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arXiv · 1504.03843

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

Clément Coine·Christian Le Merdy·Denis Potapov·Fedor Sukochev·Anna Tomskova
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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 $$ f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $$

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Clément Coine, Christian Le Merdy, Denis Potapov, Fedor Sukochev, Anna Tomskova. 2015-04-15. Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae. https://arxiv.org/abs/1504.03843

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