that the coefficient $\tilde
c_0'=C_{-1}(A,P_{1,\operatorname D})$ in the above expansions satisfies the relation $$\tilde c_0'=\tfrac12 \res A.\tag1.15$$

We shall presently investigate $C_0(A,P_{1,\operatorname{D}})$ as a candidate for a canonical trace. We establish quasi-trace properties as in (1.9), with formulas for the “value modulo local terms”. Moreover, we extract some cases where $C_0(A,P_{1,\operatorname{D}})$ is independent of $P_{1,\operatorname{D}}$ and vanishes on commutators $[A,A']$, so that it is a trace. Here we show in particular that $C_0([A,A'],P_{1,\operatorname{D}})=0$ whenever $A$ and $A'$ are singular Green operators of class zero and orders $\nu$ and $\nu'$ with $\nu+\nu'<1-n$ or $\nu+\nu'\notin \Bbb Z$; the same is true when $\nu+\nu'$ is an integer and a certain parity holds, provided we narrow down slightly the class of admissible auxiliary operators $P_1$. When $A=P_++G$ and $A'=P'_++G'$ both have nontrivial pseudodifferential part (and hence the orders are integer), however, one cannot hope for more than the quasi-trace property, see Remark 4.2 for a discussion of this point.

In contrast with the boundaryless case, the powers $P_{1,\operatorname{D}}^s$ are far from belonging to the $\psi $dbo calculus when $s\notin \Bbb Z$, so we do not expect to find residue formulas as in [@KV], [@O], [@MN].

The quasi-trace property has an interest in itself, for 