Let $E \to M$ be a vector bundle over a closed manifold $M$. Suppose $T$ is an endomorphism from $L^2$ sections of $E$ to itself. How does one prove that $T$ is trace class if the image of $T$ is contained in smooth sections of $E$?
Thanks.
EDIT
Yea I guess I should provide context since there may be assumptions I'm not stating. This is from Singer's "Recent applications of index theory for elliptic operators". Specifically the part in the middle of the second paragraph of the proof, beginning with
To show $P_j {_j S_f} P_j$ is trace class, it suffices to show that $_j S_f P_j$ maps $L_2(E_j)$ continuously into $C^\infty(E_j)$...