Strongly Order Continuous Linear Operators On Banach Lattice

In Banach lattice and operator theory, the properties of spaces and operators have been researched more widely. In this paper we mainly devote to investigate the properties of the strongly order continuous linear operators, the paper consists of three parts.In the first part, we first discuss the relationship among strong order convergence, relative uniform convergence and norm convergence. According to some known results concerning the weak order convergence, then the weak order convergence is strong order convergence while the net is relatively uniformly Cauchy net. Second, it has investigated the controllability of the strongly order continuous linear operators and given several equivalent conditions for the strongly order continuous operators, then, it has proved that the weak and strong order continuity do indeed concide when the range space was Dedekind complete. At last, we discuss the equivalent characterizations of strongly order continuous operators, order bounded operators and lattice homomorphism operators.In the second part, we discuss the direct sum of the space of strongly order continuous operators and confirm that the operator norm limits of the strongly order continuous operators are still strongly order continuous with respect to the order bounded norm, regular norm and operator norm respectly.In the third part, we first discuss the strong order convergence, weak order convergence and relative uniform convergence on the sequence space co(A), got the three convergences are equivalent. Second, it has proved that the positive operators from co(A) to any norm Riesz space are strongly order continuous through the special characterization of the space c0(A). Finally, we investigate the strong order convergence and weak order convergence on function space φ(L).