From norm derivatives to orthogonalities in Hilbert<i>C</i>*-modules

Let (X,⟨⋅,⋅⟩) be a Hilbert C∗-module over C∗-algebra A and let S(A) the set of states on A. In this paper, we first compute norm derivative for nonzero elements x y X as follows: limt→0+‖x+ty‖−‖x‖t=1‖x‖max{Reφ(⟨x,y⟩):φ∈S(A),φ(⟨x,x⟩)=‖x‖2}.We then apply it to characterize different concepts orthogonality in X. particular, present simpler proof classical characterization Birkhoff–James C∗-modules. Moreover, some generalized Daugavet equation B(H) all bounded linear operators acting space H is solved.