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

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, 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 \(3\).