Fredholm index of elliptic operator, reference request

Question

Let $E$ be a vector bundle over a smooth manifold $M$ ($M$ may need to be compact and without boundary). Furthermore let$$T:\Gamma(M,E)\to\Gamma(M,E)$$be an elliptic k-th order differential operator. I am looking for a reference (and a confirmation) for the following claims:

1. If we pick a metric $h$ on $E$, a covariant derivative $\nabla$ and set $H^l:=H^l(h,\nabla)$, then $T$ induces a family of Fredholm operators $(T_l)_{l\in\mathbb Z}$ with $T_l\in L(H^l,H^{l-k})$.
2. The index of $T_l$ is independent of the choice of $(l,h,\nabla)$.

Answer 1

1. This is proposition $2.3(3)$ in David Reutter's essay on the The Heat Equation and the Atiyah-Singer Index Theorem.

2. I will sketch an incomplete proof. Perhaps someone can complete it, so I decided to add it to this answer: As discussed here we have $\operatorname{coker}(A)\cong \operatorname{ker}(A^*)$ and in addition (this should be checked) $(T_l)^*=(T^*)_l$ (where $T^*$ is the formal adjoint of $T$). Hence $$\operatorname{ind}(T_l)=\operatorname{dim}\operatorname{ker}T_l-\operatorname{dim}\operatorname{ker}(T^*)_l$$ and the RHS is independent of $l,h,\nabla$ since the kernel of $A_l$ is simply the kernel of $A$ (this should also be verified).