Suppose that $T$ is an operator from a Banach space $X$ to a Banach space $Y$. Let $1<p<q<\infty$. If $TS$ is compact for any operator $S:l\_{p}\rightarrow X$, is $TR$ compact for any operator $R:l\_{q}\rightarrow X$? 



A question on compact operators with domain $l_{p}$
Yes. If $R:\ell\_q \to X$ and $TR$ is not compact then there is a normalized block basis $u\_n$ of the unit vector basis for $\ell\_q$ s.t. $TRu\_n$ is bounded away from zero. Let $V:\ell\_p \to \ell\_q$ be the bounded linear operator that maps $e\_n$ to $u\_n$ and set $S=RV$. Then $TS$ is not compact. 


