# Higher order differentiability of operator functions in Schatten norms

## TL;DR

This paper characterizes the higher order differentiability of operator functions in Schatten norms, linking smoothness of scalar functions to operator differentiability, and extends previous results to unbounded operators and broader function classes.

## Contribution

It provides new necessary and sufficient conditions for higher order Schatten differentiability of operator functions, extending existing theories to unbounded operators and wider function classes.

## Key findings

- Characterizes when scalar functions induce higher order Schatten differentiable operator functions.
- Extends differentiability results to unbounded operators and broader function classes.
- Provides explicit formulas for Fréchet and Gâteaux derivatives.

## Abstract

We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert space: (i) $f$ is $n$ times continuously Fr\'{e}chet $\mathcal{S}^p$-differentiable at every bounded self-adjoint operator if and only if $f\in C^n(\mathbb{R})$; (ii) if $f',\ldots,f^{(n-1)}\in C_b(\mathbb{R})$ and $f^{(n)}\in C_0(\mathbb{R})$, then $f$ is $n$ times continuously Fr\'{e}chet $\mathcal{S}^p$-differentiable at every self-adjoint operator; (iii) if $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, then $f$ is $n-1$ times continuously Fr\'{e}chet $\mathcal{S}^p$-differentiable and $n$ times G\^{a}teaux $\mathcal{S}^p$-differentiable at every self-adjoint operator. We also prove that if $f\in B_{\infty1}^n(\mathbb{R})\cap B_{\infty1}^1(\mathbb{R})$, then $f$ is $n$ times continuously Fr\'{e}chet $\mathcal{S}^q$-differentiable, $1\le q<\infty$, at every self-adjoint operator. These results generalize and extend analogous results of [10] to arbitrary $n$ and unbounded operators as well as substantially extend the results of [2,4,19] on higher order $\mathcal{S}^p$-differentiability of $f$ in a certain Wiener class, G\^{a}teaux $\mathcal{S}^2$-differentiability of $f\in C^n(\mathbb{R})$ with $f',\ldots,f^{(n)}\in C_b(\mathbb{R})$, and G\^{a}teaux $\mathcal{S}^q$-differentiability of $f$ in the intersection of the Besov classes $B_{\infty1}^n(\mathbb{R})\cap B_{\infty1}^1(\mathbb{R})$. As an application, we extend $\mathcal{S}^p$-estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fr\'{e}chet differentials and G\^{a}teaux derivatives.