Singular Traces. Volume 1. Theory

"This monograph introduces the theory of singular traces and their spectral properties for a separable Hilbert space. The first part of the text describes all traces on ideals of bounded operators in terms of invariant linear functionals on ideals of bounded sequences, and provides exact conditions when every trace on an ideal is a function of eigenvalues. For mathematical physicists and users of Alain Connes' noncommutative geometry the second part of the text provides a complete reference to positive traces and Dixmier traces on the ideal of weak trace class operators. The weak trace class operators form the most important trace ideal outside of the ideal of trace class operators. Formulas for a Dixmier trace as a residue of a spectral zeta function and as the leading term in an asymptotic expansion of a heat trace are described."--taken from back cover.