Higher order $\Sc^2$-differentiability and application to Koplienko trace formula

DOI10.1016/J.JFA.2018.09.005arXiv1712.10289MaRDI QIDQ6295839

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

Publication date: 29 December 2017

Abstract: Let $A$ be a selfadjoint operator in a separable Hilbert space, $K$ a selfadjoint Hilbert-Schmidt operator, and $fin{C}^n(mathbbR)$. We establish that $varphi(t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $mathbbR$ in the Hilbert-Schmidt norm, provided either $A$ is bounded or the derivatives $f^{(i)}$, $i=1,ldots,n$, are bounded. As an application of the second order $S{c}^2$-differentiability, we extend the Koplienko trace formula from the Besov class $B_{infty1}^2(R)$ to functions $f$ for which the divided difference $f^{[2]}$ admits a certain Hilbert space factorization.

Mathematics Subject Classification ID

Perturbation theory of linear operators (47A55) Noncommutative function spaces (46L52) Transformers, preservers (linear operators on spaces of linear operators) (47B49)

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