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

TL;DR

This paper establishes precise conditions under which functions are n-times Fréchet differentiable on Schatten classes, linking differentiability of functions to properties of their derivatives.

Contribution

It provides necessary and sufficient conditions for n-times Fréchet differentiability of functions on Schatten classes, connecting operator differentiability to classical function smoothness.

Findings

A function is n-times Fréchet differentiable on ${ m S}^p$ iff its derivatives up to order n are bounded.

The n-th derivative must be uniformly continuous for differentiability.

The results characterize operator differentiability in terms of classical function properties.

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 Fr\'echet 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.