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

TL;DR

This paper constructs a specific example demonstrating that the Koplienko-Neidhardt trace formula does not always hold for unitary operators, resolving a problem posed by Peller.

Contribution

It provides a counterexample showing the failure of the trace formula in the unitary case, addressing a longstanding open problem.

Findings

Counterexample disproves the universal validity of the trace formula for unitaries.

The constructed operators show the difference does not belong to trace class.

The result clarifies limitations of the Koplienko-Neidhardt trace formula.

Abstract

We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0} $$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries.