Real trace expansions

In this paper, we show that the trace of the operators $A\eta (t\mathcal{L})$ where $A$ and $\mathcal{L}$ are classical pseudo-differential operators on a compact manifold $M$ and $\mathcal{L}$ is elliptic and self-adjoint admits an expansion in powers of $t\to 0^{+}$. The functions $\eta$ being smooth and compactly supported on $\mathbb{R}$ have no meromorphic properties, unlike in the case of the heat trace or zeta functions. We also show that the constant coefficients in our expansions are related to the non-commutative residue and the canonical trace of $A$.