Real-valued spectral shift functions for contractions and dissipative operators

TL;DR

This paper discusses conditions under which real-valued spectral shift functions exist for pairs of contractions and dissipative operators, extending previous results on trace formulas and spectral analysis.

Contribution

It provides sufficient conditions for the existence of real-valued spectral shift functions for contractions and dissipative operators, expanding the spectral theory framework.

Findings

Existence of real-valued spectral shift functions under certain conditions

Extension of Lifshits–Krein trace formula to non-self-adjoint operators

New criteria for spectral shift functions in dissipative cases

Abstract

In recent joint papers the authors of this note solved a famous problem remained open for many years and proved that for arbitrary contractions 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. Note that in contrast with the case of self-adjoint and unitary operators it may happen that there is no {\it real-valued} integrable spectral shift function. In this note we announce results that give sufficient conditions for the existence of an integrable real-valued spectral shift function in the case of pairs of contractions. We also consider the case of pairs of dissipative operators.