Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae

Abstract

A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product ${\mathcal{S}}^2\times {\mathcal{S}}^2$ of two copies of the Hilbert-Schmidt classes into the trace class ${\mathcal{S}}^1$ is established in terms of linear Schur multipliers acting on the space ${\mathcal{S}}^{\infty }$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in {\mathcal{S}}^2$ such that $$f(A+B)-f(A)-\frac{d}{dt}(f(A+tB)){|}_{t=0}\notin {\mathcal{S}}^1.$$