Singular traces and residues of the $zeta$-function

Journal article, 2017

This paper studies the relationship between the singular trace of a weak trace class operator and the asymptotic behaviour of its ζ-function at its leading singularity. For Dixmier measurable and universally measurable operators, we describe their measurability in terms of the behaviour of the ζ- function. We use recent advances in singular trace theory and a new approach based on Tauberian theorems other than the familiar Hardy-Littlewood or weak Karamata theorems. The approach finalises the long story of results [3,5,12,32,54] relating singular traces on the weak trace class ideal, the ζ-function, and the leading term of heat semi-group expansion. The results are illustrated by a number of examples, including discussions of pseudo-differential operators and Laplacians on fractals.