Singular Traces and Compact Operators. I

TL;DR

This paper characterizes when positive compact operators admit singular traces that assign finite non-zero values, extending prior results and providing explicit descriptions using non-standard analysis.

Contribution

It establishes necessary and sufficient conditions for the existence of singular traces on positive compact operators, generalizing earlier work by Dixmier and Varga.

Findings

Provides a complete characterization of operators with singular traces.

Offers explicit descriptions of singular traces and ergodic states.

Uses non-standard analysis tools to achieve these results.

Abstract

We give a necessary and sufficient condition on a positive compact operator $T$ for the existence of a singular trace (i.e. a trace vanishing on the finite rank operators) which takes a finite non-zero value on $T$. This generalizes previous results by Dixmier and Varga. We also give an explicit description of these traces and associated ergodic states on $\ell^\infty(\Bbb N)$ using tools of non standard analysis in an essential way.