Singular Traces and Residues of the Ζ-function
نویسنده
چکیده
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 [9, 3, 32, 54, 4] 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.
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