I'm taking a course in functional analysis that is using Rudin's book. I want to ask if there is a relationship between the characteristic polynomials and their elliptic linear operators?
I understand that we call $L$ an elliptic linear operator of $n$ variables of order $N$ if its characteristic polynomial $p(x,y)=\sum_{|\alpha|=N}f_\alpha(x)y^\alpha \neq 0$ for $x\in \Omega \subset \mathbb{R}^n$ and $y\in \mathbb{R}\setminus \{0\}$.
But is there an analogous formula as in linear algebra, where we have that a characteristic polynomial of a transformation/operator/map $T$ (on vector space $V$) is the determinant of a polynomial with respect to $T$?
As it turns out, credit to a friend of mine who pointed out, that typically, $p(x,y)$ is called the principal symbol of an elliptic operator. And it has nothing to do with the linear algebra object we expect. Here is a link that says more about this on the Encyclopedia of Mathematics.