Order-to-topology continuous operators

An operator T from vector lattice E into topological vector space (F,tau) is said to be order-to-topology continuous whenever x(alpha)->(0) 0 implies Tx(alpha)->(tau)0 for each (x(alpha))(alpha)subset of E. The collection of all order-to-topology continuous operators will be denoted by L-o tau(E,F). In this paper, we will study some properties of this new class of operators. We will investigate the relationships between order-to-topology continuous operators and others classes of operators such as order continuous, order weakly compact and b-weakly compact operators. Under some sufficient and necessary conditions we show that the adjoint of order-to-norm continuous operators is also order-to-norm continuous and vice verse.