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, $1\le p<\infty$. 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.

Published in Canadian journal of Mathematics

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

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