First order Elliptic operator

Here is a simple way of producing first order elliptic operators $\newcommand{\bR}{\mathbb{R}}$ $C^\infty(\mathbb{R}^n, W)\to C^\infty(\bR^n, W)$ with constant coefficients. Denote by $L(W)$ the space of linear operators $W\to W$. Consider a map $\newcommand{\si}{\sigma}$

$$\si :\bR^n\to L(W), $$

such that $\si(x)$ is invertible for any $x\in\mathbb{R}^n\setminus 0$. Denote by $e_1,\dotsc, e_n$ the canonical basis of $\bR^n$. Now set $\si_k=\si(e_k)$ and define $\newcommand{\pa}{\partial}$

$$ D=\sum_{k=1}^n \si_k \pa_k: C^\infty(\mathbb{R}^n, W)\to C^\infty(\bR^n, W). $$

This is an elliptic operator because its symbol is the map $\si: \bR^n\to L(W)$, up to a multiplication by $\sqrt{-1}$. Let me point two things.

• All elliptic first order p.d.o.'s with constant coefficients are obtained in this fashion and
• The symbol at a point $p \in\bR^n$ of any first order p.d.o. $C^\infty(\mathbb{R}^n, W)\to C^\infty(\bR^n, W)$ is a map $\si_p:\bR^n\to L(W)$ with the above property.

The problem is then to find the lowest dimensional vector space $W$ such that there exists a linear map $\si: \bR^n\to L(W)$ with the property that $\si(x)$ is invertible for any $x\in\mathbb{R}^n\setminus 0$.If $W$ is real, the minimal dimension is computed by the so called Radon-Hurwitz numbers. For details, I refer to the book Topological Geometry, by I.R. Porteous. At the end of Chapter 13 of the book you can find a discussion on these numbers and the above problem. In any case, the solution is not simple, it is related to the problem of vector fields on spheres and was elucidated by J.F. Adams using $K$-theoretic techniques. Atiyah has a discussion where he explains that the index theorem for Dirac operators implies the index operators for arbitrary operators. I'll add the reference once I find it. I have posted on MO a similar question myself a while ago.

Note. In the paper Topology of elliptic operators, Proc. Sympo.Pure. Math., vol. 16, Amer. Math. Soc 1970, p 101-119, Atiyah essentially answers your first question in the complex case, again referring to the work of Adams.

In the paper The index of elliptic operators, Colloquium Lectures, Amer. Math. Soc. Dallas, 1973 Atiyah gives a brief explanation why the index theorem for Dirac operators implies the index theorem in general. Essentially, it has to do with the Bott periodicity theorem. You can find both papers in the 3rd volume of Atiyah's Collected Works.