Trace formula for contractions and it's representation in $\mathbb{D}$

TL;DR

This paper provides a simplified proof of the spectral shift function and trace formula for pairs of contractions, and introduces a trace formula involving the unit disc that resembles the Helton-Howe formula.

Contribution

It offers a minimal assumption proof for the spectral shift function and derives a new trace formula for contraction differences involving the unit disc.

Findings

Simplified proof of spectral shift function existence.

Trace formula for contraction differences over the unit disc.

Expression similar to Helton-Howe formula.

Abstract

The aim of this article is twofold: give a short proof of the existence of real spectral shift function and the associated trace formula for a pair of contractions, the difference of which is trace-class and one of the two a strict contraction, so that the set of assumptions is minimal in comparison to those in all the existing proofs. The second one is to find a trace formula for differences of functions of contraction and its adjoint, in which case, the integral in the formula is over the unit disc and has an expression surprisingly similar to the Helton-Howe formula.