Traces on ideals and the commutator property

TL;DR

This paper introduces a new class of spectral traces motivated by a specific trace property, explores their characteristics, and investigates when operators commute under these traces within certain ideals.

Contribution

It proposes a new class of spectral traces, examines their properties, and addresses the conditions under which operators commute under these traces in ideals.

Findings

Spectral traces depend only on spectrum and multiplicities.

The paper raises the question of operator commutativity under traces in ideals.

Provides an overview of traces on ideals and related formulas.

Abstract

We propose a new class of traces motivated by a trace/trace class property discovered by Laurie, Nordgren, Radjavi and Rosenthal concerning products of operators outside the trace class. Spectral traces, traces that depend only on the spectrum and algebraic multiplicities, possess this property and we suspect others do, but we know of no other traces that do. This paper is intended to be part survey. We provide here a brief overview of some facts concerning traces on ideals, especially involving Lidskii formulas and spectral traces. We pose the central question: whenever the relevant products, $AB$, $BA$ lie in an ideal, do bounded operators $A$, $B$ always commute under any trace on that ideal, i.e.,$\tau(AB) = \tau(BA)$? And if not, characterize which traces/ideals do possess this property.