On Riesz spaces with $b$-property and $b$-weakly compact operators

An operator $T\colon E\to X$ between a Banach lattice $E$ and a Banach space $X$ is called $b$-weakly compact if $T(B)$ is relatively weakly compact for each $b$-bounded set $B$ in $E$. We characterize $b$-weakly compact operators among $o$-weakly compact operators. We show summing operators are $b$-weakly compact and discuss relation between Dunford–Pettis and $b$-weakly compact operators. We give necessary conditions for $b$-weakly compact operators to be compact and give characterizations of $K\!B$-spaces in terms of $b$-weakly compact operators defined on them.