Reference request: trace class operator

Question

Suppose $D$ is an elliptic differential operator on a non-compact manifold $M$ of order $1$. Then $D$ has a pseudodifferential parametrix $Q$ of order $-1$ such that $$DQ-1=S_0,\qquad QD-1=S_1$$ are pseudodifferential operators with smooth Schwartz kernels.

However, if I'm not mistaken, this does not mean $S_0$ and $S_1$ extend to bounded operators on $L^2(M)$, since $M$ is non-compact.

Now suppose that $D$ is also Fredholm. Then is it possible to choose the pseudodifferential operator $Q$ so that $S_0$ and $S_1$ are:

1. bounded on $L^2(M)$?
2. compact on $L^2(M)$?
3. trace-class on $L^2(M)$?

Thanks!