$\tilde{o}$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 the vector lattice $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous whenever $x_{\alpha }\stackrel{Fo}{\to }0$ implies $T{x}_{\alpha }\stackrel{\mathrm{\parallel }\mathrm{.}\mathrm{\parallel }}{\to }0$ for each $(x_{\alpha }{)}_{\alpha }\subseteq E$. Our mean from the convergence $x_{\alpha }x$ is that there exists another net $\left(y_{\alpha }\right)$ in $F$ with the same index set satisfying $y_{\alpha }\downarrow 0$ in $F$ and $\mathrm{\mid }{x}_{\alpha }-x\mathrm{\mid }\le y_{\alpha }$ for all indexes $\alpha$. In this paper, we will study some properties of this new class of operators and its relationships with some known classifications of operators. We also define the new class of operators that named $\tilde{o}$rder weakly compact operators. A continuous operator $T:E\to X$ is said to be $\tilde{o}$rder weakly compact, if $T(A)$ in $X$ is a relatively weakly compact set for each $Fo$-bounded $A\subseteq E$. In this manuscript, we study some properties of this class of operators and its relationships with $\tilde{o}$rder-norm continuous operators.