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

TL;DR

This paper proves higher order differentiability of operator functions in Hilbert spaces and extends the Koplienko trace formula to broader classes of functions using second order $ ext{S}^2$-differentiability.

Contribution

It establishes higher order differentiability of operator functions in Hilbert spaces and extends the Koplienko trace formula to functions with specific Hilbert space factorizations.

Findings

Proves $n$-times differentiability of $(A+tK)-(A)$ in Hilbert-Schmidt norm.

Extends Koplienko trace formula to functions beyond Besov class $B_{e,1}^2( )$.

Provides conditions under which the second order $ ext{S}^2$-differentiability holds.

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 $\varphi(t)=f(A+tK)-f(A)$ 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 of the second order $\Sc^2$-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.