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

We prove the existence of a C2C2-function f:T→Cf:T→C defined on the unit circle, a unitary operator U and a self-adjoint operator Z   in the Hilbert–Schmidt class S2S2, such thatf(eiZU)−f(U)−ddt(f(eitZU))|t=0∉S1, the space of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko–Neidhardt trace formula for unitaries.