Decomposition of linear partial differential operators

I was wondering about the following:

Let $M$ be a smooth, second-countable (possibly noncompact) manifold and let $E$ and $F$ be smooth vector bundles over $M$.

1. Can every smooth linear partial differential operator $P$ from $E$ to $F$ be written as $\sum_{i=0}^n (T_i)_* \circ P_i$ for certain smooth linear partial differential operators $P_0, \dots, P_n$ from $E$ to $E$ and certain vector bundle homomorphisms $T_0, \dots, T_n$ from $E$ to $F$?

2. Can every smooth linear partial differential operator $P$ from $E$ to $E$ be written as a finite sum of compositions of smooth linear partial differential operators from $E$ to $E$ of order at most 1?

Of course, locally (i.e., on a chart domain over which the vector bundles trivialize) the answer to both questions is yes, but this does not seem to be of much help when trying to answer the questions globally.

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What are you using as your definition of partial differential operator? –  John M Aug 2 '11 at 12:01
For me, a partial differential operator from $E$ to $F$ of order at most $k$ is a local linear map $P \colon \Gamma^\infty(E) \to \Gamma^\infty(F)$ such that for every chart $(U, \kappa)$ of $M$ over which $E$ trivializes, $\left.P\right|_U$ can be written as $\sum_{|\alpha| \le k} (T_\alpha)_* \circ \partial^\alpha_\kappa$ for certain vector bundle homomorphisms $T_\alpha \colon E_U \to F_U$. (Here the partial derivatives $\partial^\alpha_\kappa$ act 'componentwise' on the sections, something which can be defined formally by using the frame that corresponds to the trivialization of $E_U$.) –  Marcel de Reus Aug 2 '11 at 15:32
I think it's a very interesting question. I suppose you've already tried patching arguments using smooth partitions of unity? –  John M Aug 3 '11 at 0:40
I found a remark in Ramanan's Global Calculus p62 which suggests, for question (2) that on compact manifolds, all differential operators are generated by 1st order operators, but that for non-compact manifolds, you can construct differential operators of "infinite order": books.google.com/… –  John M Aug 3 '11 at 3:02
The problem with using smooth partitions of unity is that you will not get finite sums on noncompact manifolds. (For compact manifolds, I believe that the answer to both questions is positive and that one can prove this by using finite smooth partitions of unity.) Your reference to Ramanan's Global Calculus made me realize that the questions are not formulated precise enough: in both questions $P$ is assumed to be of finite order. Thanks for thinking along! –  Marcel de Reus Aug 3 '11 at 14:16