Singular traces. Volume 1. Theory
A singular trace on a two-sided ideal of the algebra of bounded operators \(\mathcal L(H)\) on a Hilbert space \(H\) is a trace which vanishes on finite-rank operators. The first example of a singular trace was given by Dixmier in the 1960s. This book is the first volume of a work which consists of three volumes and is devoted to the study of traces on ideals of \(\mathcal L(H)\), expanding on previous works of the authors [Singular traces. Theory and applications. Berlin: de Gruyter (2013; Zbl 1275.47002); in: Advances in noncommutative geometry. Based on the noncommutative geometry conference, Shanghai, China, March 23 -- April 7, 2017. On the occasion of Alain Connes' 70th Birthday. Cham: Springer. 491--583 (2019; Zbl 1457.46064)]. The first part of the book contains introductory material. Topics presented include the Calkin correspondence, singular values and submajorization of operators on a Hilbert space and symmetric ideals of operators. In the second part, the general theory of traces on ideals of bounded operators is developed and a description of traces on ideals of operators in \(\mathcal L(H)\) is provided. The Pietsch correspondence for ideals and traces and the work of Kalton on spectral traces are presented. In the third part, formulas for Dixmier traces are obtained, including formulas involving eigenvalues, formulas involving the diagonal matrix elements of an operator with respect to an orthonormal basis, formulas involving the \(\zeta\)-function residue, and heat trace formulas. Each chapter has its own introduction and its own notes section. The book is nicely written and is of interest not only to specialists in the field, but to other mathematicians as well. As the authors note, Volume II concentrates on the applications of singular traces on a separable Hilbert space while Volume III describes the semifinite theory of singular traces and some applications.
scientific article; zbMATH DE number 7353467