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Let $T$ be an operator from a Banach space $X$ into a Banach space $Y$ and $1\leq p<\infty$. If $ST$ is compact for any operator $S$ from $Y$ into $l_{p}$, Is $T(X)$ separable? Or under what conditions on $X,Y$, this question is true?

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No. Consider the identity operator on $\ell_r(S)$ with $p<r<\infty$ and $S$ uncountable. Or consider any weakly compact operator with non separable range into a $C(K)$ space or an $L_1$ space and use the Dunford-Pettis property of these spaces.

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  • $\begingroup$ Thanks, Bill. It seems that there are no proper conditions on $X,Y$ ensuring the question is true. $\endgroup$ – Dongyang Chen Oct 14 '15 at 1:10

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