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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.15072 |
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| _version_ | 1866909466333544448 |
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| author | Starkey, Quinn Kiervin Xanthos, Foivos |
| author_facet | Starkey, Quinn Kiervin Xanthos, Foivos |
| contents | Let $E, F$ be Archimedean Riesz spaces, and let $F^δ$ denote an order completion of $F$. In this note, we provide necessary conditions under which the space of regular operators $\mathcal{L}^r(E, F)$ is pervasive in $\mathcal{L}^r(E, F^δ)$. Pervasiveness of $\mathcal{L}^r(E, F)$ in $\mathcal{L}^r(E, F^δ)$ implies that the Riesz completion of $ \mathcal{L}^r(E, F)$ can be realized as a Riesz subspace of $ \mathcal{L}^r(E, F^δ$. It also ensures that the regular part of the space of order continuous operators $\mathcal{L}^{oc}(E, F)$ forms a band of $\mathcal{L}^r(E, F)$. Furthermore, the positive part $T^+$ of any operator $T \in \mathcal{L}^r(E, F)$, provided it exists, is given by the Riesz-Kantorovich formula. The results apply in particular to cases where $E = \ell_0^{\infty}$, $E = c$, or $F$ is atomic, and they provide solutions to some problems posed in [3] and [16]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pervasiveness of $\mathcal{L}^r(E,F)$ in $\mathcal{L}^r(E,F^δ)$ Starkey, Quinn Kiervin Xanthos, Foivos Functional Analysis 46A40, 46B42, 47B65 Let $E, F$ be Archimedean Riesz spaces, and let $F^δ$ denote an order completion of $F$. In this note, we provide necessary conditions under which the space of regular operators $\mathcal{L}^r(E, F)$ is pervasive in $\mathcal{L}^r(E, F^δ)$. Pervasiveness of $\mathcal{L}^r(E, F)$ in $\mathcal{L}^r(E, F^δ)$ implies that the Riesz completion of $ \mathcal{L}^r(E, F)$ can be realized as a Riesz subspace of $ \mathcal{L}^r(E, F^δ$. It also ensures that the regular part of the space of order continuous operators $\mathcal{L}^{oc}(E, F)$ forms a band of $\mathcal{L}^r(E, F)$. Furthermore, the positive part $T^+$ of any operator $T \in \mathcal{L}^r(E, F)$, provided it exists, is given by the Riesz-Kantorovich formula. The results apply in particular to cases where $E = \ell_0^{\infty}$, $E = c$, or $F$ is atomic, and they provide solutions to some problems posed in [3] and [16]. |
| title | Pervasiveness of $\mathcal{L}^r(E,F)$ in $\mathcal{L}^r(E,F^δ)$ |
| topic | Functional Analysis 46A40, 46B42, 47B65 |
| url | https://arxiv.org/abs/2501.15072 |