Reference for Weyl's law for higher order operators on closed Riemannian manifolds

Question

I am looking at page 32 (beginning of Chapter 5) here. We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading symbol, such that $A_{\tilde{g}}=c^{-2l}A_g$ whenever $\bar{g}=c^2g$ is a change of metric for some $c>0$ constant. We are told that the asymptotic behaviour of eigenvalues of $A$ satisfies Weyl's formula \begin{equation}\tag{1} \lambda_j\sim C(g,A)\,j^\frac{2l}{n}~~~\mathrm{as}~j\rightarrow\infty, \end{equation} but I am struggling to find a reference for this result. It is also stated here without proof, but even in the suggested references I still can't find anything.

Lots of the literature (for instance this paper of Gilkey) also only states (1) for (higher order) Laplace-type operators, i.e. operators for which the leading symbol is (a power of) the metric. Am I correct in saying the conditions above are more general? Thanks in advance.

EDIT: by the conditions above I mean in particular the scaling property $A_{\tilde{g}}=c^{-2l}A_g$. Does this imply, or is it implied by, $A$ having its leading symbol being a power of the metric? I guess this is a second question; answers to either would be welcomed.

Answer 1

One possible reference is Seeley's paper on Complex powers of Elliptic Operators, where Seeley did it for the Laplacian (page 6). But the discussion carries over to all elliptic $\Psi DO$s without much difficulty. For a "modern" expository article, see this. Seeley's argument would imply a term of type $j^{2l/n}$, just as the article claimed. A standard textbook is Shubin's book, where the result is proved in page 130.