On $b$-order Dunford-Pettis operators and the $b$-$AM$-compactness property

Abstract

‎In this paper‎, ‎we introduce $b$-order Dunford-Pettis operators‎, ‎that is‎, ‎an operator $T$ from a normed Riesz space $E$ into a Banach space $X$ is called $b$-order Dunford-Pettis if $T$ carries each $b$-order bounded subset of $E$ into a Dunford-Pettis subset of $X$‎, ‎and we investigate its relationship with order Dunford-Pettis operators‎. ‎We also introduce the $b$-$AM$-compactness property for a Banach lattice and we study some of its topological properties and its relationships with the Dunford-Pettis property‎. ‎We show that the identity operator of Banach lattice $E$ is $b$-order Dunford-Pettis if and only if $E$ has the $b$-$AM$-compactness property‎.
‎ We characterize Banach lattices $E$ and $F$ on which the adjoint of each operator from $E$ into $F$ which is $b$-order Dunford-Pettis and weak Dunford-Pettis‎, ‎is Dunford-Pettis‎.