On automatic boundedness of some operators in ordered Banach spaces
Eduard Emelyanov
TL;DR
The paper investigates automatic boundedness of order-to-weak continuous operators from ordered Banach spaces to normed spaces. It links Lebesgue-type continuity with order-to-(weak) norm continuity and leverages closed generating normal cones to derive automatic boundedness, yielding inclusions among operator classes and equalities under norm-order-continuity assumptions. A key contribution is proving that every $\sigma$-w-Lebesgue operator from an ${\rm OBS}$ with a closed generating normal cone to a ${\rm NS}$ is bounded, and it shows that the various Lebesgue- and wLebesgue-type operator classes are norm-closed subspaces of $\mathcal{L}(X,Y)$ with several inclusions becoming equalities under additional continuity assumptions. The results extend automatic boundedness phenomena from lattice contexts to general ordered Banach spaces and clarify when order-to-weak continuity guarantees boundedness.
Abstract
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.