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

TL;DR

This paper establishes conditions for pairs of contractions and dissipative operators to have real-valued integrable spectral shift functions, extending previous results and applying them to dissipative Schrödinger operators.

Contribution

It identifies specific conditions ensuring the existence of real-valued spectral shift functions for pairs of contractions and dissipative operators, and applies these findings to Schrödinger operators.

Findings

Conditions for real-valued spectral shift functions are established.

An analogue of the Lifshits--Krein trace formula is confirmed under these conditions.

Applications to dissipative Schrödinger operators are demonstrated.

Abstract

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 ${\Bbb T}$ 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 Schr\"odinger operators.