Order structure of order-to-topological continuous operator

Let J be a vector lattice, and W be a topological vector space. An operator K: J → W is called an order-to-topological continuous operator if uα → 0 in J implies K(uα) → 0 in W for each net (uα) in J. In this study, we examine the order structure of the space of order-to-topological continuous operators in general and the order structure of order-to-norm continuous operators in particular. We study the relationships between order-to-topology continuous operators and other classes of operators, such as order weakly compact and order continuous operators. Moreover, we give solutions to two open problems posed by Jalili et al. (Order-to-topology continuous operators. Positivity 2021; 25: 1313-1322).