  In this paper we give formulae for the Dixmier trace and the noncommutative
residue (also called Wodzicki's residue) of pseudo-differential operators by
using the notion of global symbol. We consider both cases, compact manifolds
with or without boundary. Our analysis on the Dixmier trace of invariant
pseudo-differential operators on closed manifolds will be based on the Fourier
analysis associated to every elliptic and positive operator and the
quantization process developed by Delgado and Ruzhansky. In particular, for
compact Lie groups this can be done by using the representation theory of the
group in view of the Peter-Weyl theorem and the Ruzhansky-Turunen symbolic
calculus. The analysis of invariant pseudo-differential operators on compact
manifolds with boundary will be based on the global calculus of
pseudo-differential operators developed by Ruzhansky and Tokmagambetov.
