Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively compact open subsets with smooth boundaries.
If $L^2 (M, E)$ is the space of square-integrable sections in $E$, let $L : L^2 (M, E) \to L^2 (M, E)$ be the (densely defined) Friedrichs extension of $\nabla^* \nabla$ (with $\nabla^*$ the formal adjoint of $\nabla$). By the Hille-Yoshida theorem, this operator generates a $C_0$-semigroup $(\mathrm e ^{-tL}) _{t \ge 0}$ of contractions in $L^2 (M, E)$.
Similarly, one may perform the same construction on $U_j$ and obtain the operator $L_j$ and the associated semigroup $(\mathrm e ^{-tL_j}) _{t \ge 0}$ of contractions in $L^2 (M, E_j)$, for every $j \ge 0$.
Is there any way in which $\mathrm e ^{-tL_j}$ converges to $\mathrm e ^{-tL}$ for every $t \ge 0$?
I know that this is true when $E = M \times \mathbb C$ with the usual Hermitian structure, and $\nabla$ is just the differential operator acting on smooth functions. The proof, though, relies on the existence of the heat kernel, on its property of being the minimal positive fundamental solution of the heat equation, and on using the monotone convergence theorem (to define the heat kernel of $\mathrm e ^{-tL}$ as the pointwise limit of the heat kernels of $\mathrm e ^{-tL_j}$). These tools are clearly not available in general bundles, so what to do then? I have tried to use the theorems in chapter VIII of volume 1 of Reed & Simon, but the hypotheses therein are not satisfied in my setting.
