Let $M$ be a closed (compact without boundary) submanifold in $\mathbb R^N.$ Then for a second-order symmetric (say with respect to $L^2$) elliptic operator $L$ acting on $C^\infty(M),$ we have the spectral theory, saying that the eigenvalues are discrete and increasing (or decreasing, depending on the sign convention). Also, we can take the first eigenfunction being positive.
My question is as follows. Suppose now the operator is acting on vectors, for example, smooth normal vector fields. (Thus, $L\colon \Gamma(NM)\to \Gamma(NM)$ where $\Gamma(NM)$ is the space of smooth sections of $NM.$) I guess that we could still have the spectral theory, in the sense that the eigenvalues are discrete and increasing. However, I am not sure if we could have a similar statement for the positivity of the first eigenvector. A possible statement I could say is like that the first eigenvector is non-vanishing. (i.e., I guess if $L$ is a second-order symmetric elliptic operator acting on $\Gamma(NM)$ and $$LV=\mu_1V$$ where $\mu_1$ is the first eigenvalue, then either $|V|\equiv 0$ or $|V|$ is non-vanishing.
However, I am not sure this could be derived using the same argument in the scalar case, since it seems to require some maximum principle argument.
Has anyone heard of this kind of statement for differential operator acting on bundle-valued sections?