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arXiv · 2405.03046

A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces

Catalin Badea·Jochen Glück
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Abstract

A classical problem posed in 1992 by Huijsmans and de Pagter asks whether, for every positive operator $T$ on a Banach lattice with spectrum $\sigma(T) = \{1\}$, 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, Drnov\v{s}ek 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 $3$.

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Catalin Badea, Jochen Glück. 2024-05-05. A note on the Huijsmans-de Pagter problem on finite dimensional ordered vector spaces. https://arxiv.org/abs/2405.03046

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