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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 $C^\infty(E_j)$...

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Where did you encounter this statement? –  Nate Eldredge Oct 16 '12 at 2:18
    
@NateEldredge I've updated my question with context. –  Eric O. Korman Oct 16 '12 at 3:04
    
You can make nice links by using the code [nice link description](http://nicelink.direction.here/nicelink) –  Pragabhava Oct 16 '12 at 3:08
    
@Pragabhava thanks for the edit! –  Eric O. Korman Oct 16 '12 at 3:10

1 Answer 1

up vote 2 down vote accepted

I found a proof, which is available here.

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