this property, up to multiples. Moreover, the value of $\operatorname{res}A$ is determined from finitely many terms in the asymptotic expansion of the symbol of $A$ (in fact, from the component of homogeneity $-n$ in this expansion); we call such functionals “locally determined”.

It vanishes on operators of noninteger order or of order $<-n$, and it is here that Kontsevich and Vishik’s functional takes over: $\TR A$ is well-defined and nontrivial when $A$ is of noninteger order or of order $<-n$, and it vanishes on commutators $[A,A']$ of these types. It is “global” in that the value of $\TR A$ depends on the full operator and cannot be determined from the terms in the asymptotic expansion of the symbol. The canonical trace equals the standard trace when the order is $<-n$; it is moreover defined on integer-order operators with certain parity properties, see below.

In this article we shall generalize the canonical trace to pseudodifferential boundary value problems in the calculus of Boutet de Monvel [@B].

The original approach of [@KV] is based on studies of generalized zeta functions $\zeta (A,P,s)=\Tr(AP^{-s})$ and certain regularizations of them, where $P$ is an auxiliary elliptic operator. This does not extend easily to the situation 