Reflexive spaces and numerical radius attaining operators (Q2708474)

Let \(B\) be a Banach space with dual \(B^{\ast}\). The numerical range of bounded linear operator \(T\) on \(B\) is the set of scalars \(V(T) := \{ f( Tx): f(x)=1\), \(x\in B\), \(f\in B^{\ast}\), \(\|x\|=\|f\|=1\}\). The number \(v(T):=\sup\{|\lambda|: \lambda\in V(T)\}\) is called the numerical radius of \(T\). The well-known James theorem can be stated in terms of operators as follows: a Banach space is reflexive if, and only if, each rank-one operator attains the norm. NEWLINENEWLINENEWLINEThis paper deals with a version of James' theorem for numerical radius. Earlier in [Bull. Lond. Math. Soc. 31, No. 1, 67-74 (1999; Zbl 0922.47004)] the authors proved that a Banach space is reflexive provided that each rank-one operator on it attains its numerical radius. Here they give an easier proof of this result in the case that Banach space is separable. In addition, certain reverse implication in James' theorem for numerical radius are found.