$\mathcal{S}^2$-differentiability and extension of the Koplienko trace formula

Clément Coine, Christian Le Merdy, Anna Skripka, Fedor Sukochev

Published 2017-12-29Version 1

Let $A$ be a selfadjoint operator in a separable Hilbert space, $K$ a selfadjoint Hilbert-Schmidt operator, and $f\in C^n(\mathbb{R})$. We establish that $\varphi(t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $\mathbb{R}$ in the Hilbert-Schmidt norm, provided either $A$ is bounded or the derivatives $f^{(i)}$, $i=1,\ldots,n$, are bounded. As an application, we extend the Koplienko trace formula \begin{equation*} \mathrm{Tr}\Big(f(A+K) - f(A) - \dfrac{d}{dt}f(A+tK)_{|t=0}\Big) = \int_{\mathbb{R}} f"(t) \eta(t) \, dt \end{equation*} from the Besov class $B_{\infty1}^2(\mathbb{R})$ to functions $f$ for which the divided difference $f^{[2]}$ admits a certain Hilbert space factorization.