A Version of James' Theorem for Numerical Radius
Marı́a D. Acosta, Manuel Ruiz Galán · Bulletin of the London Mathematical Society · 1999
We prove that if all the rank-one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank-one operator not attaining the numerical radius. 1991 Mathematics Subject Classification 47A12, 46B10.