With $Q_{\lambda ,+}$ denoting the truncation to $X$ of the resolvent $Q_\lambda $ of $P_1$ on $\widetilde X$, one can easily check that $R_\lambda =Q_{\lambda ,+}- K_\lambda  \gamma_0Q_{\lambda ,+}$ solves the problem; it has the singular Green operator part $G_\lambda  =- K_\lambda  \gamma_0Q_{\lambda ,+}$.

We shall also study powers of the resolvent and write them in the form $$R^N_\lambda =(Q_{\lambda ,+}+G_\lambda)^{N}=(Q^N_\lambda
)_++G^{(N)}_\lambda
.\tag 2.11$$ Note that $$R_\lambda ^{N}=\tfrac 1{(N-1)!}\partial _\lambda
^{N-1}R_\lambda ,\quad Q_{\lambda ,+}^{N}=\tfrac 1{(N-1)!}\partial
_\lambda
^{N-1}Q^N_{\lambda ,+},\quad G^{(N)}_\lambda =\tfrac
1{(N-1)!}\partial _\lambda ^{N-1}G_\lambda ;
\tag 2.12$$ this allows us to replace calculations for powers of $R_\lambda $ by calculations for $\lambda $-derivatives.

We shall now start with the proof of Theorem 2.1. The case where $A=G$ is a singular Green operator of class zero and the pseudodifferential part vanishes can be attacked more directly and will be studied first.

3\. Traces on the algebra of singular Green operators

Let $G$ be a singular Green operator in $E$ of order $\nu \in\Bbb R$ and class 0, and consider $GR^N_\lambda = GQ^N_{\lambda ,+}+GG^{(N)}_\lambda$. In [@GSc] we introduced the auxiliary variable $\mu =(-\lambda )^{\frac12}$ and phrased the results in terms of $\mu$ instead of $\lambda $. In the present paper we shall keep 