õrder-norm continuous operators and ˜order weakly compact ˜operators

Let e be a sublattice of a vector lattice f. a continuous operator t from e into a normed vector space x is said to be õrder-norm continuous if xα fo → 0 implies t(xα) ∥.∥ → 0 for every (xα)α∈a ⊆ e. this paper aims to investigate the properties of this new class of operators and explore their relationships with existing classifications of operators. we introduce a new class of operators called õrder weakly compact operators. a continuous operator t : e → x is considered õrder weakly compact if t(a) in x is a relatively weakly compact set for every f o-bounded a ⊆ e. in this manuscript, we examine various properties of this class of operators and explore their connections with õrder-norm continuous operators.