Absolute continuity of spectral shift

TL;DR

This paper develops a method using double operator integrals to prove trace formulas and establish the absolute continuity of spectral shift for various classes of operators, leveraging dilation theorems.

Contribution

It introduces a novel approach combining double operator integrals with dilation theory to analyze spectral shift and absolute continuity in operator theory.

Findings

Proved trace formulas for functions of various operators.

Established absolute continuity of spectral shift using dilation theorems.

Constructed intermediate contractions for pairs with trace class difference.

Abstract

In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectrum of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.