HIGHER ORDER DIFFERENTIABILITY OF OPERATOR FUNCTIONS IN SCHATTEN NORMS

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:

1. (i) $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every bounded self-adjoint operator if and only if $f\in C^{n}(\mathbb{R})$;

2. (ii) if $f^{\prime },\ldots ,f^{(n-1)}\in C_{b}(\mathbb{R})$ and $f^{(n)}\in C_{0}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator;

3. (iii) if $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, then $f$ is $n-1$ times continuously Fréchet ${\mathcal{S}}^{p}$-differentiable and $n$ times Gâteaux ${\mathcal{S}}^{p}$-differentiable at every self-adjoint operator.

We also prove that if $f\in B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^{q}$-differentiable, $1\leqslant q<\infty$, at every self-adjoint operator. These results generalize and extend analogous results of Kissin et al. (Proc. Lond. Math. Soc. (3)108(3) (2014), 327–349) to arbitrary $n$ and unbounded operators as well as substantially extend the results of Azamov et al. (Canad. J. Math.61(2) (2009), 241–263); Coine et al. (J. Funct. Anal.; doi:10.1016/j.jfa.2018.09.005); Peller (J. Funct. Anal.233(2) (2006), 515–544) on higher order ${\mathcal{S}}^{p}$-differentiability of $f$ in a certain Wiener class, Gâteaux ${\mathcal{S}}^{2}$-differentiability of $f\in C^{n}(\mathbb{R})$ with $f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$, and Gâteaux ${\mathcal{S}}^{q}$-differentiability of $f$ in the intersection of the Besov classes $B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$. 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 Fréchet differentials and Gâteaux derivatives.