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Let $1<p<\infty$ and let $n\geq 1$. It is proved that a function $f:{\mathbbR}\to {\mathbb C}$ is $n$-times Fr\'echet differentiable on ${\mathcal S}^p$ atevery 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.

Necessary and sufficient conditions for $n$-times Fr\'echet differentiability on ${\mathcal S}^p,$ $1