I have a question about the definition of pseudodifferential operator on a manifold $M$.
In most of the cases, assume $(M_i,\phi_i)$ is a chart for $M$, $P$ is defined as a linear operator from $C^\infty(M)$ to $C^\infty(M)$ (or $\mathcal{D}^{'}(M)$) such that the restriction of $P$ on each chart $M_i$ is a pseudodifferential operator on the corresponding Euclidean space, denote by $P_i$.
I always don't understand this part, is the definition constructive? By constructive I mean that if given $P_i$, is $P$ determined? When we are given $u\in C^\infty(M)$, I guess we should first find a partition of unity $\lambda_i$ corresponding to the chart $M_i$, then $Pu=P\sum\lambda_i u=\sum P\lambda_i u$. Now $\lambda_i u\in C_0^\infty (M_i)$ and we can use $P_i$, but $P_i(\lambda_i u)$ only tells us the information of $P(\lambda_i u)$ on the chart $M_i$, how about other charts? Like if we view $P:C^\infty(M)\rightarrow C^\infty(M)$, then $P_i(\lambda_i u)$ tells us the value of $P(\lambda_i u)$ on $M_i$, then what is the value of it on other charts?