Real spectral shift functions for pairs of contractions and pairs of dissipative operators

Recently the authors solved a long-standing problem and showed that for an arbitrary pair of contractions on Hilbert space with trace class difference has an integrable spectral shift function on the unit circle $\mathbb{T}$ and an analogue of the Lifshits–Krein trace formula holds. It is also known that it may happen that there is no real-values integrable spectral shift function. In this paper we find conditions under which a pair of contractions with trace class difference has {\it a real-valued integrable} spectral shift function. We also consider a similar problem for pairs of dissipative operators. Finally, we find an application of the results in question to dissipative Schrödinger operators.