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

Author

Listed:
  • 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)

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 ′ .

Suggested Citation

  • Fu Zhang & Hanhan Shen & Zili Chen, 2023. "Property ( h ) of Banach Lattice and Order-to-Norm Continuous Operators," Mathematics, MDPI, vol. 11(12), pages 1-16, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:12:p:2747-:d:1173396
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