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

Abstract

Let $1<p<\infty$ and let $n\ge 1$. 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, $f^{\prime },f^{\prime \prime },\dots ,f^{(n)}$ are bounded and $f^{(n)}$ is uniformly continuous.

Accepted for publication in the Proccedings of the American Mathematical Society