Asymptotic Behaviour of Semigroup Traces and Schatten Classes of Resolvents

TL;DR

This paper investigates the asymptotic behavior of semigroup traces and Schatten class properties of resolvents, providing new characterizations of Gibbs semigroups relevant to physics and noncommutative geometry.

Contribution

It introduces a novel characterization of Gibbs semigroups and explores the relationship between resolvent Schatten classes and trace behavior near zero.

Findings

Established a link between Schatten class of resolvents and trace asymptotics.

Applied Tauberian theorems to analyze asymptotic behaviors.

Provided a new characterization of Gibbs semigroups.

Abstract

Motivated by examples from physics and noncommutative geometry, given a generator $A$ of a Gibbs semigroup, we reexamine the relationship between the Schatten class of its resolvents and the behaviour of the norm-trace $\norm{e^{-tA}}_1\,$ when $t$ approaches zero. In addition to applying Tauberian results, we specifically investigate the compatibility of asymptotic behaviours with derivations and perturbations. Along the course of our study, we present a novel characterisation of Gibbs semigroups.