An operator $T:X\rightarrow Y$ is said to be completely continuous if $T$ maps weakly convergent sequences to norm convergent sequences. 


Let $Q: l\_{1}\rightarrow l\_{2}$ be any surjection and $J:l\_{1}\rightarrow Y$ be an isomorphic embedding. 


Question. Is there a completely continuous operator $S:Y\rightarrow l\_{2}$ such that $Q=SJ$?


Thank you!



A question on completely continuous operators
By Grothendieck's theorem, $Q$ is absolutely summing, in particular $2$-summing. As such, it has a $2$-summing extension to $Y$, by Pietsch's factorisation theorem. But $2$-summing operators are completely continuous (again by Pietsch's theorem).


