[Paper Review] Some characterizations of almost limited sets and applications

Nabil Machrafi, Aziz Elbour+1 more|arXiv (Cornell University)|Dec 10, 2013

Advanced Banach Space Theory8 references19 citations

TL;DR

This paper introduces and characterizes almost limited sets in Banach lattices, generalizing prior work by Chen et al. It establishes equivalent conditions for the wDP* property and defines almost limited operators—those mapping the unit ball into almost limited sets. The key contribution is a characterization showing that in σ-Dedekind complete Banach lattices with order continuous norm, every almost limited operator is compact, and the lattice operations on the dual are weak* sequentially continuous if and only if all almost limited operators are limited.

ABSTRACT

Recently, J.X. Chen et al. introduced and studied the class of almost limited sets in Banach lattices. In this paper we establish some characterizations of almost limited sets in Banach lattices (resp. wDP* property of Banach lattices), that generalize some results obtained by J.X. Chen et al.. Also, we introduce and study the class of the almost limited operators, which maps the closed unit bull of a Banach space to an almost limited subset of a Banach lattice. Some results about the relationship between the class of almost limited operators and that of limited (resp. L- and M-weakly compact, resp. compact) operators are presented.

Motivation & Objective

• To generalize characterizations of almost limited sets in Banach lattices beyond those established by Chen et al.
• To define and study the class of almost limited operators, which map the unit ball of a Banach space into almost limited subsets of a Banach lattice.
• To investigate the relationship between almost limited operators and other operator classes, including limited, L-weakly compact, M-weakly compact, and compact operators.
• To establish conditions under which almost limited operators are compact or limited, particularly in σ-Dedekind complete Banach lattices with order continuous norm.

Proposed method

• Introduces the concept of almost limited sets in Banach lattices as norm-bounded sets where every disjoint weak* null sequence in the dual converges uniformly to zero.
• Uses the notion of almost order bounded sets and their approximation by limited or almost limited sets to extend characterizations beyond order intervals.
• Applies the property (d): if (fn) is a disjoint weak* null sequence in E*, then |fn| → 0 weak* in E*.
• Defines almost limited operators as those for which T(BX) is an almost limited subset of E.
• Employs the adjoint operator T* to analyze convergence of sequences and derive contradictions when assuming failure of weak* sequential continuity.
• Utilizes the structure of ℓ¹ and the representation of operators via series to construct counterexamples and prove equivalence results.

Experimental results

Research questions

• RQ1 Under what conditions on a Banach lattice E is every almost order bounded subset almost limited?
• RQ2 When is every almost limited operator between Banach lattices necessarily limited?
• RQ3 What is the relationship between almost limited operators and compact, M-weakly compact, or L-weakly compact operators?
• RQ4 How does the weak* sequential continuity of lattice operations on E* relate to the properties of almost limited operators?
• RQ5 What characterizes Banach lattices in which every almost limited operator is compact?

Key findings

• A Banach lattice E has the property (d) if and only if every almost order bounded subset of E is almost limited.
• The lattice operations of E* are weak* sequentially continuous if and only if every almost order bounded subset of E is limited.
• In a σ-Dedekind complete Banach lattice with order continuous norm, every almost limited operator is compact.
• For a σ-Dedekind complete Banach lattice E, the lattice operations of E* are weak* sequentially continuous if and only if every almost limited operator from ℓ¹ into E is limited.
• An order bounded M-weakly compact operator T: E → F is almost limited if E* has weak* sequentially continuous lattice operations or F has property (d).
• An order bounded operator T: E → F is almost limited if and only if fn(T(xn)) → 0 for every norm bounded disjoint sequence (xn) in E+ and every weak* null disjoint sequence (fn) in F*.