Volume 8, Issue 2 (Vol. 8,No. 2, 2022)                   mmr 2022, 8(2): 129-137 | Back to browse issues page

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Zabeti O. Some results about unbounded convergences in Banach lattices. mmr 2022; 8 (2) :129-137
URL: http://mmr.khu.ac.ir/article-1-3090-en.html
University of Sistan and Baluchestan , o.zabeti@gmail.com
Abstract:   (1089 Views)
Suppose E is a Banach lattice. A net (xα) in E is said to be unbounded absolute weak convergent (uaw-convergent, for short) to xE provided that the net (xα-x˄u) convergences to zero, weakly, whenever uE+. In this note, we further investigate unbounded absolute weak convergence in E. We show that this convergence is stable under passing to and   from ideals and sublattices. Compatible with un-convergenc, we show that uaw-convergence is topological, which means that E with uaw-topology forms a topological vector space. We consider some closedness properties for this type of convergence. Some examples  are given to make the context more understandable. Finally, we introduce the notion of strongly continuous operators between Banach lattices and investigate some properties about them. Specially, we characterize Banach lattices with a strong unit in tems of this type of operators.
Material and methods
In this paper, we combine the order structure and the norm structure in a Banach lattice to consider the unbounded convergences in the category of all Banach lattices.
Results and discussion
We shall show the following main results.
1. The uaw-convergence in a Banach lattice is topological.
2. In an order continuous Banach lattice, uaw-convergence is stable under passing to and from sublattices and ideals.
3. Introduce strongly continuous operators between Banach lattices and investigate some properties of them.
The following main conclusions were drawn from this research.
Theorem 2. Theorem 4.  Proposition 10. Theorem 11.
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Type of Study: S | Subject: Mat
Received: 2020/05/9 | Revised: 2022/11/16 | Accepted: 2020/08/3 | Published: 2022/05/21 | ePublished: 2022/05/21

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