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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.19768 |
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| _version_ | 1866912050142248960 |
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| author | Glück, Jochen |
| author_facet | Glück, Jochen |
| contents | An ordered Banach space $X$ is said to have the Levi property or to be regular if every increasing order bounded net (equivalently, sequence) is norm convergent. We prove four theorems related to this classical concept:
(i) The Levi property follows from the - formally weaker - assumption that every increasing net that has a minimal upper bound is norm convergent. This motivates a discussion about in which sense the Levi property resembles the notion of order continuous norm from Banach lattice theory.
(ii) If $X$ is separable and has normal cone, then the assumption that every increasing order bounded sequence has a supremum implies the Levi property. This generalizes a classical result about Banach lattices, but requires new ideas since one cannot work with disjoint sequences in the proof.
(iii) A version of Dini's theorem for ordered Banach spaces that is more general than what is typically stated in the literature. We use this to derive a sufficient condition for the space of all compact operators between two Banach lattices to have the Levi property.
(iv) Dini's theorem never holds on reflexive ordered Banach spaces with non-normal cone - i.e., on such a space one can always find an increasing sequence that converges weakly but not in norm.
We illustrate our results by various examples and counterexamples and pose four open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Increasing sequences in ordered Banach spaces -- new theorems and open problems Glück, Jochen Functional Analysis 46B40 An ordered Banach space $X$ is said to have the Levi property or to be regular if every increasing order bounded net (equivalently, sequence) is norm convergent. We prove four theorems related to this classical concept: (i) The Levi property follows from the - formally weaker - assumption that every increasing net that has a minimal upper bound is norm convergent. This motivates a discussion about in which sense the Levi property resembles the notion of order continuous norm from Banach lattice theory. (ii) If $X$ is separable and has normal cone, then the assumption that every increasing order bounded sequence has a supremum implies the Levi property. This generalizes a classical result about Banach lattices, but requires new ideas since one cannot work with disjoint sequences in the proof. (iii) A version of Dini's theorem for ordered Banach spaces that is more general than what is typically stated in the literature. We use this to derive a sufficient condition for the space of all compact operators between two Banach lattices to have the Levi property. (iv) Dini's theorem never holds on reflexive ordered Banach spaces with non-normal cone - i.e., on such a space one can always find an increasing sequence that converges weakly but not in norm. We illustrate our results by various examples and counterexamples and pose four open problems. |
| title | Increasing sequences in ordered Banach spaces -- new theorems and open problems |
| topic | Functional Analysis 46B40 |
| url | https://arxiv.org/abs/2409.19768 |