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

Abstract

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 $\phi (t)=f(A+tK)-f(A)$ is $n$-times continuously differentiable on $\mathbb{R}$ in the Hilbert-Schmidt norm, provided either $A$ is bounded or the derivatives $f^{(i)}$, $i=1,\dots ,n$, are bounded. As an application of the second order $\Sc^2$-differentiability, we extend the Koplienko trace formula from the Besov class $B_{\infty 1}^2(\mathbb{R})$ to functions $f$ for which the divided difference $f^{[2]}$ admits a certain Hilbert space factorization.

Minor editorial changes, to appear in J. Funct. Anal

References in corpus (2)

• Operator Lipschitz Functions
• Operator Lipschitz functions (English translation)