A functional representation approach to vector lattice covers for spaces of compact operators
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912263566262272 |
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| author | van Gaans, Onno Glück, Jochen Kalauch, Anke |
| author_facet | van Gaans, Onno Glück, Jochen Kalauch, Anke |
| contents | For ordered normed vector spaces $X, Y$, we consider the space $\mathcal{L}(X,Y)$ of bounded linear operators and characterize when its cone of positive operators has non-empty interior. When this is satisfied, we give a functional representation of the closure $\mathcal{C}(X,Y)$ of the finite rank operators in $\mathcal{L}(X,Y)$. This space is particularly interesting since it coincides in many cases with the space of compact operators from $X$ to $Y$.
Our functional representation has very good order properties in the sense that it is a so-called vector lattice cover of $\mathcal{C}(X,Y)$. This can be used to characterize disjointness of operators in $\mathcal{C}(X,Y)$ and to determine which operators have a modulus in $\mathcal{C}(X,Y)$. We demonstrate how our results can be applied to a variety of concrete spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A functional representation approach to vector lattice covers for spaces of compact operators van Gaans, Onno Glück, Jochen Kalauch, Anke Functional Analysis 46B40, 47B65 For ordered normed vector spaces $X, Y$, we consider the space $\mathcal{L}(X,Y)$ of bounded linear operators and characterize when its cone of positive operators has non-empty interior. When this is satisfied, we give a functional representation of the closure $\mathcal{C}(X,Y)$ of the finite rank operators in $\mathcal{L}(X,Y)$. This space is particularly interesting since it coincides in many cases with the space of compact operators from $X$ to $Y$. Our functional representation has very good order properties in the sense that it is a so-called vector lattice cover of $\mathcal{C}(X,Y)$. This can be used to characterize disjointness of operators in $\mathcal{C}(X,Y)$ and to determine which operators have a modulus in $\mathcal{C}(X,Y)$. We demonstrate how our results can be applied to a variety of concrete spaces. |
| title | A functional representation approach to vector lattice covers for spaces of compact operators |
| topic | Functional Analysis 46B40, 47B65 |
| url | https://arxiv.org/abs/2503.04958 |