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Differentiability of the operator norm on $\ell _p$
spaces
- Part of
-
- Journal:
- Canadian Journal of Mathematics / Volume 78 / Issue 5 / October 2026
- Published online by Cambridge University Press:
- 16 December 2024, pp. 1405-1424
- Print publication:
- October 2026
-
- Article
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- You have access
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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.
