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!