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Abstract

Let $E$ be an ideal of $L^0$ over a $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$. For a real Banach space $(X,\|\cdot\|_X)$ let $E(X)$ be a subspace of the space $L^0(X)$ of $\mu$-equivalence classes of strongly $\Sigma$-measurable functions $f: \Omega\to X$ and consisting of all those $f\in L^0(X)$ for which the scalar function $\|f(\cdot)\|_X$ belongs to $E$. Let $E(X)^\sim$ stand for the order dual of $E(X)$. For $u\in E^+$ let $D_u$ $(=\{f\in E(X): \|f(\cdot)\|_X\leq u\})$ stand for the order interval in $E(X)$. For a real Banach space $(Y,\|\cdot\|_Y)$ a linear operator $T: E(X)\to Y$ is said to be order-bounded whenever for each $u\in E^+$ the set $T(D_u)$ is norm-bounded in $Y$. In this paper we examine order-bounded operators $T: E(X)\to Y$. We show that $T$ is order-bounded iff $T$ is $(\tau(E(X), E(X)^\sim), \|\cdot\|_Y)$-continuous. We obtain that every weak Dunford-Pettis operator $T: E(X)\to Y$ is order-bounded. In particular, we obtain that if a Banach space $Y$ has the Dunford-Pettis property, then $T$ is order-bounded iff it is a weak Dunford-Pettis operator.