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DOMINATION BY POSITIVE WEAK$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}^{*}$ DUNFORD–PETTIS OPERATORS ON BANACH LATTICES

Part of: Normed linear spaces and Banach spaces; Banach lattices Special classes of linear operators

Published online by Cambridge University Press:  05 June 2014

JIN XI CHEN ,
ZI LI CHEN  and
GUO XING JI
JIN XI CHEN*
Affiliation:
Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China email jinxichen@home.swjtu.edu.cn
ZI LI CHEN
Affiliation:
Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China email zlchen@home.swjtu.edu.cn
GUO XING JI
Affiliation:
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China email gxji@snnu.edu.cn
*
✉jinxichen@home.swjtu.edu.cn
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Abstract

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Recently, H’michane et al. [‘On the class of limited operators’, Acta Math. Sci. (submitted)] introduced the class of weak$^*$ Dunford–Pettis operators on Banach spaces, that is, operators which send weakly compact sets onto limited sets. In this paper, the domination problem for weak$^*$ Dunford–Pettis operators is considered. Let $S, T:E\to F$ be two positive operators between Banach lattices $E$ and $F$ such that $0\leq S\leq T$. We show that if $T$ is a weak$^{*}$ Dunford–Pettis operator and $F$ is $\sigma $-Dedekind complete, then $S$ itself is weak$^*$ Dunford–Pettis.

Keywords

limited setdomination propertyweak∗ Dunford–Pettis operatorpositive operatorBanach lattice

MSC classification

Secondary: 46B42: Banach lattices 47B65: Positive operators and order-bounded operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 311 - 318
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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