On automatic boundedness of some operators in ordered Banach spaces

We study order-to-weak continuous operators from an ordered Banach space to a normed space. It is proved that under rather mild conditions every order-to-weak continuous operator is bounded.

💡 Research Summary

**
The paper investigates automatic boundedness of operators acting from ordered Banach spaces (OBS) into normed spaces (NS). The central objects are order‑to‑weak continuous operators, also called σ‑order‑to‑weak (or σ‑w‑Lebesgue) operators, and their relationship with Lebesgue‑type and order‑to‑norm continuous operators. After fixing notation, the authors recall that for positive operators between a normal ordered normed space Y and an ordered vector space X, being Lebesgue is equivalent to being order‑to‑norm continuous (Lemma 1.2) and that a positive order‑continuous operator is Lebesgue if and only if it is weak‑Lebesgue (Lemma 1.3).

The first substantial result, Lemma 2.1, shows that if X is a normal OBS and Y an arbitrary NS, any σ‑w‑Lebesgue operator T: X→Y must be order‑to‑norm bounded; that is, T maps every order interval