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

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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