Asymptotic behaviour of semigroup traces and Schatten classes of resolvents

Motivated by examples from mathematical physics and noncommutative geometry, given a generator A of a Gibbs semigroup ${\left\{U_A\right(t):=e^{-tA}\}}_{t\ge 0}$, we re-examine the relationship between the Schatten class of its resolvents and the behaviour of the norm-trace ${\Vert e^{-tA}\Vert }_1$ when t approaches zero. Besides the applications of the Tauberian results, we specifically investigate the compatibility of asymptotic behaviours with the semigroup derivations and perturbations. Along the course of our study, we present a novel characterisation of Gibbs semigroups.