Does the partial order on $\mathbb{C}$ induced by $[0,\infty)$ have any non-trivial uses or application?

Question

We can define a binary relation on the complex numbers as follows.

• $z \leq z'$ iff there exists $x \in [0,\infty)$ with $z+x=z'.$

Equivalently,

• $z \leq z'$ iff $\mathrm{Re}(z) \leq \mathrm{Re}(z')$ and $\mathrm{Im}(z) = \mathrm{Im}(z')$.

It is straightforward to show that $\leq$ is a partial order on $\mathbb{C}$, extending the usual (total) order on $\mathbb{R},$ and that the following holds for all $z,z' \in \mathbb{C}.$

If $z \leq z',$ then:

1. Given $w \in \mathbb{C}$, we have $w+z \leq w+z'$.
2. Given $r \in [0,\infty)$, we have $rz \leq rz'.$
3. $-z' \leq -z$.
4. $\bar{z} \leq \bar{z'}$

Despite these formulae, I find it hard to believe that the relation $\leq$ could actually be useful. There just aren't enough points comparable to each other.

Conjecture. The partial order $\leq$ on $\mathbb{C}$ lacks a (non-trivial) use or application.

Question. Does anyone know of a counterexample to this "conjecture"?

Answer 1

For a non-negative matrix, i.e. a matrix with all entries greater or equal to zero, the set of eigenvalues has a very special form. This is explained in the Perron-Frobenius theorem:

http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem

One consequence of this theorem is e.g.: Let $A$ be an irreducible non-negative $n \times n$ matrix with period $h$ and spectral radius $\rho(A) = r$, then $A$ has exactly $h$ (where $h$ is the period) complex eigenvalues with absolute value $r$. Each of them is a simple root of the characteristic polynomial and is the product of $r$ with an $h$th root of unity.

For $\rho(A)=1$, this means that you can decompose the vector space in a part where the action of the matrix tends to zero and a part where the action of the matrix is periodic.

This classical result of Perron and Frobenius was subsequently generalized to infinite dimensional systems (ordered topological vector spaces, Banach lattices). A standard reference is:

H.H. Schaefer, Banach lattices and positive operators , Springer (1974).

Whenever you need eigenvalues (spectral theory) you usually work over $\mathbb{C}$ (as opposed to $\mathbb{R})$ equipped with the order you describe. Non-negative matrices (operators) then have a "nice" set of eigenvalues.