Let $M$ be a manifold. Does there exist a theorem that says something like: "given two pseudodifferential operators $P$ and $Q$ on $M$ of the same order $k$, there exists an $L^2$-bounded operator $B$ such that $P-BQ$ has order at most $k-1$?"
I feel like this should be true, but do not know of a proof/reference. Any help finding either would be appreciated.