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.

Mathematics Subject Classification (2010):

• Primary: 47B10, 47B37
• Secondary: 46B45, 46A45

Key words:

• Operator ideal
• sequence ideal
• quasi-norm
• trace
• invariant linear form
• shift operator
• spectrum