Norm aúaining and numerical radius attaining operators

ABSTRAer. In Ihis note we discusa sorne results oit numerical radius altaining operators paralleling carlier results Oit norm attaining operatora. Eorarbitrary Banach spacesXand Y, the set of (bounded, linear) operatora from Xto Ywhose adjoints altain [heir norms is norm-dense ita [hespaee of ah operators. This theorem. due toW. Zizíer, improves an earlier result by J. Lindenstrauss on the denseness ofoperalors whose secotad adjoints attain Iheir norms, and is also related loa recent result by C. Stegall where it it assumed thai Ihe dual space r has the Radon-Nikodym property lo obtain a stronger asscrtíon. Numerical radios attaining operators behave in quite a similar way. It it also true thai the set of operatora oit ata arbitrary Banach space whose adjoints atlain Iheir numerical radii it normdense ita Ihe space of ah operators. HoweVer no exantple is known of a Banach space X such that the numenical radius auaining operaíors o,, X are nol dense. Wc can prove thai such an space A’ musí fail tite Radon-Nikodym property. The eontent of ihís paper it merely expvsitory. Complete proofs wilI be published elsewhere.