Some results about unbounded convergences in Banach lattices

Author
Omid Zabeti
University of Sistan and Baluchestan
Abstract
Introduction

Suppose E is a Banach lattice. A net (xα) in E is said to be unbounded absolute weak convergent (uaw-convergent, for short) to x∈E provided that the net (xα-x˄u) convergences to zero, weakly, whenever u∈E+. 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.

Conclusion

The following main conclusions were drawn from this research.

Theorem 2. Theorem 4. Proposition 10. Theorem 11.
Keywords
Unbounded absolute weak convergence
ideal
sublattice
closedness
unbounded norm convergence
strongly continuous operator
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Mathematical Research
Volume 8, Issue 2 - Serial Number 21
Spring 2022
Pages 129-137