Norm-attaining operators between Marcinkiewicz and Lorentz spaces

Bishop and Phelps proved that the set of norm-attaining functionals on any Banach space is dense in the topological dual. After that, the study of the same kind of problems for operators was initiated by Lindenstrauss, and several general positive results were proved. It was then consistently continued for different classes of spaces including L(1)(mu) or C( K). Here a similar problem is studied in the context of classical interpolation Marcinkiewicz and Lorentz spaces, M(W)(0) and Lambda(1,upsilon), in both the real and the complex cases. We show that if w upsilon is an element of L(1) then the identity operator between these spaces is bounded, but it is not possible to approximate it by norm-attaining operators. We also prove that every compact operator from M(W)(0) to Lambda(1,upsilon), can be approximated by finite-rank norm-attaining operators.