A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910460853354496 |
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| author | Badea, Catalin Glück, Jochen |
| author_facet | Badea, Catalin Glück, Jochen |
| contents | A classical problem posed in 1992 by Huijsmans and de Pagter asks whether, for every positive operator $T$ on a Banach lattice with spectrum $σ(T) = \{1\}$, the inequality $T \ge \operatorname{id}$ holds true. While the problem remains unsolved in its entirety, a positive solution is known in finite dimensions. In the broader context of ordered Banach spaces, Drnovšek provided an infinite-dimensional counterexample. In this note, we demonstrate the existence of finite-dimensional counterexamples, specifically on the ice cream cone and on a polyhedral cone in $\mathbb{R}^3$. On the other hand, taking inspiration from the notion of $m$-isometries, we establish that each counterexample must contain a Jordan block of size at least $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03046 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces Badea, Catalin Glück, Jochen Functional Analysis Spectral Theory 47B65 A classical problem posed in 1992 by Huijsmans and de Pagter asks whether, for every positive operator $T$ on a Banach lattice with spectrum $σ(T) = \{1\}$, the inequality $T \ge \operatorname{id}$ holds true. While the problem remains unsolved in its entirety, a positive solution is known in finite dimensions. In the broader context of ordered Banach spaces, Drnovšek provided an infinite-dimensional counterexample. In this note, we demonstrate the existence of finite-dimensional counterexamples, specifically on the ice cream cone and on a polyhedral cone in $\mathbb{R}^3$. On the other hand, taking inspiration from the notion of $m$-isometries, we establish that each counterexample must contain a Jordan block of size at least $3$. |
| title | A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces |
| topic | Functional Analysis Spectral Theory 47B65 |
| url | https://arxiv.org/abs/2405.03046 |