$ in a region$$\Lambda =\{ \,\lambda \in\Bbb C\mid \arg\lambda \in [\tfrac \pi
2-\varepsilon ,\tfrac{3\pi }2+\varepsilon] \text{ or }|\lambda |\le
r(\varepsilon )\,\}\tag2.3$$ with suitable $\varepsilon>0$ and $r(\varepsilon)>0$. Moreover, we take a strongly elliptic second-order pseudodifferential operator $S_1$ on $C^\infty(X',E')$ and set $
B=\pmatrix P_{1,D}& 0\\ 0&S_1\endpmatrix.
$ Then $$\aligned
&\Tr \Bigg(\pmatrix P_++G & K\\ T&S
\endpmatrix(B-\lambda)^{-N}\Bigg)\\
&\qquad\qquad= \Tr_X \left((P_++G)(P_{1,D}-\lambda)^{-N}\right)
+\Tr_{X'}\left(S(S_1-\lambda)^{-N}\right)\endaligned\tag2.4$$ is well-defined for $N>(n+\nu)/2$. The behavior of $\Tr_{X'}\left(S(S_1-\lambda)^{-N}\right)$ is well-known from the theory of $\psi $do’s on manifolds without boundary, so we shall restrict the attention to $$\Tr_X \left((P_++G)(P_{1,D}-\lambda)^{-N}\right)\text{ if $\nu
\in\Bbb Z$},\text{ and }~~ \Tr_X
\left(G(P_{1,D}-\lambda)^{-N}\right)\text{ if $\nu
\in\Bbb R\setminus \Bbb Z$}.\tag2.5$$ We henceforth denote $P_++G=A$.

A natural candidate for an extension of the canonical trace is the functional $A\mapsto C_0(A,P_{1,\operatorname D})$ which associates with the $\psi$dbo $A=P_++G$ the coefficient of $(-\lambda )^{-N}$ in (1.10), equal to the coefficient of $s^0$ in the Laurent expansion of $\zeta(A,P_{1,\operatorname D},s)$ at $s=0$ (1.14). We shall show that this is indeed a good choice at least in two cases:

\(i) $A=G$ is a singular Green operator of noninteger order $\nu$ and class zero with vanishing pseudodifferential part;

\(ii) $A=P_++G$ is of integer order $\nu$ with a pseudodifferential operator $P$ of normal order $0$ 