Perturbation formulas for traces on normed ideals

TL;DR

This paper develops perturbation formulas for traces on normed ideals in semifinite von Neumann algebras, including Dixmier traces, establishing spectral shift measures and a linearization formula for perturbed operators.

Contribution

It introduces new perturbation results for traces, especially for Dixmier traces, and proves a linearization formula not valid for normal traces.

Findings

Existence of spectral shift measures with potential singular components.

Linearization formula for Dixmier traces on perturbed operators.

Perturbation results applicable to dissipative or bounded initial operators.

Abstract

We prove perturbation results for traces on normed ideals in semifinite von Neumann algebra factors. This includes the case of Dixmier traces. In particular, we establish existence of spectral shift measures with initial operators being dissipative or bounded, and show that these measures can have singular components in the case of Dixmier traces. We also establish a linearization formula for a Dixmier trace applied to perturbed operator functions, a result that does not typically hold for normal traces.