Geometry of \({\mathfrak J}\)-spaces and properties of reversing operators (Q5951063)

The authors study the existence of linear and continuous functionals and operators on a Banach space which attain their norms on the unit sphere. In particular, a normed space \(N\) is said to be a James space iff for any \(f\in N^*\) there is some \(x_0\in N\), \(\|x_0\|=1\), such that \(f(x_0)= \|f\|\). The paper contains connections of these properties with reflexivity of a Banach space, and uniform positiveness of a subspace in a Krein space. The results are used to get strict contractions in James spaces which are operators of norm \(<1\).