Gateaux derivative of 𝐵(𝐻) norm

Abstract

We prove that for Hilbert space operators X X and Y Y , it follows that \[ lim t → 0 + | | X + t Y | | − | | X | | t = 1 | | X | | inf ε > 0 sup φ ∈ H ε , | | φ | | = 1 Re ⁡ ⟩ Y φ , X φ ⟩ , \lim _{t\to 0^+}\frac {||X+tY||-||X||}t=\frac 1{||X||} \inf _{\varepsilon >0}\sup _{\varphi \in H_\varepsilon ,||\varphi ||=1} \operatorname {Re}\left >Y\varphi ,X\varphi \right >, \] where H ε = E X ∗ X ( ( | | X | | − ε ) 2 , | | X | | 2 ) H_\varepsilon =E_{X^*X}((||X||-\varepsilon )^2,||X||^2) . Using the concept of φ \varphi -Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in B ( H ) B(H) , and to give an easy proof of the characterization of smooth points in B ( H ) B(H) .