Differentiability of the operator norm on $\ell _p$ spaces

Abstract 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 Fréchet 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 Fréchet 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.