Some Notes on b -Weakly Compact Operators
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Some Notes on b-Weakly Compact Operators Masoumeh Mousavi Amiri1 · Kazem Haghnejad Azar1 Received: 8 January 2019 / Revised: 23 November 2019 / Accepted: 5 December 2019 © Iranian Mathematical Society 2019
Abstract The main aim of this paper is studying the family Wb (E, F) of b-weakly compact operators between two Banach lattices. For an order dense sublattice G of a vector lattice E, if T : G → F is a b-weakly compact operator between two Banach lattices, then T ∈ Wb (E, F) whenever the norm of E is order continuous and T : E → F is a positive operator. We also investigate the relationship between Wb (E, F) and some (1) (2) other classes of operators like L c (E, F) and L c (E, F). Keywords Banach lattice · Order continuous norm · b-weakly compact operator Mathematics Subject Classification 46B42 · 47B60
1 Introduction and Preliminaries An operator T from a Banach lattice E to a Banach space X is said to be b-weakly compact, if the image of every b-order bounded subset of E (that is, order bounded in the topological bidual E of E) under T is relatively weakly compact. The authors in [8] proved that an operator T from a Banach lattice E into a Banach space X is bweakly compact if and only if {T xn }n is norm convergent for every positive increasing sequence {xn }n of the closed unit ball B E of E. The class of all b-weakly compact operators between E and X will be denoted by Wb (E, X ). The class of b-weakly compact operators was firstly introduced by Alpay et al. [3]. One of the interesting properties of the class of b-weakly compact operators is that it satisfies the domination property. Some more investigations on Wb (E, X ) were done by [3–5,7,8].
Communicated by Hamid Reza Ebrahimi Vishki.
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Kazem Haghnejad Azar [email protected] Masoumeh Mousavi Amiri [email protected]
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Department of Mathematics and Applications, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
123
Bulletin of the Iranian Mathematical Society
In this paper, we continue the investigation on Wb (E, X ). In the first section, we provide some prerequisites. The second section is devoted to the main results. We mainly focus on the inclusion relationship between Wb (E, X ) with some known class of operators. We also study those Banach lattices for which the modulus of an order bounded operator is b-weakly compact. 1.1 Some Basic Definitions Let E be a vector lattice. An element e > 0 in E is said to be an order unit whenever for each x ∈ E there exists a λ > 0 with |x| ≤ λe. A sequence (xn ) in a vector lattice is said to be disjoint whenever |xn | ∧ |xm | = 0 holds for n = m. A vector lattice is called Dedekind complete whenever every nonempty bounded above subset has a supremum. For an operator T : E → F between two vector lattices, we shall say that its modulus |T | exists whenever |T | := T ∨ (−T ) exists in the sense that |T | is the supremum of the set {−T , T }. An operator T : E → F between two vector lattices is called order bounded if it maps order bounded subsets of E into order bounded subs
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