Traces of Hilbert space operators and their recent history

Abstract

This paper surveys the recent theory of traces on ideals of Hilbert space operators. The main purpose is to summarize a new approach, which was developed by the author in the last decade. We sincerely hope that our simple and elegant methods will be helpful for many people working in non-commutative geometry, pseudo-differential operators, and mathematical physics, where traces have become useful tools. Historical comments and open problems should serve to complete the picture.

Key words: Operator ideal, sequence ideal, quasi-norm, trace, invariant linear form, shift operator, spectrum.