Necessary and sufficient conditions for 𝑛-times Fréchet differentiability on 𝒮^{𝓅}, 1<𝓅<∞

Let 1 &gt; p &gt; ∞ 1&gt;p&gt;\infty and let n ≥ 1 n\geq 1 . It is proved that a function f : R → C f:\mathbb {R}\to \mathbb {C} is n n -times Fr&#233;chet differentiable on S p \mathcal {S}^p at every self-adjoint operator if and only if f f is n n -times differentiable, f ′ , f , … , f ( n ) f’,f,\ldots ,f^{(n)} are bounded and f ( n ) f^{(n)} is uniformly continuous.