of manifolds with boundary; in the case without boundary, $P^{-s}$ is a classical $\psi $do again (by Seeley [@S]), whereas complex powers of elliptic boundary problems fall outside the calculus of Boutet de Monvel. However, the introduction of $\TR A$ can instead be based on trace expansions of heat operators $ Ae^{-tP}$ (Lesch [@L]) or resolvents $A(P-\lambda )^{-N}$ (Grubb [@G4]), where the latter admit a direct generalization to manifolds with boundary (Grubb and Schrohe [@GSc]).

Therefore, let us explain the functional of [@KV] from the resolvent point of view: Let $A$ be a classical $\psi $do of order $\nu$; its symbol has an expansion in local coordinates $a\sim
\sum_{j\ge 0}a_{\nu -j}$ with $a_{\nu -j}(x,\xi )$ smooth in $(x,\xi )$ and homogeneous of degree $\nu -j$ in $\xi $ for $|\xi |\ge 1$. When $P$ is an auxiliary elliptic operator of integer order $m>0$ with no principal symbol eigenvalues on $\Bbb R_-$, a calculation in local coordinates shows that the operator family $A(P-\lambda )^{-N}$ has for $N>(n+\nu )/m$ a trace expansion $$\Tr (A(P-\lambda )^{-N})
\sim
\sum_{j\ge 0}\tilde c_{j}(-\lambda )^{\frac {n+\nu -j}m-N}
+\sum_{k\ge 0}
\bigl(\tilde c'_{k}\log (-\lambda )+\tilde
c''_{k}\bigr)(-\lambda )^{-k-N},\tag 1.1$$ for $\lambda \to \infty$ in a small sector around $\Bbb R_-$, 