Krein–Rutman theorem

Remarks

Proof idea: for $\lambda > r(T)$ the resolvent $(\lambda I - T)^{-1} = \sum_{n \ge 0} T^n / \lambda^{n+1}$ maps $K$ into $K$. Using totality of the cone one shows that $r(T)$ lies in the spectrum and that the resolvent blows up along vectors of $K$ as $\lambda$ decreases to $r(T)$. Normalizing $(\lambda I - T)^{-1} x$ for suitable $x \in K$ and using compactness of $T$ to pass to a limit gives a nonzero eigenvector in $K$.

Example: let $X = \R^n$, let $K$ be the vectors with nonnegative entries, and let $T$ be a matrix with nonnegative entries and $r(T) > 0$. The theorem gives the weak form of the Perron–Frobenius theorem: $r(T)$ is an eigenvalue with a nonnegative eigenvector.

The hypothesis $r(T) > 0$ is needed: the Volterra operator $(Vf)(x) = \int_0^x f(t)\,dt$ on $C[0, 1]$ is compact and maps nonnegative functions to nonnegative functions, but $r(V) = 0$ and $V$ has no eigenvalue at all.

A stronger form holds when $K$ has nonempty interior and $T$ maps $K \setminus \set{0}$ into the interior of $K$. Then $r(T) > 0$, it is a simple eigenvalue with an eigenvector in the interior of $K$, and every other eigenvalue $\lambda$ satisfies $\abs{\lambda} < r(T)$.

Going deeper

For a student

The theorem is a main tool for principal eigenvalues of elliptic operators. The inverse of such an operator, with suitable boundary conditions, is compact and positive on a space of continuous functions, so it has a positive eigenfunction. This gives the existence of a positive first eigenfunction of the Dirichlet Laplacian and similar results in population dynamics and bifurcation theory.