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: (i) $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^p$-differentiable at every bounded self-adjoint operator if and only if $f\in C^n(\mathbb{R})$; (ii) if $f^{\prime },\dots ,f^{(n-1)}\in C_b(\mathbb{R})$ and $f^{(n)}\in C_0(\mathbb{R})$, then $f$ is $n$ times continuously Fréchet ${\mathcal{S}}^p$-differentiable at every self-adjoint operator; (iii) if $f^{\prime },\dots ,f^{(n)}\in C_b(\mathbb{R})$, 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, $1\le q<\infty$, 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, Gâteaux ${\mathcal{S}}^2$-differentiability of $f\in C^n(\mathbb{R})$ with $f^{\prime },\dots ,f^{(n)}\in C_b(\mathbb{R})$, 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.

to appear in J. Inst. Math. Jussieu