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 x 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-infinity 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 is an element of S-2 such that f(A + B) - f(A) - d/dt(fA + tB))vertical bar(t=0) is not an element of S-1.