combinations of the given operators and $\log P$; cf. [@KV] and Okikiolu [@O] for $C_0(A,P)-C_0(A,P')$, cf. Melrose and Nistor [@MN] for $C_0([A,A'],P)$. (In works of Melrose et al., notation such as $\widehat{\operatorname{Tr}}(A)$ or $\Tr_P(A)$ is used for $C_0(A,P)$; it is called a regularized trace there.)

In order to extend the definition of the canonical trace to operators in the calculus of Boutet de Monvel [@B] on an $n$-dimensional compact $C^\infty$-manifold $X$ with boundary $\partial X=X'$, we shall rely on resolvent expansions analogous to (1.1). To this end we choose an auxiliary operator $P_{1,\operatorname{D}}$, which is the (invertible) Dirichlet realization of a strongly elliptic principally scalar second-order differential operator.

It was shown in [@GSc] that when $A=P_++G$ is a pseudodifferential boundary operator ($\psi$dbo) in this calculus of order $\nu \in\Bbb Z$ with $G$ of class 0, then there is a trace expansion for $N>(n+\nu )/2$: $$\Tr(A(P_{1,\operatorname{D}}-\lambda )^{-N})\sim \sum_{j\ge
0}\!\tilde c_{j}(-\lambda)
^{\frac{n+\nu - j}{2}-N}
+\sum_{k\ge
0}(\tilde c'_k\log (-\lambda )+\tilde c''_k)(-\lambda )
^{-\frac k2-N}.
\tag 1.10$$ The proof depends on a reduction to parameter-dependent $\psi $do’s on $X'$, where [@GS1] can be used; it is easily generalized to cases where $A=G$ of noninteger order $\nu $ and class 0. When $\nu
\notin\Bbb Z$, 