Necessary and sufficient conditions for $n$-times Fréchet differentiability on $\mathcal {S}^p,$ $1 <p<\infty$

Abstract:

Let $1<p<\infty$ and let $n\geq 1$. It is proved that a function $f:\mathbb {R}\to \mathbb {C}$ is $n$-times Fréchet differentiable on $\mathcal {S}^p$ at every self-adjoint operator if and only if $f$ is $n$-times differentiable, $f’,f'',\ldots ,f^{(n)}$ are bounded and $f^{(n)}$ is uniformly continuous.