Operators defined on L_1 which "nowhere" attain their norm - Mykhaylyuk, V. V.; Krasikova, I. V.; Popov, M. M. (Methods of Functional Analysis and Topology, 2010)
Let E be either _1 of L_1. We consider E-unattainable continuous linear operators T from L_1 to a Banach space Y, i.e., those operators which do not attain their norms on any subspace of L_1 isometric to E. It is not hard to see that if T: L_1 → Y is _1-unattainable then it is also L_1-unattainable. We find some equivalent conditions for an operator to be _1-unattainable and construct two operators, first _1-unattainable and second L_1-unattainable but not _1-unattainable. Some open problems remain unsolved.