Order-norm continuous operators and Order weakly compact operators

Let E be a sublattice of a vector lattice F. In this paper, we will introduce and study some properties of this new class of operators as F-order-norm continuousoperators and its relationships with some known classifications of operators. We also define the new class of operators that named order weakly compact operators. A continuous operator T : E → X is said to be F-order weakly compact, if T(A) in X is arelatively weakly compact set for each Fo-bounded A ⊆ E. In this manuscript, we study some properties of this class of operators and its relationships with F-order-norm continuous operators.