TITLE: A question on compact operators with domain $l_{p}$
QUESTION [5 upvotes]: 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 $R:l_{q}\rightarrow X$?

REPLY [4 votes]: 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.