On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

Abstract

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

Generalizations of L- and M-weakly compact operators

“…and, as a consequence of (2), that (f n ) in X ′ converges uniformly on B X iff it converges uniformly on B X ′′ (3) under the identification of f ∈ X ′ with f ∈ (X ′ ) ′′ . By [4, Def.1.1]: a subset A ⊆ E is called b-order bounded whenever  is order bounded in E ′′ , where E → E ′′ is the natural embedding of E into its bi-dual E ′′ ; and E has bproperty, whenever every b-order bounded subset of E is order bounded.…”

Some Properties of Unbounded M -Weakly and Unbounded L -Weakly Compact Operators

“…For example, he proved that L-weakly and M-weakly compact operators are weakly compact (Teorem 5.61 in [3]). Te properties of these classifcations of operators have been investigated and extended to some general cases by some authors; see [5][6][7][8]. In this paper, we introduce an unbounded version for these classifcations of operators as unbounded L-weakly and M-weakly (in short u-Land u-M-weakly) compact operators.…”