Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case

We prove the existence of a C-2-function f : T -> C defined on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class S-2, such that f(e(iz) U) - f(U) - d/dt (f(e(itz)U)) (vertical bar t=0) is not an element of S-1, the space of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries. (C) 2016 Elsevier Inc. All rights reserved.