Gateaux derivative of $B(H)$ norm

Abstract:

We prove that for Hilbert space operators $X$ and $Y$, it follows that \[ \lim _{t\to 0^+}\frac {||X+tY||-||X||}t=\frac 1{||X||} \inf _{\varepsilon >0}\sup _{\varphi \in H_\varepsilon ,||\varphi ||=1} \operatorname {Re}\left <Y\varphi ,X\varphi \right >,\] where $H_\varepsilon =E_{X^*X}((||X||-\varepsilon )^2,||X||^2)$. Using the concept of $\varphi$-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 $B(H)$.