Injective Tauberian Operators on $L_1$ and Operators with Dense Range on $ell_infty$

摘要

There exist injective Tauberian operators on $L_1(0,1)$ that havedense, nonclosed range. This gives injective, nonsurjectiveoperators on $ell_infty$ that have dense range. Consequently, thereare two quasi-complementary, noncomplementary subspaces of$ell_infty$ that are isometric to $ell_infty$.