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.
Highlights
Introduction and NotationIn this article, we denote real Banach spaces by X and Y and real Banach lattices by E and F
If T : E ⟶ F is an operator with modulus, the regular norm of T is given by kTkr = kjTjk
We show that the class of almost L-weakly compact operators satisfies the domination property
Summary
Introduction and NotationIn this article, we denote real Banach spaces by X and Y and real Banach lattices by E and F. 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. The class of compact (resp., weakly compact) operators does not satisfy the domination property [1, 2]. An operator T : E ⟶ Y is called M-weakly compact if for each norm bounded disjoint sequence ðxnÞ in E, we have kTxnk ⟶ 0.
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