Real Trace Expansions

In this paper, we investigate trace expansions of operatorsof the form Aη(tL) where η: R→C is a Schwartz function, A and L are classical pseudo-differential operators on a compact manifold M with L elliptic. In particular, we show that, under certain hypotheses,this trace admits an expansion in powers of t→0+. We also relate the constant coefficient to the non-commutative residue and the canonical trace of A. Our main tool is the continuous inclusion of the functional calculus of L into the pseudo-differential calculus whose proof relies on the Helffer-Sjöstrand formula.