commutators $[A,A']$ modulo local terms, and has a specific value independent of the auxiliary operator, modulo local terms. The local “errors” vanish when $A$ is a singular Green operator of noninteger order, or of integer order with a certain parity; then $C_0(A,B)$ is a trace of $A$. They do not in general vanish when the interior ps.d.o. part of $A$ is nontrivial.

[*Keywords:*]{} Canonical trace — Nonlocal invariant — Pseudodifferential boundary value problems — Boutet de Monvel calculus — Asymptotic resolvent trace

[*Math. classification:*]{} 58J42 – 35S15

1\. Introduction

In their work on the geometry of determinants of elliptic operators, Kontsevich and Vishik [@KV] introduced the [*canonical trace*]{}, a novel functional $\operatorname{TR} A$ defined for a class of pseudodifferential operators ($\psi$do’s) $A$ on a closed $n$-dimensional manifold $X$. What makes it remarkable is that it extends the standard operator trace, and hereby complements the noncommutative residue.

The noncommutative residue $\operatorname{res}A$ was discovered by Wodzicki [@W] and, independently, by Guillemin [@Gu]; it is a trace (i.e., a nontrivial linear functional which vanishes on commutators) on the full algebra ${\Cal A}$ of all classical pseudodifferential operators on $X$. The noncommutative residue turns out to be the only trace on $\Cal A$ with 