Background: Let $M$ be a smooth manifold, $E$ a vector bundle over $M$, and $L$ a linear differential operator of order $k$ over $E$, that is:
$$\begin{align} L: \ \Gamma(E) &\to \Gamma(E) \\ &u \mapsto \sum_{|\alpha| \leq k}( L_{\alpha} \partial^{\alpha})(u) \end{align}$$
If $L$ is a second order linear differential operator over $E$, for example, then in local coordinates:
$${L}(u)=\left(\left(\lambda_{i j}\right)_{\ell}^{k} \frac{\partial^{2} u^{\ell}}{\partial x^{i} \partial x^{j}}+\left(\mu_{i}\right)_{\ell}^{k} \frac{\partial u^{\ell}}{\partial x^{i}}+\nu_{\ell}^{k} u^{\ell}\right) e_{k}$$
where $\lambda \in \Gamma(\text{Sym}_2 (T^{*}M) \otimes \text{Hom}(E, E))$, while $\mu \in \Gamma(T^{*}M \otimes \text{Hom}(E, E))$ and $\nu \in \Gamma(\text{Hom}(E, E))$ and $(e_k)$ is a local frame for $E$ over a neighbourhood of $p \in M$ with local coordinates $(x^i)$.
What I'm having trouble understanding is the following:
- Why must we impose $\lambda \in \Gamma(\text{Sym}_2 (T^{*}M) \otimes \text{Hom}(E, E))$? The $\text{Sym}_2 (T^{*}M)$ part I get (since partial derivatives commute, we can assume w.l.o.g that $\lambda_{ij} = \lambda_{ji}$), but where does the $\text{Hom}(E, E)$ part come from? My understanding is that we're seeing $\lambda$ "depending" on $\partial_i$, $\partial_j$, the components of $u$ and the frame $(e_k)$, so I also get why we want the $T^{*}M$ part in $\mu \in \Gamma(T^{*} M \otimes \text{Hom}(E, E))$, but again the $\text{Hom}(E, E)$ confuses me.
I've been giving this a lot of thought over the last few days but haven't been able to figure it out. I'd appreciate some help.