I am reading the book Index Theorem and the Heat Equation written by Peter.B.Gilkey. Here is my question:
Let E be a hermitian vector bundle on a compact smooth manifold M. Let $D : \mathcal{C}^{\infty}(E) \rightarrow \mathcal{C}^{\infty}(E)$ be an elliptic operator of positive order. Then it extends to the completion of $\mathcal {C}^{\infty}(E)$ to $L^2(E)$. It has been said in the first paragraph of chapter 3 of the book mentioned above that this is an unbounded operator and admits a sequence of eigenvalues $0 \leq \lambda\_n \rightarrow \infty$. Any reference or proof of this will be helpful.



Elliptic operator are unbounded
The fact that (non-trivial) elliptic operators are not bounded on $L^2$ is a special case of the fact that differential operators generally are not continuous (i.e., not bounded) on $L^2$. This much has little to do with the compactness of the manifold.


Note that such operators do not extend "to the completion of $C^\infty(E)$ to $L^2(E)$", as in the question. Yes, such operators have extensions to distribution-valued operators, but that's not immediately to the point.


The fact that symmetric elliptic differential operators have self-adjoint extensions is also a fairly general fact, not depending on compactness, since Friedrichs' self-adjoint extensions always exist for semi-bounded operators. (The von Neumann or Krein general classification of self-adjoint extensions of symmetric unbounded operators requires that we check some conditions (equality of dimensions of deficiency indices...) before knowing there's *any* self-adjoint extension.)


The property of eigenvalues mentioned in the question is much more specific, depending for proof on having compact resolvent, which follows for compact manifolds. That argument basically reduces/compares to the case of flat tori, that is, products of circles, where Fourier series give a straightforward proof of the relevant Rellich compactness assertion.


