In this note we discuss some results on numerical radius attaining operators paralleling earlier results on norm attaining operators.

For arbitrary Banach spaces and , the set of (bounded, linear) operators from X to Y whose adjoints attain their norms is norm-dense in the space of all operators. This theorem, due to W. Zizler, improves an earlier result by J. Lindenstrauss on the denseness of operators whose second adjoints attain their norms, and is also related to a recent result by C. Stegall where it is assumed the the dual space has the Radon-Nikodým property to obtain a stronger assertion.

Numerical radius attaining operators behave in a quite similar way. It is also true that the set of operators on an arbitrary Banach space whose adjoints attain their numerical radii is norm-dense in the space of all operators. However no example is known of a Banach space such that the numerical radius attaining operators on are not dense. We can prove that such space must fail the Radon-Nikodým property.

The content of this paper is merely expository. Complete proofs will published elsewhere.

1980 Mathematics Subject Classification (1985 revision): 46B20.