and $G$ of class zero (we say that a $\psi
$do symbol has normal order $d$ when it is $O(\xi _n^{d})$ at the boundary).

There are two conditions involved, namely on the class of the singular Green part and on the normal order of the pseudodifferential part in (ii). They are both natural, and related:

If $G$ is effectively of class $r>0$, i.e., if it can be written in the form $G=G_0+\sum_{j=0}^{r-1} K_j\gamma_j$, where $G_0$ is a singular Green operator of class zero, $\gamma_ju=(\partial _n^ju)|_{X'}$, and the $K_j$ are Poisson operators with $K_{r-1}\ne 0$, then $G$ will not be bounded on $L^2(E)$, much less of trace class, even if its order is arbitrarily low, cf. e.g. . As we are looking for a functional coinciding with the standard trace for operators of sufficiently low order, we shall exclude operators of positive class. An expansion analogous to (1.10) still exists, with the sum over $k\ge 0$ replaced by a sum over $k\ge k_0$ with a possibly negative starting index $k_0$. In this case, however, the coefficient $c'_0$ will not in general coincide with the noncommutative residue, cf. [@GSc; @Remark; @1.2].

For operators $P$ having the transmission property, we find that $C_0(P_+, 