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

TL;DR

This paper derives a formula for bilinear Schur multipliers' norms and uses it to solve Peller's problem by providing a counterexample involving a twice differentiable function and specific operators.

Contribution

The paper establishes a new formula for the norm of bilinear Schur multipliers and applies it to resolve Peller's problem on trace formulae.

Findings

Established a formula for the norm of bilinear Schur multipliers.

Constructed a counterexample disproving Peller's conjecture.

Showed existence of functions and operators with specific trace properties.

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 $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $$