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

TL;DR

This paper investigates the Huijsmans-de Pagter problem, showing finite-dimensional counterexamples on specific cones and establishing that such counterexamples must include a Jordan block of size at least 3.

Contribution

The paper provides the first finite-dimensional counterexamples to the problem and links their structure to Jordan blocks of size at least 3.

Findings

Finite-dimensional counterexamples exist on the ice cream cone and polyhedral cone in R^3.

Counterexamples must contain a Jordan block of size at least 3.

The problem remains open in infinite dimensions, with known counterexamples.

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 $3$.