  Classical pseudodifferential operators A on closed manifolds are considered.
It is shown that the basic properties of the canonical trace TR A introduced by
Kontsevich and Vishik are easily proved by identifying it with the leading
nonlocal coefficient C_0(A,P) in the trace expansion of A(P-\lambda)^{-N} (with
an auxiliary elliptic operator P), as determined in a joint work with Seeley
1995. The definition of TR A is extended from the cases of noninteger order, or
integer order and even-even parity on odd-dimensional manifolds, to the case of
even-odd parity on even-dimensional manifolds.
  For the generalized zeta function \zeta (A,P,s)=\Tr(AP^{-s}), extended
meromorphically to C, C_0(A,P) equals the coefficient of s^0 in the Laurent
expansion at s=0 when P is invertible. In the mentioned parity cases, \zeta
(A,P,s) is regular at all integer points. The higher Laurent coefficients
C_j(A,P) at s=0 are described as leading nonlocal coeficients C_0(B,P) in trace
expansions of resolvent expressions B(P-\lambda)^{-N}, with B
log-polyhomogeneous as defined by Lesch (here -C_1(I,P)=C_0(\log P,P) gives the
zeta-determinant). C_0(B,P) is shown to be a quasi-trace in general, a
canonical trace TR B in restricted cases, and the formula of Lesch for TR B in
terms of a finite part integral of the symbol is extended to the parity cases.
