Differentiability of the operator norm on $\ell_p$ spaces

In this paper, we present a characterization of strong subdifferentiability of the norm of bounded linear operators on $\ell_p$ spaces, $1\leq p<\infty$.

Furthermore, we prove that the set of all bounded linear operators in ${B}(\ell_p, \ell_q)$ for which the norm of ${B}(\ell_p, \ell_q)$ is strongly subdifferentiable is dense in ${B}(\ell_p, \ell_q)$.

Additionally, we present a characterization of Frechet differentiability of the norm of bounded linear operators from $\ell_p$ to $\ell_q$, where $1 < p, q < \infty$.

Applying this result, we will show that the Frechet differentiability and the Gateaux differentiability of the norm of bounded linear operators on $\ell_p$ spaces coincide, extending a known theorem regarding the operator norm on Hilbert spaces.