COPIES OF $c_0$ AND $\ell_\infty$ INTO A REGULAR OPERATOR SPACE

Abstract

For an Orlicz function $\varphi$ and a Banach lattice $X$, let $\ell_\varphi$ denote the Orlicz sequence space associated to $\varphi$, ${\mathcal L}^r(\ell_\varphi, X)$ denote the space of regular operators from $\ell_\varphi$ to $X$, and ${\mathcal K}^r(\ell_\varphi, X)$ denote the linear span of positive compact operators from $\ell_\varphi$ to $X$. In this paper, we show that if $\varphi$ and its complementary function $\varphi^\ast$ satisfy the $\Delta_2$-condition, then (a) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $\ell_\infty$; and (b) ${\mathcal K}^r(\ell_\varphi, X)$ contains no copy of $c_0$ if and only if ${\mathcal L}^r(\ell_\varphi, X)$ contains no copy of $\ell_\infty$ if and only if $X$ contains no copy of $c_0$ and each positive linear operator from $\ell_\varphi$ to $X$ is compact.