Property (h) of Banach Lattice and Order-to-Norm Continuous Operators

• Published in: Mathematics (Basel) Vol. 11; no. 12; p. 2747
• Main Authors: Zhang, Fu, Shen, Hanhan, Chen, Zili
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.06.2023
• Subjects: Banach lattices; Banach spaces; Food science; Lattice theory; Lattices; Mathematical research; Norms; Operator theory; Operators (mathematics); order weakly compact operators; order-to-norm continuous operators; property (h); Sequences; σ-order continuous operators; China
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11122747

In this paper, we introduce the property (h) on Banach lattices and present its characterization in terms of disjoint sequences. Then, an example is given to show that an order-to-norm continuous operator may not be σ-order continuous. Suppose T:E→F is an order-bounded operator from Dedekind σ-complete Banach lattice E into Dedekind complete Banach lattice F. We prove that T is σ-order-to-norm continuous if and only if T is both order weakly compact and σ-order continuous. In addition, if E can be represented as an ideal of L0(μ), where (Ω,Σ,μ) is a σ-finite measure space, then T is σ-order-to-norm continuous if and only if T is order-to-norm continuous. As applications, we extend Wickstead’s results on the order continuity of norms on E and E′.