Higher order [formula omitted]-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∈Cn(R). We establish that φ(t)=f(A+tK)−f(A) is n-times continuously differentiable on R in the Hilbert–Schmidt norm, provided either A is bounded or the derivatives f(i), i=1,…,n, are bounded. This substantially extends the results of [3] on higher order differentiability of φ in the Hilbert–Schmidt norm for f in a certain Wiener class. As an application of the second order S2-differentiability, we extend the Koplienko trace formula from the Besov class B∞12(R)[20] to functions f for which the divided difference f[2] admits a certain Hilbert space factorization.