Compactness of positive semi-compact operators on Banach lattices

Abstract

An operator T from a Banach space E into a Banach lattice F is said to be semi-compact if for each e > 0, there exists some u ∈ F+ such that T (BE) ⊂ [−u, u] + eBF where BH is the closed unit ball of H = E or F and F+ = {x ∈ F : 0 ≤ x}. Contrarily to compact operators, the class of semi-compact operators satisfies the domination problem. Indeed, if S and T are operators from a Banach lattice E into F such that 0 ≤ S ≤ T and T is semi-compact, then S is semi-compact (Theorem 18.20 of [3]). It is clear that each compact operator is semi-compact but a semicompact operator is not necessary compact. In fact, the operator identity Idl∞ : l∞ −→ l∞ is semi-compact but it is not compact. Let us recall that Dodds and Fremlin ([11], Theorem 125.5) proved that if E and F are two Banach lattices such that the norms of F and of the topological dual E ′ are order continuous, then each semi-compact operator T from E into F which is AM-compact (i.e. a regular operator such that the image of each order bounded subset of E is a relatively compact subset of F ) is compact. In [5], we studied the compactness of positive Dunford-Pettis operators, and we gave several characterizations of Banach lattices on which each