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

Fu Zhang, Hanhan Shen () and Zili Chen
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Fu Zhang: School of Mathematics, Xipu Campus, Southwest Jiaotong University, Chengdu 611756, China
Hanhan Shen: School of Mathematics, Xipu Campus, Southwest Jiaotong University, Chengdu 611756, China
Zili Chen: School of Mathematics, Xipu Campus, Southwest Jiaotong University, Chengdu 611756, China

Mathematics, 2023, vol. 11, issue 12, 1-16

Abstract: 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 L 0 ( 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 ′ .

Keywords: Banach lattices; property ( h ); order weakly compact operators; order-to-norm continuous operators; ? -order continuous operators (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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