topic | Functional Analysis Primary 47A07, 47B25, 47A67, Secondary 47A06, 47B65 |
| url | https://arxiv.org/abs/2403.19041 |