  The aim of this paper is to show that various known characterizations of
traces on classical pseudodifferentials operators (PsiDOs) can actually be
obtained by very elementary considerations on PsiDOs, using only basic
properties of these operators. Thereby, we give a unified treatment of the
determinations of the space of traces (i) on PsiDOs of noninteger orders or of
regular parity-class, (ii) on integer order PsiDOs, (iii) on nonpositive order
PsiDOs in dimension greather than or equal to 2, and (iv) on nonpositive order
PsiDOs in dimension 1.
