Order compact and unbounded order compact operators

Abstract

We investigate properties of order compact, unbounded order compact and relatively uniformly compact operators acting on vector lattices. An operator is said to be order compact if it maps an arbitrary order bounded net to a net with an order convergent subnet. Analogously, an operator is said to be unbounded order compact if it maps an arbitrary order bounded net to a net with uo-convergent subnet. After exposing the relationships between order compact, unbounded order compact, semicompact and GAM-compact operators; we study those operators mapping an arbitrary order bounded net to a net with a relatively uniformly convergent subnet. By using the nontopological concepts of order and unbounded order convergences, we derive new results related to these classes of operators. Banach lattices can be equipped with various canonical convergence structures such as order, relatively uniform, unbounded order and unbounded norm convergences. Although some of these convergences are not topological, they share the common property that the underlying order structure plays a dominant role in deriving properties related to operators acting on these lattices. The notion of unbounded order convergence was initially introduced in [13] under the name individual convergence, and, ?uo-convergence? was proposed firstly in [7]. Recently in [4, 6, 8?11, 17], see also the references therein, further properties of various types of unbounded convergences are investigated. In the present paper, we study compactness properties of operators between vector lattices by utilizing