Gateaux derivative of B(H) norm

Abstract

We prove that for Hilbert space operators X and Y, it follows that lim t→0+ ||X+tY|| - ||X||/t=1/||X||inf ε>0 φsupφ ∈Hε||φ||=1 Re〈Yφ,Xφ〉, where H ε =E X*X ((||X|| -ε) 2 , ||X|| 2 ). Using the concept of φ-Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in B(H), and to give an easy proof of the characterization of smooth points in B(H). © 2005 American Mathematical Society.