Real-valued spectral shift functions for contractions and dissipative operators

Recently the authors of this note solved a famous problem that remained open during many years and proved that for arbitrary contractions on Hilbert space with trace class difference there exists an integrable spectral shift function, for which an analogue of the Lifshits–Krein trace formula holds. Similar results were also obtained for pairs of dissipative operators. It turns out that in contrast with the case of self-adjoint or unitary operators, it can happen that there is no real-valued integrable spectral shift function. In this note we state results that give sufficient conditions for the existence of a real-valued integrable spectral shift function for pairs of contractions and pairs of dissipative operators.