Differentiability of the operator norm on ${\mathrm{\ell }}_p$ spaces

In this paper, we present a characterization of strong subdifferentiability of the norm of bounded linear operators on ${\mathrm{\ell }}_p$ spaces, $1\le p<\mathrm{\infty }$. Furthermore, we prove that the set of all bounded linear operators in $B({\mathrm{\ell }}_p,{\mathrm{\ell }}_q)$ for which the norm of $B({\mathrm{\ell }}_p,{\mathrm{\ell }}_q)$ is strongly subdifferentiable is dense in $B({\mathrm{\ell }}_p,{\mathrm{\ell }}_q)$. Additionally, we present a characterization of Frechet differentiability of the norm of bounded linear operators from ${\mathrm{\ell }}_p$ to ${\mathrm{\ell }}_q$, where $1<p,q<\mathrm{\infty }$. Applying this result, we will show that the Frechet differentiability and the Gateaux differentiability of the norm of bounded linear operators on ${\mathrm{\ell }}_p$ spaces coincide, extending a known theorem regarding the operator norm on Hilbert spaces.