The Spectrum of Shift Operators and the Existence of Traces

Abstract

This is a counterpart of the paper ‘Commutator structure of operator ideals’ written by Dykema, Figiel, Weiss, and Wodzicki. However, contrary to them, I consistently use quasi-norms instead of gauges and replace symmetric sequence ideals as well as their characteristic sets by so-called shift-monotone sequence ideals. In this way, the Boyd index theorem of the four authors turns into a spectral radius theorem of shift operators, which can be proved in a few lines. In particular, the concept of e-completeness is no longer needed. We present several open problems, which show that substantial work remains to be done.