\tilde{O}rder-norm continuous operators and \tilde{o}rder 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 \tilde{o}rder-norm continuous if x_\alpha\xrightarrow{Fo}۰ implies T(x_\alpha)\xrightarrow{\Vert.\Vert}۰ for every (x_{\alpha})_{\alpha \in A}\subseteq 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 \tilde{o}rder weakly compact operators. A continuous operator T: E \rightarrow X is considered \tilde{o}rder weakly compact if T(A) in X is a relatively weakly compact set for every Fo-bounded A\subseteq E. In this manuscript, we examine various properties of this class of operators and explore their connections with \tilde{o}rder-norm continuous operators.

کلیدواژه ها:

Vector lattice ، property (F) ، \tilde{o}-convergence ، order-to-norm continuous operator ، \tilde{o}rder-norm continuous operator ، \tilde{o}rder weakly compact