Injective Tauberian operators on $L_1$ and operators with dense range on   $\ell_\infty$
  There exist injective Tauberian operators on $L_1(0,1)$ that have dense, non closed range. This gives injective, non surjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, non complementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$. 
William B. Johnson, Amir Bahman Nasseri, Gideon Schechtman and Tomasz   Tkocz
math.FA