Collective order boundedness of sets of operators between ordered vector spaces
Eduard Emelyanov, Nazife Erkursun-Ozcan, Svetlana Gorokhova
TL;DR
This work develops a unified framework for collective boundedness and continuity properties of families of operators between ordered vector spaces. It proves that, when the domain is Archimedean with a generating cone, collectively order continuous operator families are collectively order bounded, and that collectively order-to-norm bounded families from an ordered Banach space with a closed generating cone to a normed space are norm bounded, yielding continuity of individual order-to-norm bounded operators. The paper also establishes precise links between ru-continuity and order boundedness, and extends these ideas to operator semigroups, showing conditions under which lco-/lcru-/lcon-bounded semigroups are (weakly) compact or converge to projections, with mean ergodicity type results and concrete examples illustrating the necessity of normality/closed-cone assumptions for certain implications.
Abstract
It is proved that: each collectively order continuous set of operators from an Archimedean OVS with a generating cone to an OVS is collectively order bounded; and each collectively order to norm bounded set of operators from an ordered Banach space with a closed generating cone to a normed space is norm bounded. Several applications to commutative operator semigroups on ordered vector spaces are given.