Resolution of Peller's problem concerning Koplienko–Neidhardt trace formulae
Clément CoineLaboratoire de Mathématiques Université de Franche‐Comté 25030 Besançon cedex FranceChristian Le MerdyLaboratoire de Mathématiques Université de Franche‐Comté 25030 Besançon cedex FranceDenis PotapovSchool of Mathematics & Statistics University of New South Wales Kensington NSW 2052 AustraliaFedor SukochevSchool of Mathematics & Statistics University of New South Wales Kensington NSW 2052 AustraliaAnna TomskovaSchool of Mathematics & Statistics University of New South Wales Kensington NSW 2052 Australia
2016en
ABI
Annotatsiya
A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product S 2 × S 2 of two copies of the Hilbert–Schmidt classes into the trace class S 1 is established in terms of linear Schur multipliers acting on the space S ∞ 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 ∈ S 2 such that f ( A + B ) − f ( A ) − d d t ( f ( A + t B ) ) t = 0 ∉ S 1 .
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